Table of Content
1. Representation of Data
2
a. Stem and leaf diagram
2
b. Histogram
7
c. Cumulative Frequency Curve
20
2. Measure of Location
32
a. Median and Quartiles
32
b. Box and Whisker Plot
39
c. Mean
45
3. Measure of Variation
50
a. Range and IQR
50
b. Standard Deviation
52
c. Combined Mean and SD
58
4. Permutation and Combination
70
a. Permutation
70
b. Combination
87
5. Probability
102
a. Basic
102
b. Mutually Exclusive Events
105
c. Independent Events
109
d. Conditional Probability
115
e. Tree Diagram
119
6. Probability Distribution
131
7. Binomial Distribution
157
8. Geometric Distribution
169
9. Normal Distribution
181
10.Formulae list and Table
217
11. Revision Checklist
219
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 1
1.Representation of Data
Statistics: statistics is the study of the collection, analysis, interpretation, presentation, and
organization of data .
Data : piece of numerical and other information are called data .
Variable; In order to collect data you need to observe or to measure some property. This property is
called variable. Example weight of student in AS level ., obtaining marks in optional mathematics.
Continuous variable: A quantitative variable which take any value in the given interval.
Discrete variable: A quantitative variable which has clear steps between its possible values.
Stem and leaf plots
A Stem and Leaf Plot is a special table where each data value is split into a "stem" (the first digit or
digits) and a "leaf" (usually the last digit)
How do you do a stem and leaf diagram?
The stem and leaf diagram is formed by splitting the numbers into two parts - in this case tens
and units. The tens form the 'stem' and the units form the 'leaves'. The numbers are usually ordered, so
the row: shows the numbers 21, 23, 24, 24, 25, and 27 in order.
How do you read a stem and leaf plot?
A stem-and-leaf diagram is a graph that shows the shape of the data according to the data place values.
The "leaf" of a number is usually the last digit in a number. The "stem" is the remaining digit or digits to
the left. Look at the stem part of the diagram and find the highest number.
Characters of stem and leaf diagram
➢ The entire interval must be equal width. So it seems sensible to choose intervals 10-19 , 20-29 ,
30-39 ..................... 80-89
➢ Stem part represent the tens and the leaf to represent the units.
➢ A key is essential.
➢ Not use any decimals and fraction.
➢ Ignore the common property while plotting the stem and leaf diagram but key represent the
common property.
➢ Not use comma in leaf part.
Advantages
➢ It shows all the original data
➢ It shows the shape of the distribution
➢ The mode , median and quartiles can be found from the diagram
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 2
➢ It is useful for comparing two sets of data .
Disadvantages
➢ It is not suitable for large amounts of data.
Exercise:
1. The stem and leaf diagram below shows the price of bottle of milk in 13 different shops
5
9
6
0 0 2 5 5
7
0 0 0 5
8
3 4 5
Key: 5/ 9 means 59p
a. How much was the most expensive bottle of milk?
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b. How much was the cheapest bottle of milk?
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c. What was the range of the price of the bottles of milk?
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d. What was the modal price of a bottle of milk in 13 shops?
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Ans: 85 ; 59 ;26 ; 70
2. The stem and leaf diagram below shows the mass of 16 pumpkins in a pumpkin growing
competition
2
0 3 7
3
3 3 5 6
4
5 7 8 9
5
0 1 2 4 7
Key: 2/ 7 means 2.7 kg
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 3
a. What was the range of the mass of the pumpkins?
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b. What was the modal mass of the pumpkins?
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c. How many of the pumpkins have mass less than 3.4 kg ?
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Ans: 3.7; 3.3; 5
3. The weights , correct to the nearest kg , of 30 men are shown below
74 52 67 68 71 76 86 81 73 68
64 75 71 61 63 57 67 57 59 72
79 64 70 74 77 79 65 68 76 83
i.
Draw a stem-and –leaf diagram to show the data
ii.
Write down the mode.
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 4
4. The following are the annual amounts of money spent on clothes, to the nearest $10, by 27
people.
10
40
60
80
100
130
140
140
140
150
150
150
160
160
160
160
170
180
180
200
210
250
270
280
310
450
570
Construct a stem-and-leaf diagram for the data.
5. Cartons are advertised as containing 2 litres of milk. In a quality control test, the volume of milk
in 20 randomly selected cartons was measured in litres, to the nearest millilitre and gave the
following results.
2.015 2.019 1.982 2.003 1.986 2.024 2.017 2.001 2.004 1.988 2.033 1.990
2.011
2.018
2.023 2.019 2.022 2.008 1.985
i.
Draw a stem-and –leaf diagram to show the data
ii.
What percentage of cartons contained less than the advertised volume of milk?
Ans: 25%
6. The lengths of the diagonals in metres of the 9 most popular flat screen TVs and the 9 most
popular conventional TVs are shown below.
Flat screen : 0.85
0.94
0.91
0.96
1.04
0.89
1.07
0.92
0.76
Conventional : 0.69
0.65
0.85
0.77
0.74
0.67
0.71
0.86
0.75
Represent this information on a back-to-back stem-and-leaf diagram.
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 5
7. The weights in kilograms of 11 bags of sugar and 7 bags of flour are as follows.
Sugar: 1.961
1.983 2.008 2.014 1.968 1.994 2.011
Flour: 1.945
1.962 1.949 1.977 1.964 1.941
2.017 1.977
1.984
1.953 1.989
Represent this information on a back-to-back stem-and-leaf diagram with sugar on the left-hand
side.
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 6
Histogram
Properties of histogram:
i.
Histogram uses continuous data.
ii.
There is no gap between any two bars.
iii.
Area of the bar is proportional to the frequency that it represents.
iv.
Y-axis represents frequency density and x-axis represents independent variable.
𝐹. 𝐷 =
v.
𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦
𝑐𝑙𝑎𝑠𝑠 𝑤𝑖𝑑𝑡ℎ
The modal class is the interval with the greatest frequency density.
Advantages
i.
It can represent group of different widths.
ii.
It shows whether the distribution is symmetrical or skew.
iii.
The mean and standard deviation can be estimated from the histogram.
Disadvantages
i.
The visual impact can be altered by choosing different groups.
ii.
Two distributions cannot be shown on the same diagram.
iii.
It cannot use discrete data.
Exercise :
1. A teacher asked some year 10 students how long they spent doing homework each night. The
histogram was drawn from this information.
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 7
Use the histogram to complete the table.
2. Some students at GEMS School took a mathematics examination. The unfinished table and histogram
show some information about their marks.
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 8
a. Use the information in the table to complete the histogram.
b. Use the information in the histogram to complete the table.
3. Data on the travel times of students getting
to school has been collected and is summarised
in a histogram below.
Use the histogram to fill in the
in the grouped frequency table.
4. The table and histogram show some data on the amount spent by shoppers during the Christmas
period. Complete the table and histogram
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 9
5. Use the histogram below to answer the following questions.
The score required for a grade C was 45%.
How many students achieved at least a grade C?
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Which group was the modal class?
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The score required for a grade B was 75%.
Estimate the number of students that achieved a grade A or B assuming A is the highest grade.
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6. Samir has a part-time job delivering Pizza. On a number of days, he noted the time, correct to the
nearest minute, that it took him to do his job. Samir used his results to draw up the following table;
three of the values in the table are denoted by 𝑎 ; 𝑏 and 𝑐.
Time (t minutes)
60-62
62-64
64-67
67-c
Frequency (No of days)
4
8
b
20
Frequency density
2
a
4
5
Find the values of , 𝑏 𝑎𝑛𝑑 𝑐 .
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 10
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Ans: 4 ; 12 ; 71
7.The histogram shows some information about the weights of a sample of apples.
i. Show that 11 apples having weight between 180 and 200 grams .
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ii.
Complete the following frequency table for the data.
Weight (grams)
Frequency
100-
140-
160-
180-
200-
140
160
180
200
240
11
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 11
8.
i.
A histogram is drawn to represent a set of data.
The first two classes have boundaries 2.0 and 2.2 and 2.2 and 2.5 , with frequencies 5 and 12 .
The height of the bar drawn is 2.5 cm . What is the height of the second bar?
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ii. The class boundaries of the third bar are 2.5 and 2.7 . What is the corresponding frequency if bar
drawn has height 3.5 cm ?
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iii.
The fourth bar has a height of 3 cm and corresponding frequency is 9 . The lower class boundary
for this bar is 2.7 cm . find the upper class boundary.
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Ans: 4 𝑐𝑚
; 7 ; 3.0
9. The following table shows the distribution of times, in minutes, that some students spent taking a
shower.
Time
2-4
4-6
6-7
7-8
8-10
10-16
Frequency
20
44
34
30
30
36
Draw the histogram on graph paper to represent these results.
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 12
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10. In a survey, the percentage of meat in a certain type of take-away meal was found. The results, to
the nearest integer, for 193 take-away meals are summarised in the table.
Percentage of meat
1-5
6-10
11-20
21-30
31-50
Frequency
59
67
38
18
11
Draw, on graph paper, a histogram to illustrate the information in the table.
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 13
11. The table summarises the times that 112 people took to travel to work on a particular day.
Time
frequency
19
0<𝑡
10 < 𝑡
15 < 𝑡
20 < 𝑡
25 < 𝑡
400 < 𝑡
≤ 10
≤ 15
≤ 20
≤ 25
≤ 40
≤ 60
12
28
22
18
13
On graph paper, draw a histogram to represent the data.
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 14
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.
12.
The weights of 220 sausages are summarised in the following table.
Weight <20
<30
<40
<45
<50
<60
<70
CF
20
50
100
160
210
220
0
On graph paper, draw a histogram to represent the weights of the sausages.
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 15
13. The times taken by 57 athletes to run 100metres are summarised in the following cumulative
frequency table.
i.
State how many athletes ran 100 metres in a time between 10.5 and 11.0 seconds.
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ii.
Draw a histogram on graph paper to represent the times taken by these athletes to run 100 metres.
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 16
iii.
Calculate estimates of the mean of the times taken by these athletes.
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Ans: 6, 11.7
14. The weights in grams of a number of stones, measured correct to the nearest gram, are
represented in the following table.
A histogram is drawn with a scale of 1 𝑐𝑚 to 1 unit on the vertical axis, which represents frequency
density. The 1 − 10 rectangle has height 3 𝑐𝑚.
i.
Calculate the value of 𝑥 and the height of the 51 − 70 rectangle
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 17
ii.
Calculate an estimate of the mean weight of the stones.
Ans:15 , 0.75 , 26.6
15. The following table gives the marks, out of 75, in a pure mathematics examination taken by 234
students.
i.
Draw a histogram on graph paper to represent these results.
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 18
ii.
Calculate estimates of the mean mark and the standard deviation.
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Ans: 37.5 ; 16.9
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 19
Cumulative frequency curve
A cumulative frequency graph is a visual representation of ranked categorical data. An example is
grouping people by height categories: under five feet, five feet to six feet, and above six feet. The number
of people in each category is the frequency.
Properties of cumulative frequency Graph
i.
Cumulative frequency graph use continuous data.
ii.
Y-axis represents cumulative frequency and x-axis represents independent variable.
iii.
Position of lower quartile (𝑁/4) , position of median( 2 ), position of upper quartile( 4 ) lies
𝑁
3𝑁
on y-axis and actual value of 𝑄1 ; 𝑄2 (𝑚𝑒𝑑𝑖𝑎𝑛 )𝑎𝑛𝑑 𝑄3 lies on x-axis.
iv.
Points are plotted upper class boundaries against cumulative frequency .
How to use cumulative frequency curve
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 20
For example, given a value of x you can estimate the number of values less than or greater than x by
drawing a straight line that meets the cumulative frequency curve and then drawing a corresponding
line to the value.
Advantages
i.
The median and quartiles can be estimated from the graph.
ii.
Set of data can be compared by drawing graphs on the same diagram.
Disadvantages
i.
The visual impact can be altered by using different scales.
Exercise:
1. Use each cumulative frequency graph to find an estimate for the median.
2. The graph shows information about the time taken to solve a puzzle.
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 21
a.
How many people took less than 30 seconds?
b.
How many people took less than 10 seconds?
c.
How many people took longer than 25 seconds?
d.
How many people took longer than 35 seconds?
e.
The fastest 10 people completed the puzzle in under how many seconds?
f.
The slowest 2 people completed the puzzle in longer than how many seconds?
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3. The graph shows information about the speed of cars on a road.
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 22
a) How many cars travelled under 50km/h?
b) How many cars travelled over 110km/h?
c) 42 cars were exceeding the speed limit. What is the speed limit?
d) Mr Thapa says 18% of the cars were travelling too slowly on this road. Below what speed does he
feel is too slow?
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4. Some students complete a quiz.
The cumulative frequency graph shows their results
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 23
a. How many students completed the quiz?
b. Complete the frequency table alongside.
c. What percentage of the students scored above 20 marks?
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5. The cumulative frequency graph alongside
shows the salaries of 80 teachers.
The lowest salary is £4,000 and
the highest salary is £39,000.
A teacher is picked at random
to answer a survey.
a. Find the probability that the
teacher selected is paid less that £15,000.
b. Find the probability that the
teacher selected is paid over £25,000
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 24
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6. The distribution of the time taken when a certain task was performed by a large group of people was
noted . it was found that 20% performed the task in less than 30 minutes , 40% in less than 38 minutes ,
60% in less than 45 minute s and 80% in less than 53 minutes. The shortest time was 10 minutes and
the greatest time wad 69 minutes.
i.Draw a cumulative frequency graph to illustrate the data.
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 25
ii.
Estimate the percentage of people who performed the task in less than 50 minutes.
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Ans: 73%
7.
In a recent survey 640 people were asked about the length of time each week that they spent
watching television .the median time was found to be 20 hours, and the lower and upper quartile s were
15 hours and 35 hours respectively. The least amount of time that anyone spent was 3 hours. And the
greatest amount was 60 hours.
i. On graph paper , show these result using a fully labelled cumulative frequency graph.
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ii.
Use your graph to estimate how many people watched more than 50 hours of television each week.
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 26
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8. There are 5000 schools in a certain country. The cumulative frequency table shows the number of
pupils in a school and the corresponding number of schools.
Number of pupils in
≤ 100
≤ 150
≤ 200
≤ 250
≤ 350
2100
4100
≤ 450
≤ 600
a school
Cumulative
200
800
1600
4700
5000
frequency
a. Draw a cumulative frequency graph with a scale of 2 cm to 100 pupils on the horizontal axis and a
scale of 2 cm to 1000 schools on the vertical axis. Use your graph to estimate the median number of
pupils in a school.
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 27
b.
80% of the schools have more than n pupils. Estimate the value of n correct to the nearest ten.
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c.
Find how many schools have between 201 and 250 (inclusive) pupils.
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Ans :270 , 160 , 500,
9. A hotel has 90 rooms. The table summarises information about the number of rooms occupied each
day for a period of 200 days.
Number of room
1-20
21-40
41-50
51-60
61-70
71-90
10
32
62
50
28
18
occupied
Frequency
i.Draw a cumulative frequency graph on graph paper to illustrate this information.
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ii.
Estimate the number of days when over 30 rooms were occupied.
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 28
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iii . On 75% of the days at most n rooms were occupied. Estimate the value of n.
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Ans : 174-180 , 59 to 60
10. The weekly maximum temperatures in a certain town were recorded , to the nearest degree
Celsius , over a period of two years and group in the following table .
Temperatures -5to-1
0-4
5-9
5-14
15-19
15-24
25-29
Frequency
12
17
48
23
32
4
i.
8
State the boundaries and width of the interval 0-4.
........................................................................................................................................................................................................
........................................................................................................................................................................................................
...............................................................................................................................................................................................
ii.
Draw the cumulative frequency graph
........................................................................................................................................................................................
........................................................................................................................................................................................
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 29
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........................................................................................................................................................................................................
........................................................................................................................................................................................................
........................................................................................................................................................................................................
.......................................................................................................................................................................................
iii.
A week is classified as `extremely warm’ when the weekly maximum is at least 21°𝑐 . use your
graph to estimate the percentage of week that are classified as ` extremely warm’
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Ans: (-0.5 - 4.5) ; 10%
11. Ina survey, two groups of 200 students were asked to keep a record of the number of text
messages they sent during a certain period of time.
The results are as follows.
Number of text
1-5
6-10
11-15
16-25
26-34
35-40
Group1-frequecy
4
8
11
35
104
38
Group2-frequecy
11
17
27
87
50
8
message sent
On the same diagram, draw two cumulative frequency graphs to represent the data.
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 30
12.
A random sample of people was asked how old they were when they first met their partner. The
histogram represents this information.
SSSSSS
i.
What is the modal age group?
ii.
How many people took part in the survey?
iii.
Draw a cumulative frequency curve for the data.
Ans: 20 − 30 ; 250
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 31
2.Measure of Location
Median
The median is a simple measure of central tendency. To find the median, we arrange the observations
in order from smallest to largest value. If there is an odd number of observations, the median is the
middle value. If there is an even number of observations, the median is the average of the two middle
values.
Advantages of Median:
i.
It is very simple to understand and easy to calculate. In some cases it is obtained simply by
inspection.
ii.
Median lies at the middle part of the series and hence it is not affected by the extreme values.
iii.
It is a special average used in qualitative phenomena like intelligence or beauty, which are not
quantified, but ranks are given. Thus we can locate the person whose intelligence or beauty is
the average.
iv.
In grouped frequency distribution, it can be graphically located by drawing ogives.
v.
It is especially useful in open-ended distributions since the position rather than the value of
item that matters in median.
Disadvantages of Median:
i.
In simple series, the item values have to be arranged. If the series contains large number of
items, then the process becomes tedious.
ii.
It is a less representative average because it does not depend on all the items in the series.
iii.
It is not capable of further algebraic treatment. For example, we cannot find a combined median
of two or more groups if the median of different groups are given.
iv.
It is affected more by sampling fluctuations than the mean as it is concerned with on1y one item
i.e. the middle item.
v.
It is not rigidly defined. In simple series having even number of items, median cannot be exactly
found. Moreover, the interpolation formula applied in the continuous series is based on the
unrealistic assumption that the frequency of the median class is evenly spread over the
magnitude of the class interval of the median group.
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 32
Exercise:
1. Find the median of each of these sets of numbers:
i.
7, 7, 2, 3, 4, 2, 7, 9, 31
ii.
36, 41, 27, 32, 29, 39, 39, 43
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Ans: (𝑖) 𝑚𝑒𝑑𝑖𝑎𝑛 = 7 (𝑖𝑖)𝑚𝑒𝑑𝑖𝑎𝑛 = 37.5
2. The stem and leaf diagram below shows the mass of 16 pumpkins in a pumpkin growing
competition
2
0 3 7
3
3 3 5 6
4
5 7 8 9
5
0 1 2 4 7
Key: 2/ 7 means 2.7 kg
Find median and inter quartile range of the mass of 16 pumpkins.
........................................................................................................................................................................................................................
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Ans:𝑚𝑒𝑑𝑖𝑎𝑛 = 4.6 ; 𝐼𝑄𝑅 = 1.75
3. The stem-and-leaf diagram below represents data collected for the number of hits on an internet
site on each day in July 2018. There is one missing value, denoted by x.
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 33
0
0 1 5 6
(4)
1
1 3 5 6 8
2
1 1 2 3 4 4 4 8 9
(9)
3
1 2 2 2 x 8 9
(7)
4
2 5 6 7 9
(5)
9
(6)
Key: 1/5 means 15 hits
i.
Find the median and lower quartile for the number of hits each day.
................................................................................................................................................................................................................
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................................................................................................................................................................................................................
................................................................................................................................................................................................................
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................................................................................................................................................................................................................
ii.
The inter quartile range is 19. Find the value of x.
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...............................................................................................................................................................................................................
Ans: 𝑙. 𝑄 = 16 𝑚𝑒𝑑𝑖𝑎𝑛 = 24 ; 𝑥 = 5
4. The masses, in grams, of components made in factory A and components made in factory B are
shown below.
Factory A: 0.049 0.050 0.053 0.054 0.057 0.058 0.058
0.059 0.061 0.061 0.061 0.063 0.065
Factory B : 0.031 0.056 0.049 0.044 0.038 0.048 0.051
0.064
i.
0.035 0.042 0.047 0.054 0.058
Draw a back-to-back stem-and-leaf diagram to represent the masses of components made in the
two factories.
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 34
ii.
Find the median and the inter quartile range for the masses of components made in factory B.
................................................................................................................................................................................................................
................................................................................................................................................................................................................
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iii.
Find the median and quartiles for the masses of components made in factory A.
................................................................................................................................................................................................................
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................................................................................................................................................................................................................
................................................................................................................................................................................................................
iv.
Make two comparisons between the masses of components made in factory A and the masses of
those made in factory B.
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 35
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
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................................................................................................................................................................................................................
...............................................................................................................................................................................................................
Ans: 𝑚𝑒𝑑𝑖𝑎𝑛 = 0.048 g IQR = 0.015
median = 0.058 IQR = 0.0085
5. The cumulative frequency graph shows the annual salaries, in thousands of euros, of a random
sample of 500 adults with jobs, in France. It has been plotted using grouped data. You may assume that
the lowest salary is 5000 euros and the highest salary is 80 000 euros.
Estimate the median and the inter quartile range of the salary.
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 36
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
Ans:, 𝑀𝑒𝑑𝑖𝑎𝑛 = 18, 𝐼𝑄𝑅 = 11
13.
The time to the nearest minute ,taken by 120 students to write a timed essay, were recorded
.The result is shown in the table .
Time
40-44
45-49
50-54
55-59
60-64
24
32
30
26
(minute )
Frequency 8
i. Construct the cumulative frequency table and draw cumulative frequency graph .
ii. Use your graph to estimate the lower quartile and the median.
Another group of 40 students wrote the same essay and of them took at least 1 hour to complete it .
iii. Use your graph to estimate the lower quartile of the 160 students.
iv. Explain why it is not possible to estimate the interquartile range of the time spent by all 160
students.
Answer : 49 ,54, 50.5
14.
The following histogram represents the lengths of worms in a garden.
i. Calculate the frequencies represented by each of the four histogram columns.
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 37
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
.......................................................................................................................................................................................................................
ii. On the grid, draw a cumulative frequency graph to represent the lengths of worms in the garden.
iii. Use your graph to estimate the median and inter quartile range of the lengths of worms in the
garden.
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 38
iv. Calculate an estimate of the mean length of worms in the garden.
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Box and whisker plots
A graphic way to display the median, quartiles, and extremes of a data set on a number line to show the
distribution of the data.
Advantages
i.
It is easy to see whether the distribution is symmetrical or whether there is a tail to the left or
right .
ii.
It can be used to investigate extreme values (outlier)
iii.
It is easy to see the range and inter quartile range.
iv.
Easy to compare two or more sets of data
Disadvantages
It does not show frequency
Exercise:
1
A survey on the heights of all the girls in a particular year group in a school gave the following
information
Minimum height :
144 cm
Lower quartile :
159 cm
Median :
165 cm
Upper quartile :
Maximum heights :
169 cm
181cm
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 39
Represent the information by a box and whisker plot.
2
In a recent survey, 480 people were asked about the length of time each week that they spent
watching television. The median time was found to be 10 hours, and the lower and upper quartiles
were 6 hours and 18 hours respectively. The least amount of time that anyone spent was 2 hours
and the greatest amount was 25 hours.
On graph paper, draw a box and whisker plot to illustrate these times spent.
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 40
3
The back-to-back stem-and-leaf diagram shows the values taken by two variables A and B.
i.
Find the median and the inter quartile range for variable A.
ii.
You are given that, for variable B, the median is 0.171, the upper quartile is 0.179 and the
lower quartile is 0.164. Draw box-and-whisker plots for A and B in a single diagram on graph
paper.
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 41
Ans: 𝑚𝑒𝑑𝑖𝑎𝑛 = 0.186 ; 𝐼𝑄𝑅 = 0.019
4
The birth weights of random samples of 900 babies born in country A and 900 babies born in country B
are illustrated in the cumulative frequency graphs.
i.
Find the median and the interquartile range for country A.
ii.
Draw box-and-whisker plots for country A and country B in a single diagram on graph paper.
iii.
Make two comparisons between the weight of babies in country A and country B.
Ans: median A = 2.0 – 2.1 ; IQ range A = 0.9 ;
Country B has heavier babies on average ; Country B has greater spread of weights
Outliers
An outlier is an observation point that is distant from other observations. An outlier may be due to
variability in the measurement or it may indicate experimental error; the latter are sometimes excluded
from the data set.
An item of data x may be identified as an outlier if it is more than 𝟏. 𝟓 × 𝑰𝑸𝑹 beyond the lower or
upper quartile, i.e. if 𝒙 < 𝑸𝟏 − 𝟏. 𝟓 × (𝑸𝟑 – 𝑸𝟏) 𝒐𝒓 𝒙 > 𝑸𝟑 + 𝟏. 𝟓 × (𝑸𝟑 − 𝑸𝟏).
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 42
Exercise :
1.
The times in minutes for seven students to become proficient at a new computer game were
measured. The results are shown below.
15 10
i.
48 10 19 14 16 -58
Find the median and the quartiles for the given data.
An ‘outlier’ is defined as any data value which is more than 1.5 times the interquartile range above the
upper quartile, or more than 1.5 times the interquartile range below the lower quartile.
ii.
List the outliers.
Ans: 𝑚𝑒𝑑𝑖𝑎𝑛 = 14.5 ; 𝐿. 𝑄 = 10 ; 𝑈𝑄 = 17.5 ; 𝑜𝑢𝑡𝑙𝑖𝑒𝑟𝑠 = −58; 48
2.
The following are the annual amounts of money spent on clothes, to the nearest $10, by 27 people.
10
40 60
80 100 130 140 140 140
150 150 150 160 160 160 160 170 180
180 200 210 250 270 280 310 450 570
i.
Construct a stem-and-leaf diagram for the data.
ii.
Find the median and the interquartile range of the data.
An ‘outlier’ is defined as any data value which is more than 1.5 times the interquartile range above the
upper quartile, or more than 1.5 times the interquartile range below the lower quartile.
iii.
List the outliers.
Ans: 𝐼𝑄𝑅 = 70 ; 𝑜𝑢𝑡𝑙𝑖𝑒𝑟𝑠 = 10 ,450,570
3.
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 43
In an open-plan office there are 88 computers. The times taken by these 88 computers to access a
particular web page are represented in the cumulative frequency diagram.
i.
Find the median and the interquartile range of the data
An ‘outlier’ is defined as any data value which is more than 1.5 times the interquartile range above
the upper quartile, or more than 1.5 times the interquartile range below the lower quartile.
ii.
Show that there are no outliers.
Ans: 𝑚𝑒𝑑𝑖𝑎𝑛 = 3.9 ; 𝐼𝑄𝑅 = 3.8
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 44
Mean
The mean id the most commonly used average. It is calculated by dividing the sum of all the observation
by the number of observations.
Advantages
i.
Arithmetic mean is simple to understand and easy to calculate.
ii.
It is rigidly defined.
iii.
It is suitable for further algebraic treatment.
iv.
It is least affected fluctuation of sampling.
v.
It takes into all the values in the series.
Disadvantages
i.
It is highly affected by the present of a few abnormally high or abnormally low scores.
ii.
In absence of a single item, its value becomes inaccurate.
iii.
It cannot be determined by inspection.
Formulas:
Combine mean
̅̅̅̅
∑𝑥
𝑋̅ = 𝑛
̅̅̅̅
𝑛 𝑋 +𝑛 𝑋
𝑐𝑜𝑚𝑏𝑖𝑛𝑒 𝑚𝑒𝑎𝑛 (𝑋̅) = 1𝑛1 +𝑛2 2
1
2
∑ 𝑓𝑥
𝑋̅ =
𝑁
∑(𝑥−𝑎)
𝑋̅ = 𝑛 + 𝑎
Exercise
1. Rachel measured the lengths in millimeters of some of the leaves on a tree. Her results are
recorded below.
32
35
45
37
38
44
33
39
36
45
Find the mean of the lengths of these leaves.
Ans: 38.4 mm
2. Find mean:
a. ∑20
𝑖=1 𝑥𝑖 = 226
b.
b. ∑18
𝑖=4 𝑦𝑖 = 66
𝐴𝑛𝑠: 11.3 ; 4.4
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 45
3. Find the mean from the given frequency distribution table
Number of instrument ,
1
2
3
4
5
11 10 5
3
1
x
Frequency
Ans 2.1
4.
The table summarizes the times that 112 people took to travel to work on a particular day.
Time to travel to
0<t≤
10 < t ≤
15 < t ≤
20 < t ≤
25 < t ≤
40 < t ≤
work
10
15
20
25
40
60
Frequency
19
12
28
22
18
13
Calculate an estimate of the mean time to travel to work.
5.
Ans :22.0
The following table gives the marks, out of 75, in a pure mathematics examination taken by 234
students.
Marks
1–20
Frequency
40
21–30
31–40
34
41–50
56
51–60
54
Calculate estimates of the mean mark.
6.
29
61–75
21
Ans:20/49
The length of time, t minutes, taken to do the crossword in a certain newspaper was observed on
12 occasions. The results are summarised below.
𝛴(𝑡 − 35) = −15
Calculate the mean of these times taken to do the crossword.
7.
Ans:33.75
The amounts of money, x dollars, that 24 people had in their pockets are summarised by
∑(𝑥 − 36) = −60 and. Find ∑ 𝑥 .
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Ans :804
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 46
8. A summary of 30 values of x gave the following information:
Σ(x – c) = 234, where c is a constant.
Given that the mean of these values is 86, find the value of c.
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Ans: c = 78.2
9. The following table shows the mean of the height of 20 boys and 30 girls.
Numbers
Mean
160 cm
30
Girls
Boys
20
155 cm
Find the mean of the height of 50 children.
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Ans: 157 cm
10. The number of errors x. on each of 200 pages of typescript was monitored. The result where
summarised as follows.
∑ 𝑥 = 920
. Calculate the me.an of the number of errors on a page.
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 47
........................................................................................................................................................................................................................
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A further 50 pages where monitored and it was found that, for these pages, the mean was 4.4 errors.
Find the mean of the number of errors per pages for the 250 pages.
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Ans: 4.6 , 4.56
11.
The heights, x cm, of a group of 28 people were measured. The mean height was found to be
172.6 cm. A person whose height was 161.8 cm left the group. Find the mean height of the remaining
group of 27 people.
Ans : 173 𝑐𝑚
12.
The ages, x years, of 18 people attending an evening class are summarised by the following
totals: 𝛴𝑥 = 745,
i.
Calculate the mean of the ages of this group of people.
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ii. One person leaves the group and the mean age of the remaining 17 people is exactly 41 years. Find
the age of the person who left.
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 48
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Ans: 41.38 , 48
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 49
3.Measure of variation
Range and Interquartile Range:
1. Find the range and inter quartile range of the following data.
7 9 4 6 3 2 8 1 10 15 11
2. 15 students do mathematics test. Their marks are shown opposite.
7
4 9 7 6 10 12 11 3 8 5 9 8 7 3
a. Find the value of the median.
b. Find Q1 and Q3.
c. Work out the inter quartile range.
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A group of works were asked to write down their weekly wage. The wages were:
$550 $400 $260 $320 $500 $450 $460 $480 $510 $490 $505
a. works out the range for these wages
b. Find Q1 and Q3.
c. Work out the inter quartile range.
3. A super store record the number of hours overtime worked by their employees in one particular
week . The results are shown in the table.
Number of hours
Frequency
0
25
1
10
2
20
3
10
4
25
5
10
Cumulative frequency
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 50
a. Fill in the cumulative frequency column and works out how many employees the superstore
had in that week.
b. Find Q1 and Q3.
c. Work out the inter quartile range.
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4. A month trap was set every night for five weeks. The number of month caught in the trap was
recorded. The results are shown in the table.
Number of months
Frequency
7
2
8
5
9
9
10
14
11
5
5. The weight of 31 Jersey cows was recorded to the nearest kilogram. The weights are shown in
the table.
Weight of cattle (kg)
Frequency
300-349
3
350-339
6
400-449
10
450-449
7
500-549
5
Cumulative frequency
a. Compute the cumulative frequency in the column in the table.
b. State which class contain the lower quartile and which class contain the upper quartile .Hence
find the least and greatest possible value of inter quartile range.
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
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6. A typing test is taken by 111 people. The numbers of typing errors they make in the test are
summarised in the table below.
Number of typing error
1-5
6-20
21-35
36-60
61-80
Frequency
24
9
21
15
42
State, which class contains the lower quartile and which class contains the upper quartile. Hence find
the least and greatest possible value of the inter quartile range.
Standard deviation :
In statistics, the standard deviation (SD), also represented by the Greek letter sigma σ or the Latin
letter s) is a measure that is used to quantify the amount of variation or dispersions of a set of data
values. A low standard deviation indicates that the data points tend to be close to the mean (also called
the expected value) of the set, while a high standard deviation indicates that the data points are spread
out over a wider range of values.
Advantages
i.It is calculated using all the data and so represents every item.
ii.It is calculated using a mathematical formula, so calculators can be programmed to find it
iii.It is very useful for further analysis .
iv.It is very useful in comparing two sets of data , for example by showing which is more consistent.
Disadvantages
i.It can be unduly affected by one or two extreme values .
ii.For a single set of data, its value is difficult to interpret .
Formulas :
∑ 𝑥2
𝜎=√
𝑛
combine S.D
− (𝑋̅)2
∑ 𝑓𝑥 2
𝜎=√
𝑁
∑ 𝑥 2 +∑ 𝑦 2
𝑛1 +𝑛2
− (𝑚𝑒𝑎𝑛)2
− (𝑋̅)2
∑(𝑥−𝑎)2
𝜎=√
𝑐𝑜𝑚𝑏𝑖𝑛𝑒 𝑆. 𝐷(𝜎) = √
𝑛
∑(𝑥−𝑎) 2
−(
𝑛
)
∑(𝑋−𝑋̅)2
𝜎=√
𝑛
Exercise:
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 52
1. Given hat for a variable x:
∑ 𝑥 2 = 78 ;
∑ 𝑥 = 24 ;
𝑛=8
Find
a. The mean.
b. The variance.
c. The standard deviation.
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2.
Ten collie dogs are weighed (w kg) .the following summary data for the weights are shown below.
∑ 𝑤 2 = 5905
∑ 𝑤 = 241
Use this summary data to find the standard deviation of the collies' weights.
3. The times in minutes for seven students to become proficient at a new computer game were
measured. The results are shown below.
15
10
48
10
19
14
16
i.
Find the mean and standard deviation of these times.
ii.
State which of the mean, median or mode you consider would be most appropriate to use as a
measure of central tendency to represent the data in this case.
iii.
For each of the two measures of average you did not choose in part (ii), give a reason why you
consider it inappropriate.
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 53
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Ans ; 18.9 12.3 , median mode is 10 , inappropriate because it is smallest number , mean
inappropriate because it is affected by the outliers ,48
4. Rachel measured the lengths in millimetres of some of the leaves on a tree. Her results are recorded
below.
32
35
45
37
38
44
33
39
36
45
Find the mean and standard deviation of the lengths of these leaves.
Ans: 38.4 , 4.57
5. Sweets are packed into begs with a nominal weight of 75 grams . ten begs are picketed at random
from the production line and weighed . their weights , in grams are
76.0 , 74.2 , 75.1 ,73.7 , 72.0 , 74.3 , 75.4 , 75.4, 74.0, 73.1, 72.8
Find the mean and standard deviation
Ans :74.06 , 1.17
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 54
6. There are two routes for a worker to get to his office. Both the routes involve hold ups due to traffic
lights. He records the time it takes over a series of six journeys for each route .the results are shown in
the table.
Route 1
15
15
11
17
14
12
Route 2
11
14
17
15
16
11
a.
Work out the mean time taken for each route.
b.
Calculate the variance and standard deviation of each of the two routes.
c.
Using your answer to a and b suggest which route you would recommend. State your reason
clearly.
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Coded variance or standard deviation
1.
Data is coded using =
𝑥−50
.
50
The standard deviation of the coded data is 2.5 . Find the standard
deviation of the original data.
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 55
2.
The coding 𝑦 =
𝑆−8
10
produced a standard deviation for y of 0.65 write down the standard
deviation of s.
3. The coding 𝑦 = 𝑥 − 140 gives a standard deviation for y of 2.38 write down the standard
deviation of x.
4. The weekly income, 𝑖 of 100 women workers was recorded. The data where coded using
𝑦=
𝑖−90
and the
100
following summation where obtained.
∑ 𝑦 = 131 ,
∑ 𝑦 2 = 176.84
Work out estimate for the standard deviation of the actual women workers' weekly income.
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Standard deviation using ∑(𝒙 − 𝒂) 𝒂𝒏𝒅 ∑(𝒙 − 𝒂)𝟐
1. The values, x, in a particular set of data are summarised by
∑(𝑥 – 25) = 133 , ∑(𝑥 – 25)2 = 3762
.
The mean, x, is 28.325.
i. Find the standard deviation of x.
ii. 𝐹𝑖𝑛𝑑 ∑ 𝑥 2 .
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Page 56
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𝑎𝑛𝑠 ∶ 9.11 , 35412
2.
The heights, x cm, of a group of 82 children are summarised as follows.
∑(𝑥 − 130) = −287, 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑥 = 6.9.
i.
Find the mean height.
ii.
Find ∑(𝑥 – 130)2 .
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Ans: 127 , 4910
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 57
3. A summary of 24 observations of x gave the following information:
∑(𝑥 – 𝑎) = −73.2 , ∑(𝑥 – 𝑎)2 = 2115.
The mean of these values of x is 8.95.
i.
Find the value of the constant a.
ii. Find the standard deviation of these values of x.
Ans: 12 , 8.88
4. The length of time, t minutes, taken to do the crossword in a certain newspaper was observed on
12 occasions. The results are summarised below.
∑(𝑥 – 35) = −15 , ∑(𝑥 – 35)2 = 82.23.
Calculate the mean and standard deviation of these times taken to do the crossword.
Ans :33.8 ; 2.3
Combine mean and standard deviation
1. For a set of 10 numbers:
∑ 𝑥 = 50
∑ 𝑥 2 = 310
For a set of 15 numbers:
∑ 𝑦 = 86
∑ 𝑦 2 = 568
Find the mean and the standard deviation of the combined set of 25 numbers.
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 58
2. Barry weighs 20 oranges and 25 lemons. For the oranges, the mean weight is 220 g and the
standard deviation is 32 g. For the lemons, the mean weight is 118 g and the standard deviation
is 12 g.
i.
Find the mean weight of the 45 fruits.
ii.
The individual weights of the oranges in grams are denoted by 𝑥0 , and the individual weights
of the lemons in grams are denoted by 𝑥𝑙 . By first finding∑ 𝑥0 2 and ∑ 𝑥𝑙 2 , find the variance of
the weights of the 45 fruits.
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Ans : 163 , New var = 3100 – 3120
3. A group of 10 married couples and 3 single men found that the mean age 𝑥𝑤 of the 10 women
was 41.2 years and the standard deviation of the women’s ages was 15.1 years. For the 13 men,
the mean age 𝑥𝑚 was 46.3 years and the standard deviation was 12.7 years.
i.
Find the mean age of the whole group of 23 people.
ii.
The individual women’s ages are denoted by 𝑥𝑤 and the individual men’s ages by 𝑥𝑚 . By first
finding ∑ 𝑥𝑤 2 and ∑ 𝑥𝑚 2 find the standard deviation for the whole group.
Ans; 44.1, 14.0
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 59
4. Red Street Garage has 9 used cars for sale. Fairwheel Garage has 15 used cars for sale. The mean
age of the cars in Red Street Garage is 3.6 years and the standard deviation is 1.925 years. In
Fairwheel Garage, ∑ 𝑥 = 64 𝑎𝑛𝑑 ∑ 𝑥 2 = 352, where x is the age of a car in years.
i.
Find the mean age of all 24 cars.
ii.
Find the standard deviation of the ages of all 24 cars.
Ans: 4.02 , 2.19
5. A test is taken by 30 student . their score ,x , have a mean of 60 and a student deviation is 20
i.
Find ∑ 𝑥 𝑎𝑛𝑑 𝑠ℎ𝑜𝑤 𝑡ℎ𝑎𝑡 ∑ 𝑥 2 = 120000
Another 20 student taken the test . their scores ,y , are such that
∑ 𝑥 = 1400 𝑎𝑛𝑑 ∑ 𝑥 2 = 100000
ii.
Show that the mean score of the 50 students is 60.
iii.
Calculate the standard deviation of 50 students.
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Ans :1800 , 17.436
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 60
6. The number of errors x. on each of 200 pages of typescript was monitored . the result where
summarised as follows .
∑ 𝑥 = 920
i.
∑ 𝑥 2 = 5032
Calculate the mean and standard deviation of the number of errors on a page .
A further 50 pages where monitored and it was found that ,for these pages , the mean was 4.4 errors
and standard deviation was 2.2 errors .
ii.
Find the mean and standard deviation of the number of errors per pages for the 250
pages .
Ans: 4.6 , 2 , 4.56, 2.043
7. The manager of a car showroom monitored the number of cars sold during two successive five
day periods. During the first five days the numbers of cars sold per day had mean 1.8 and
standard deviation 0.6 . During the next five days numbers of cars sold per day had mean 2.8 and
standard deviation 0.81. then find the mean and standard deviation of the numbers of cars sold
per day during the ten days period .
Ans :2.3 , 0.871
Mean and standard deviation when one item is added or subtracted
8.
A sample of 36 data values, x, gave∑(𝑥 – 45) = −148 𝑎𝑛𝑑 ∑ (𝑥 – 45)2 = 3089.
i.
Find the mean and standard deviation of the 36 values.
ii.
One extra data value of 29 was added to the sample. Find the standard deviation of all 37 values.
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Ans :40.9 , 8.30 , 8.41
9.
The heights, x cm, of a group of 28 people were measured. The mean height was found to be 172.6
cm and the standard deviation was found to be 4.58 cm. A person whose height was 161.8 cm left the
group.
i.
Find the mean height of the remaining group of 27 people.
ii.
Find ∑ 𝑋 2 for the original group of 28 people. Hence find the standard deviation of the heights of
the remaining group of 27 people.
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Ans : 173 , ∑ 𝑋 2 = 834728.6 (835000), sd of remaining = 4.16
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 62
10. The ages, x years, of 18 people attending an evening class are summarised by the following totals:
𝛴𝑥 = 745, ∑ 𝑥 2 = 33951
i.
Calculate the mean and standard deviation of the ages of this group of people.
ii.
One person leaves the group and the mean age of the remaining 17 people is exactly 41 years. Find
the age of the person who left and find the standard deviation of the ages of the remaining 17 people .
Ans :13.2 , 48 , 13.4
Finding standard deviation using
∑(𝒙−𝒙
̅) 𝟐
𝒏
1.The ages, x years, of 150 cars are summarised by ∑ 𝑥 = 645 and ∑ 𝑋 2 = 8287.5.
Find ∑(𝑋 − 𝑋̅)2 , where 𝑋̅ denotes the mean of x.
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Ans : 5514 (5510)
2.
The heights, x cm, of a group of young children are summarised by
∑(𝑥 – 100) = 72 , ∑ (𝑥 – 100)2 = 499.2 .
The mean height is 104.8 cm.
i.
Find the number of children in the group.
ii.
Find∑ (𝑥 – 104.8)2
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 63
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Ans :15 , 154
3.
For a certain set of data:
∑ 𝑓𝑥 = 1975
∑ 𝑓𝑥 2 = 52325
𝑛 = 100
Work out the variance for these data.
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4.
For a certain set of data:
∑ 𝑓𝑥 = 264
∑ 𝑓𝑥 2 = 6456
𝑛 = 12
Work out the variance for these data.
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 64
5.
In a student group, a record was kept of the number of day absence each student had over one
particular team. The results are shown in the table.
Number days absent (x)
Number of students (f)
0
12
1
20
2
10
3
7
4
5
𝑓𝑥 2
𝑓𝑥
a. Complete the table.
b. Calculate the variance for these data.
c. Work out the standard deviation for these data.
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6.
Elan records the spent out of school during the lunch hour to the nearest minute, x , of the
female students in her year. The results are as follows.
X
35
36
37
38
39
Number of students
3
17
29
34
26
,f
Calculate the mean and standard deviation of the time spent out of school.
Ans: 37.6 ; 1.09
7. Each of 200sportsmen was asked to state the distance ,x km ,he needs to travel to obtain access to
suitable training facilities. The results are summarised in the table below.
Distance
Frequency
0≤𝑥<4
5
4 ≤ 𝑥 < 10
10
10 ≤ 𝑥 < 20
39
20 ≤ 𝑥 < 35
95
35 ≤ 𝑥 < 60
51
i.Estimate the mean and standard deviation of the distance travelled .
ii.State which class contain the lower quartile and which class contain the upper quartile .Hence find the
least and greatest possible value of inter quartile range
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 65
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8.
The engine sizes 𝑥 𝑐𝑚3 of a sample of 80 cars are summarised in the table below.
A student claims that the midrange is 2750 𝑐𝑚3. Discuss briefly whether he is likely to be correct.
Calculate estimates of the mean and standard deviation of the engine sizes. Explain why your answers
are only estimates.
9.
The times taken by 57 athletes to run 100 metres are summarised in the following cumulative
frequency table.
a.
State how many athletes ran 100 metres in a time between 10.5 and 11.0 seconds.
b.
Estimate the mean and standard deviation of the time taken.
c.
State which class contain the lower quartile and which class contain the upper quartile .Hence
find the least and greatest possible value of inter quartile range
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 66
10.
In a quality –control survey the length of life , measured to the nearest hour , of 100 light bulbs is
noted .the results are summarised in the table .
Length of
650-
life
669
Frequency
6
670-679
980-689
690-699
700-729
14
40
34
6
Estimate the mean and standard deviation.
Ans ; 686.8 , 11.8
11.
Ina survey ,two groups of 200 students were asked to keep a record of the number of text
messages they sent during a certain period of time.
The results are as follows.
Number of text
1-5
message sent
6-
11-15
16-25
26-34
35-40
10
Group1-frequecy
4
8
11
35
104
38
Group2-frequecy
11
17
27
87
50
8
i.
For the group -1 and for the group -2 , calculate estimates of mean and standard.
ii.
Make two comparisons between the weight of babies in country A and country B.
Ans:𝑔𝑟𝑜𝑢𝑝 − 1(27.41 ; 8.34 );
12.
𝑔𝑟𝑜𝑢𝑝 − 2 ( 20.52 ; 8.48
The values, x, in a particular set of data are summarised by
∑(𝑥 – 25) = 133 , ∑(𝑥 – 25)2 = 3762
The mean of x is 28.325.
iii.
Find the standard deviation of x.
.....................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
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........................................................................................................................................................................................................................
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........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
𝐹𝑖𝑛𝑑 ∑ 𝑥 2 .
.....................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 67
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
9.11 , 35412
13.
A summary of 24 observations of x gave the following information:
∑(𝑥 – 𝑎) = −73.2 , ∑(𝑥 – 𝑎)2 = 2115.The mean of these values of x is 8.95.
i.
Find the value of the constant a.
........................................................................................................................................................................................................
........................................................................................................................................................................................................
........................................................................................................................................................................................................
........................................................................................................................................................................................................
........................................................................................................................................................................................................
ii.
Find the standard deviation of these values of x.
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......................................................................................................................................................................................................
Ans: 12 , 8.88
14. A group of 10 married couples and 3 single men found that the mean age 𝑥𝑤 of the 10 women was
41.2 years and the standard deviation of the women’s ages was 15.1 years. For the 13 men, the mean
age 𝑥𝑚 was 46.3 years and the standard deviation was 12.7 years.
i. Find the mean age of the whole group of 23 people.
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 68
ii.
The individual women’s ages are denoted by 𝑥𝑤 and the individual men’s ages by 𝑥𝑚 . By first finding
∑ 𝑥𝑤 2 and ∑ 𝑥𝑚 2 find the standard deviation for the whole group.
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.......................................................................................................................................................................................................................
Ans; 44.1 , 14.0
15.
The number of errors x. on each of 200 pages of typescript was monitored . the result where
summarised as follows .
∑ 𝑥 = 920
∑ 𝑥 2 = 5032
i. Calculate the mean and standard deviation of the number of errors on a page .
A further 50 pages where monitored and it was found that ,for these pages , the mean was 4.4 errors
and standard deviation was 2.2 errors .
ii. Find the mean and standard deviation of the number of errors per pages for the 250 pages .
Ans: 4.6 , 2 , 4.56, 2.043, 8.41
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 69
Permutations and Combinations
Multiplication principle (Fundamental Principle of Counting)
Suppose an event E can occur in m different ways and associated with each way of occurring of E,
another event F can occur in n different ways, then the total number of occurrence of the two events in
the given order is m × n .
Example In a class, there are 27 boys and 14 girls. The teacher wants to select 1boy and 1 girl to
represent the class for a function. In how many ways can the teacher make this selection?
Solution Here the teacher is to perform two operations:
(i) Selecting a boy from among the 27 boys and
(ii) Selecting a girl from among 14 girls.
The first of these can be done in 27 ways and second can be performed in 14 ways. By the fundamental
principle of counting, the required number of ways is
27 × 14 = 378.
Addition principle
If an event E can occur in m ways and another event F can occur in n ways, and suppose that both
cannot occur together, then E or F can occur in m + n ways.
Permutations : A permutation is an arrangement of objects in a definite order.
Permutation of n different objects The number of permutations of n objects taken all at a time,
denoted by the symbol 𝑛𝑃𝑛, is given by
𝒏𝑷𝒏 = 𝒏!
When repetition of objects is allowed: The number of permutations of n things taken all at a time,
when repletion of objects is allowed is 𝑛𝑛 .
The number of permutations of n objects, taken r at a time, when repetition of objects is allowed, is 𝑛𝑟 .
Permutations when the objects are not distinct The number of permutations of n objects of which p1
are of one kind, p2 are of second kind, ...,𝑃𝑘 are of 𝐾 𝑡ℎ kind and the rest if any, are of different kinds is
𝒏!
𝑷𝟏 ! × 𝑷𝟐 ! … . .× 𝑷𝒌 !
Exercise
1. Find how many of the four digits numbers that can be made from 1, 2, 3 and 5.if repetition is not
allowed
i.
If there is no restriction.
ii.
Even number.
iii.
Divisible by 5.
iv.
Begin with 3 and end with 1.
Ans: 24 ; 6 ; 6 ; 2
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 70
2.
How many plates of the vehicles consisting of 4 digits can be made out of the integers
4,5,6,7,8,9?
Ans 360
3.
How many numbers between 6000 to 7000 can be made using the digits 1,2,3,6,7,9.
i. If no digit being repeated ?
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In which how many are even number?
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.............................................................................................................................
Ans :60, 12
4.
If the repetition are not allowed, how many numbers can be formed with the digit 3,4,5,6,7.
i.
Using three of the digit.
ii.
Using one or more of the digits?
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Ans :60 ,325
5. Find how many numbers between 3000 and 5000 can be formed from the digits 1, 3, 4, 5, 6 and 8.
i.
If no digits are repeated.
ii.
If repeated digits are allowed.
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 71
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Ans: 120 , 432
6.
Find how many different numbers greater than 300 can be made using some or all of the digits
1, 3, 4 and 5.
Ans: 42
7.
Find how many different numbers greater than 500 and divisible by 5 can be made using some
or all of the digits 1, 2, 5 and 6 with no digit being repeated.
Ans: 8
8. (i) How many numbers are there between 99 and 1000 having 7 in the unit’s place If repetition of
digits are allowed.
(ii) How many numbers are there between 99 and 1000 having at least one of their digits 7 If
repetition of digits are allowed.
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Ans: 90 ; 252
9. Six fair dice each with faces marked 1 , 2 , 3 ,4 , 5 , 6 are thrown and placed in a line . Find the
number of possible arrangements where the sum of the numbers at each of the line add up to 5.
Ans: 5184
10. Find how many different numbers can be made from some or all of the digits of the number 1 345
789 if
a.
all seven digits are used, the odd digits are all together and no digits are repeated,
b.
the numbers made are even numbers between 3000 and 5000, and no digits are repeated,
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 72
c.
the numbers made are multiples of 5 which are less than 2000 and digits can be repeated .
Ans : Total 720 ways , 1260 ways
11.
Each of the letters of the word CAMBRIDGE is written on a card and the cards are placed in a
line.
i.
How many different arrangements are there?
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ii.
How many arrangements begin with C?
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iii.
How many arrangements begin with C and end with G?
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iv.
How many arrangements begin with C and not end with G?
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v.
Exactly 5 Letters between M and A?
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vi.
Find the number of different arrangements, which starts and finished with vowels.
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vii.
Find the number of different arrangements if the 3 vowels A, I, E must all be together.
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 73
........................................................................................................................................................................................................................
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viii.
Find the number of different arrangements which do not have all 3 vowels (A, I , E ) next to each
other.
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ix.
Find the number of different arrangements if vowels are not together.
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Ans:
362880 ; 40320 ; 5040 ; 35280 ; 15120 ; 30240 ; 30240 ; 151200
12.
There are 4 books on Mathematics, 5 books on English and 6 books on Science. In how many
ways can you arrange them so that books on the same subject are together and they are arranged in
the order Mathematics , English , Science.
Ans: 2073600
13.
There are 3 Physics books, 4 Chemistry books, 5 Botany books and 3 Zoology books. In how
many ways can you arrange them so that the books on the same subject are together?
Ans: 2488320
14.
Find the number of different ways that 5 boys and 4 girls can stand in a line if
i.
Any individual can stand in any position?
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i.Particular boy Mihir stand at third position.
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ii.All 5 boys stand next to each other
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 74
iii.Not all 5 boys stand next to each other
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iv.No girls stand next to another girls .
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v.Each arrangements is to be symmetrical.
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vi.All boys and all girls stands next to each other but four particular boys are refuse to stand either sides
of girls .
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vii.Exactly three girls stand next to each other
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 75
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Ans: 9! ; 8! ; 1440 ; 348480 ; 43200 ; 69120 ; 1152 ; 21600
15.
A group of 7 students are staying in a hall in a hostel and they are allotted 7 beds. Among them,
Soren does not want a bed next to Krishma because she snores. Then, in how many ways can you allot
the beds?
Ans: 3600
16.
Three married couples are to be seated in a row having six seats in a cinema hall. If spouses are
to be seated next to each other, in how many ways can they be seated?
Ans:48
17.
Find the number of different arrangements which can be made of all 9 letters of the word
EVERGREEN if
i.
there are no restrictions,
ii.
There are exactly four letters between the two Rs.
iii.
The first letter is N and second last letter is E.
iv.
All vowels next to each other
v.
Exactly three vowels are next to each other
vi.
No two vowels may come together.
vii.
The Es are all together but Rs are not next to each other
viii. The letter V is at one end and there is an N at the other end.
ix.
The first letter is a consonant, the second letter is a vowel , the third letter is a consonant , the
fourth letter is a vowel , and so on alternately.
Ans:
7560 ; 840 ; 420 ; 360 ; 1800 ; 900 ; 240 ; 210 ; 60
18.
The 11 letters of the word REMEMBRANCE are arranged in a line.
(i)
Find the number of different arrangements if there are no restrictions.
........................................................................................................................................................................................................................
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(ii)
Find the number of different arrangements if there are exactly six letters between the two Rs.
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 76
........................................................................................................................................................................................................................
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.
(iii)
Find the number of different arrangements which start and finish with the letter M.
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(iv)
Find the number of different arrangements which start with A and not finish with the letter E.
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.
(v)
Find the number of different arrangements if exactly three vowels are next to each other.
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(vi)
Find the number of different arrangements which do not have all 4 vowels (E, E, A, E) next to
each other.
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Ans: 1663200 ; 13305600 ; 3326400 ; 44100 ; 2822240 ; 40320
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 77
19.
Three identical mathematics book, 2 identical economic book and 2 identical English books are
arranged in a row. Calculate the number of arrangements if
i.the first and last Book in the row are the same subject of book,
ii.the 3 books of mathematics are all next to each other and the 2 books of economics are
not next to each other.
Ans :50 ,18
Additional Questions:
1. If three people enter a bus in which there are ten vacant seats. Find in how many ways they can sit.
Ans 720
2. How many plates of the vehicles consisting of 4 digits can be made out of the integers 4,5,6,7,8,9?
Ans 360
3. In particular minibus there are 16 seats for passenger .how many possible seating arrangements are
there for 5 passengers?
Ans 210
4. How many numbers between 6000 to 7000 can be made using the digits 1, 2, 3,6,7,9? If no digit being
repeated? In which how many are even number?
Ans :60, 12
5. If the repetition is not allowed, how many numbers can be formed with the digit 3, 4,5,6,7.
i. Using three of the digit.
ii. Using one or more of the digits?
Ans :60 ,325
iii. A security code consists of 3 digit chosen from 4,5,6,7,8, followed by the 2 letters from P,Q,R,S,T. How
many different codes are possible, if repetition is not allowed?
Ans :1200
6. A. In how many ways can 5 boys and 3 girls stand in a straight line
i.
if there are no restrictions,
ii.
if the boys stand next to each other?
Ans 40320 , 2880
7. In a sweet shop 5 identical packets of toffees, 4 identical packets of fruit gums and 9 identical
packets of chocolates are arranged in a line on a shelf. Find the number of different arrangements of the
packets that are possible if the packets of chocolates are kept together.
Ans
:1260
8. Seven friends together with their respective partners all meet up for a meal. To commemorate the
occasion they arrange for a photograph to be taken of all 14 of them standing in a line.
i.
How many different arrangements are there if each friend is standing next to his or her partner?
ii.
How many different arrangements are there if the 7 friends all stand together and the 7 partners all
stand together?
9.
Ans; 645120, 50803200 (50800000)
Find the number of different ways in which the 12 letters of the word STRAWBERRIES can be
arranged
i.
if there are no restrictions,
ii. if the 4 vowels A, E, E, I must all be together
Ans: 19958400, 362880
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 78
10.Three identical cans of cola, 2 identical cans of green tea and 2 identical cans of orange juice are
arranged in a row. Calculate the number of arrangements if
i.
The first and last cans in the row are the same type of drink,
ii.
The 3 cans of cola are all next to each other and the 2 cans of green tea are not next to each other.
Ans :50 ,18
11.
Three identical yellow balloon, two identical red balloons and two identical blue balloons are
strung in a celebrate Shema ‘s birthday . Calculate the number of arrangements if
i. The balloon at each end is of the same colour.
ii.The yellow balloons are next to each other the blue balloon is not next to each other. ans 50 ,18
12.
find how many different arrangements there are of the eleven letters of the word PROBABILITY
if the two letters B are at the beginning and the two letters I are at the end .
ans :5040
13.
Find the number of ways in which all eight letters of the word ADVANCED can be arranged if the
arrangement must begin and end with an A.
14.
Ans : 360
Find how many different numbers can be made from some or all of the digits of the number
1 345 789 if
a. all seven digits are used, the odd digits are all together and no digits are repeated,
b. the numbers made are even numbers between 3000 and 5000, and no digits are repeated,
Ans : 720 ways , 1260 ways
15.
Find the number of ways of arranging 6 women and 3 men in a row so that no two men are
standing next
to each other.
16.
Ans :151200
The letters of the word CONSTANTINOPLE are written on 14 cards, one on each card.
The card are shuffled and then arranged in a straight line.
i.
How many arrangements begin with P?
ii.
How many arrangements are there where no two vowels are not next to each other?
Ans: 259,459,200
17.
457,228,800
Eight women and five men are standing in a line.
i.
How many arrangements are possible if any individual can stand in any position?
ii.
In how many arrangements will all five men be standing next to one another?
iii.
In how many arrangements will no two men be standing next to one another?
Ans :13!
43,545,600
609,638,400
18. Find the total number of permutations of all the 9 letters of the word ISOSCELES.Find the number
of these permutations which
i. Starting and finish with E.
ii. Have the letters S,S,S together?
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 79
iii. Have no two of the letters C, L, S, S, S consecutive.
Ans : 30,240 840 2520
19. If the letters of the word MINIMUM are arranged in a line at random , what is the probability that
the arrangement begins with MMM?
Ans : 1/35
20. On a self there are 4 maths books and 8 English books .
i. If the books sre arranged so that the maths books are together , in how many ways can this be done ?
ii. What is the probability that all the maths book will not be together?
Ans: 8,709,120
54/55
21. Calculate the number of permutations of the word URUGUAY when
i. There are no restriction ,
ii. U is the first letter and R is the second letter.
iii. The three Us all together.
Ans ;840, 60, 120
22. Find how many different four-digit numbers can be made using only the digits 1, 3, 5 and 6
with no digit being repeated.
i. Find how many different odd numbers greater than 500 can be made using some or all of the digits 1,
3, 5 and 6 with no digit being repeated.
ii. Six cards numbered 1, 2, 3, 4, 5, 6 are arranged randomly in a line. Find the probability that the cards
numbered 4 and 5 are not next to each other.
Ans 24 ,28 ,2/3
23. Puspa has 3 types of garnish. She has 6 different Monkeys,4 different dogs and 3 different leopards
.She displays 12 of the 13 garnishes in a row on her window-sill.
Find the number of different arrangements that are possible if there is no restriction
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She lets her daughter Amrusha displays 10 of the 13 garnishes in a row on her doorsill. Find the
number of different arrangements that are possible if
i. She has a leopard at each end of the row and no leopard anywhere else.
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 80
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ii. she has a leopard at each end of the row and monkeys and Doges are placed alternately in the positions
in between.
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Ans: 6227020800; 10886400; 103680
24.A meeting hall has seats for 28 people, consisting of 4 rows with 7 seats in each row.
Aryaman,Sushank, Alex, Arbin and Anikit are the first to arrive in the meeting hall and no seats are
taken before they arrive.
i. How many possible arrangements are there of seating Aryaman,Sushank, Alex, Arbin and Anikit
assuming there is no restriction?
ii. How many possible arrangements are there of seating Aryaman,Sushank, Alex, Arbin and Anikit if
Aryaman and Anikit sit together in the second row and the other three sit together in the last rows?
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 81
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iii. How many possible arrangements are there of seating Aryaman,Sushank, Alex, Arbin and Anikit if
Sushank,Alex and Anikit sit together in the front row and the other two sit together in one of the other
rows?
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........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
.......................................................................................................................................................................................................................
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........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
Ans: 11793600; 360 ; 1080
25. A Bus park has spaces for 10 Buses, arranged in a line. On one day there are 7 Buses, of different
makes, parked in randomly chosen positions and 3 empty spaces.
i.
Find the number of possible arrangements of the 7 Buses in the Bus park.
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........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 82
ii.
Find the probability that the 7 Buses are not all next to each other.
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Ans: 604800; 0.967
26. A minibus has seats for the driver (D) and ten passengers, as shown.
Back
Front
D
When ten passengers are seated in random order, find the probability that
a. Five particular students sitting same side of minibus.
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................................................................................................................................................................................................................
b. Two particular passenger, Priyam and Sushank ,are sitting on .The same side of the minibus.
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 83
........................................................................................................................................................................................................................
.......................................................................................................................................................................................................................
c. Mr and Mrs Anikit sits in the back row and Mr Sushank sit directly behind the driver seat.
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........................................................................................................................................................................................................................
Ans: 28800; 1612800; 10080
27. A group of 11friends travels to the airport in two taxis, P and Q. In taxi P can take 6 passengers and
sin taxi Q can take 5 passengers.
i.
The 11 friends divide themselves into two groups of 6 and 5, one group for taxi P and one group
for taxi Q, with Sushank and Aryaman travelling in the same taxi. Find the number of different
ways in which this can be done.
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Window
Window
back
back
Path
Window
Taxi P
Path
Taxi Q
In taxi 𝑃 have 6 seats for passengers. The seats are arranged in 2 rows of 3 seats and in taxi 𝑄 has 5
seats for passengers. The seat is arranged in two rows in front row has 2 seats back row of 3 seats (see
diagram).
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 84
ii.
Find the number of different seating arrangements that are now possible for the 11 friends in
which Sushank and Aryaman are in back row in taxi P.
........................................................................................................................................................................................................................
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…………………………………………………………………………………………………………………...........................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
.......................................................................................................................................................................................................................
These 11 passengers consist of 2 married couples (Mr and Mrs Alex Mr. and Mrs. Priyam ), 2 students
(Sushank and Aryaman) ,two teacher and 3 business people.
iii.
The 3 business people sit in the back row in Taxi Q. The 2 students each sit at a window seat in
taxi P. Mr and Mrs Alex sit in the same row on the same side of the Path in taxi P , Mr and Mrs
Priyam sit in another row on the same side of the Path in taxi P. The two teachers are in same
taxi .How many possible seating arrangements are there.
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........................................................................................................................................................................................................................
Ans: 210 ; 1451520 ; 192
28. The diagram shows the seating plan for passengers in a minibus, which has 21 seats arranged in 5
rows. The back row has 5 seats and the other 4 rows have 2 seats on each side. 15 passengers get on the
Windows
minibus.
Back
Pathway
Front
Windows
i.
How many possible seating arrangements are there for the 15 passengers?
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 85
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
ii.
How many possible seating arrangements are there if 4 particular people Aryaman,Sushank, Arbin
and Anikit sit together in the back row?
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........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
....................................................................................................................................................................................................................
Of the 15 passengers, 7 are unmarried and the other 8 consist of 4 married couples.
iii. Four married couples sit in the same row on the same side of the pathway. How many possible
seating arrangements are there?
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........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
iv. The seven unmarried passengers each sit at a window seat. How many possible seating
arrangements are there?
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........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
……………………………………………………………………………………………………………………………………………………………
…………………………………………………………………………………………………………………...........................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
Ans: 7.10 × 1016 ; 2.37 × 1013 ; 2.32 × 1011 ; 7.32 × 1013
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 86
Combination
1. In a class, there are 27 boys and 14 girls. The teacher wants to select 1 boy and 1 girl to represent
the class for a function. In how many ways can the teacher make this selection?
........................................................................................................................................................................................................................
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........................................................................................................................................................................................................................
.......................................................................................................................................................................................................................
........................................................................................................................................................................................................................
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........................................................................................................................................................................................................................
Ans: 378
2. A student has to answer 10 questions, choosing at least 4 from each of Parts A and B. If there are 6
questions in Part A and 7 in Part B, in how many ways can the student choose 10 questions?
........................................................................................................................................................................................................................
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Ans: 266
3. In a small village, there are 87 families, of which 52 families have at most 2 children. In a rural
development programme, 20 families are to be chosen for assistance, of which at least 18 families
must have at most 2 children. In how many ways can the choice be made?
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........................................................................................................................................................................................................................
Ans: 2.5 × 1016
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 87
4. A candidate is required to answer 7 questions out of 12 questions, which are divided into two
groups, each containing 6 questions. He is not permitted to attempt more than 5 questions from
either group. Find the number of different ways of doing questions.
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Ans: 780
5. We wish to select 6 persons from 8, but if the person A is chosen, then B must be chosen. In how
many ways can selections be made?
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........................................................................................................................................................................................................................
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Ans: 22
6. How many committees of five persons with a chairperson can be selected from 12 persons?
........................................................................................................................................................................................................................
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........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
Ans: 330
7. A bag contains 5 black and 6 red balls. Determine the number of ways in which 2 black and 3 red
balls can be selected from the lot.
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........................................................................................................................................................................................................................
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Ans: 200
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 88
8. In how many ways can a football team of 11 players be selected from 16 players? How many of them
will
i.
include 2 particular players?
ii.
exclude 2 particular players?
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Ans: 2002 ; 364
9. A bag contains six white marbles and five red marbles. Find the number of ways in which four
marbles can be drawn from the bag if
(a) they can be of any colour
(b) two must be white and two red and
(c) they must all be of the same colour.
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Ans: 330; 150; 20
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 89
10. A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be selected if the
team has
i.
no girls
ii.
at least one boy and one girl
iii.
at least three girls.
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Ans: 21 ; 441 ; 91
11. There are 10 professors and 20 lecturers out of whom a committee of 2 professors and 3 lecturer is
to be formed. Find :
a.
In how many ways committee can be formed.
b.
In how many ways a particular professor is included.
c.
In how many ways a particular lecturer is included.
d.
In how many ways a particular.
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Ans: 51300; 10260 ; 7695
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 90
12. Four of 8 letters of the word CHAMPION are selected. How many possible selection contain
i. No restriction
ii. Exactly one P
iii. Exactly two vowels
iv. More consonants than vowels.
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Ans: 70 ; 35 ; 30; 35
13. Five of 9 letters of the word EDUCATION are selected. How many possible selection contain
i.
Exactly three vowels
ii.
More consonants than vowels .
iii.
At most two vowels .
iv.
At least three vowels .
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 91
........................................................................................................................................................................................................................
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Ans: 60; 45 ; 45; 81
14. How many different selections of 3 letters can be made from the 5 letters of the word EVENT if
i. There are no Es,
ii. There is exactly one E ,
iii. There are no restrictions?
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Ans: 1; 3; 7
15. How many different selections of 4 letters can be made from the 7 letters of the word CORRECT if
i. There are no Rs,
ii. There is exactly two R ,
iii. There are no restrictions?
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 92
........................................................................................................................................................................................................................
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Ans: 4; 4; 18
16. How many different selections of 4 letters can be made from the 9 letters of the word HAMMERMAN
if
i. There are no A s,
ii. There is exactly 1 M,
iii. There are no restrictions?
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Ans: 15; 14; 41
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 93
17. How many different selections of 4 letters can be made from the 9 letters of the word ECONOMICS if
i. There are no O s,
ii. There is exactly 1 C,
iii. There are no restrictions?
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Ans: 20; 25; 71
18. A password for Lawati’s computer consists of 4 characters in a particular order. The characters are
chosen from the following.
➢
The 26 capital letters A to Z
➢
The 9 digits 1 to 9
➢
The 5 symbols # ~ * ? !
The password must include at least one capital letter, at least one digit and at least one symbol.
No character can be repeated. Find the number of different passwords that Lawati can make.
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 94
........................................................................................................................................................................................................................
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........................................................................................................................................................................................................................
Ans: 519480
19. Puspa has 2 necklaces, 8 rings and 4 bracelets, all different. She chooses 4 pieces of jewellery. How
many possible selections can she make if she chooses at least 1 necklace and at least 1 bracelet?
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Ans: 366
20. A committee of 7 people, which must contain at least 4 men and at least 2 women, is to be chosen
from 11 men and 8 women.
(i)
Find the number of possible committees that can be chosen.
(ii)
Find the probability that one particular man, Priyam, and one particular woman, Anshu, are
both on the committee.
(iii)
Find the number of possible committees that include either Reeshav or Elan but not both.
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 95
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Ans: 31416; 3990; 14784
21. A Football team of 11 players is to be chosen from 21 players consisting of 10 Central Defenders, 9
Forwards and 2 Goalkeepers. The team must include at least 5 Central Defenders, at least 4
Forwards and exactly 1 Goalkeeper. Find the number of different ways in which the team can be
chosen.
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Ans: 116424
22. Amrusha saves her digital images on her computer in three separate folders named ‘School’,
‘Birthday’ and ‘Home’. Her School folder contains 5 images, her birthday folder contains 5 images
and her home folder contains 8 images. All the images are different. Find the number of different
ways in which Amrusha can choose 6 of these images if there are at least 2 images from the birthday
folder and at least 1 image from each of the other two folders.
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 96
........................................................................................................................................................................................................................
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........................................................................................................................................................................................................................
.
Ans: 8800
23. A group of 11 people consists of 2 Business men, 4 Teachers and 5 Students. In how many ways can
a team of 5 be chosen if
(i)
both Business men are in the team,
(ii)
the 5 Students are either all in the team or all not in the team,
(iii)
at least 2 Teachers are in the team?
(iv)
At least 1 from each groups.
(v)
At least one students and equal number of teacher and Business men.
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Ans: 84; 7; 301; 310 ; 111
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 97
24. An Statistics examination consists of 10 questions in Part A and 4 questions in Part B. Candidates
must choose 6 questions. The order in which questions are chosen does not matter. Find the number
of ways in which the 6 questions can be chosen in each of the following cases.
(i)
There are no restrictions on which questions can be chosen.
(ii)
Candidates must choose at least 3 questions from Part B.
(iii)
Candidates must either choose both question 1 and question 2 in Part A, or choose neither
of these questions.
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Ans: 3003 ; 525; 1419
25. There are 11 spaniels, 15 retrievers and 7 poodles at a dog show. 7 dogs are selected to go through
to the final. How many selections of 9 different dogs can be made if there must be at least 1 spaniel,
at least 2 retrievers and at least 3 poodles?
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 98
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
Ans: 6373675
26. A committee of 7 people is to be chosen from 9 men and 11 women. In how many ways can this be
done
(i)
(ii)
if there are more women than men on the committee,
if the committee consists of 4 men and 3 women but two particular men refuse to be on
the committee together?
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Ans: 48840; 17325
Additional Questions:
1. 12 points lie on a circle. How many cyclic quadrilaterals can be drawn by using these points?
Ans: 495
2. In a box, there are 5 black pens, 3 white pens and 4 red pens. In how many ways can 2 black pens, 2
white pens and 2 red pens can be chosen?
Ans:180
3. A question paper consists of 10 questions divided into two parts A and B. Each part contains five
questions. A candidate is required to attempt six questions in all of which at least 2 should be from
part A and at least 2 from part B. In how many ways can the candidate select the questions if he can
answer all questions equally well?
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Ans: 200
Page 99
4. A committee of 5 persons is to be formed from 6 men and 4 women. In how many ways can this be
done when
i.at least 2 women are included?
ii.At most 2 women are included?
Ans: 186
5. The Indian Cricket team consists of 16 players. It includes 2 wicket keepers and 5 bowlers. In how
many ways can a cricket eleven be selected if we have to select 1 wicket keeper and at least 4 bowlers?
Ans: 1092
6. A team of 6 people is to be chosen from 5 swimmers, 7 athletes and 4 cyclists. There must be at
least 1 from each activity and there must be more athletes than cyclists. Find the number of different
ways in which the team can be chosen.
7.
Samir wishes to plant 15 flowers in a flowerbed. He can choose from 9 different geraniums, 5
different roses and 8 different lilies. He wants to have at least 5 geraniums and also to have the same
number of roses and lilies. Find the number of different selections of flowers he can make.
Ans: 20216
8. Three letters are selected at random from the letters of the word EDUCATION Find the total
number of selections.
Ans: 84
9. How many different selections of 4 letters can be made from the 8 letters of the word ECONOMICS
if
i. There are no Os,
ii.There is exactly 1 O,
iii.There are no restrictions?
10.
Ans: 35 ; 35 ; 91.
How many different selections of 4 letters can be made from the 8 letters of the word
NOTEBOOK if
i.
There are no Os,
ii. There is exactly 1 O,
iii. There are no restrictions?
Ans : 5 ; 10 ; 30
11.
How many different selections of 4 letters can be made from the 9 letters of the word
GREENGAGE if
i. There are no Es,
ii.There is exactly 1 G,
iii. There are no restrictions?
Ans : 7 ; 8 ; 26
12.How many different selections of 4 letters can be made from the 8 letters of the word EVERGREEN if
i.
There are no Rs and exactly 2 E,
ii. There are no Es,
iii. There are no restrictions?
Ans: 8 ; 4 ; 23
13. How many different team of 5 people can be chosen, without regarded to order, from a squad of
12 people.
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Ans : 792
Page 100
14. A spelling – bee team of 5 students is to be chosen from a class of 12 boys and 9 girls. in how
many ways can a team be chosen if the team consists of
-
3 boys and 2 girls .
-
3 girls and 2 boys
-
At least 3 girls?
Ans :7920 ,5544 ,7182
15. A group of 9 people consists of 2 boys, 3 girls and 4 adults. In how many ways can a team of 4 be
chosen if
i. both boys are in the team,
ii. the adults are either all in the team or all not in the team,
iii. at least 2 girls are in the team?
Ans :21 , 6, 51
16. prem wishes to plant 25 flowers in a flower-bed. He can choose from 15 different geraniums, 10
different roses and 8 different lilies. He wants to have at least 11 geraniums and also to have the
same number of roses and lilies. Find the number of different selections of flowers he can make.
Ans: 1,941,912
17. Issam has 11 different CDs, of which 6 are pop music, 3 are jazz and 2 are classical.
Issam makes a selection of 2 pop music CDs, 2 jazz CDs and 1 classical CD. How many different
possible selections can be made?
Ans : 90
18. .Four letters are to be selected from the letters in the word RIGIDITY. How many different
combinations are there?
Ans :30
19. Four letters are to be selected from the letters in the word SCHOOL. How many different
combinations are there?
ANS
:11
20. How many different selections of 4 letters can be made from the 9 letters of the word
TELEPHONE
if
(i)
There are no Es,
(ii)
There is exactly 1 E,
(iii)
There are no restrictions?
Ans : 15 , 20, 56
21. .Find the number of different selections of 4 letters from the 9 letters of the word HAPPINESS
which contain no Ps and either one or two Ss.
Ans 20
22. Four letters from the letters of the word REMEMBRANCE are chosen. Find the number of
different selections which contain no Ms and no Rs and at least 2 Es.
Ans : 10
23. Four of the 8 letters of the word TANZANIA are selected. How many possible selections contain
(i) exactly 1 N and 1 A,
(ii) exactly 1 N?
Ans : 8
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 101
5.Probability
Probability: the probability of an event is the measure of likelihood that it will happen .it is given on a
numerical scale from 0 to 1 and the numbers representing probability can be written as decimals,
fraction or percentages.
A probability of 0 indicates that event is impossible.
A probability of 1 indicates that the event is certain to happen.
All other events have a probability between 0 and 1.
Random experiments; any experiment whose outcome cannot be predicted or determined in advance is
called a random experiment. For example tossing a coin, rolling a die.
Outcomes; the result of a random experiment is called outcome or event. For example, while Tossing a
coin, the occurrence of head or tail is the outcome or event.
Sample space; the set of all possible outcome of a random experiment is known as sample space.it is
denoted by S. For example
S={ H,T} or S={1,2,3,4,5,6}
Event: a subset of sample space is called the event of the random experiment. For example
While tossing a coin 3 times, S={HHH,HHT,HTH,HTT,THH,THT,TTH,TTT}
Let A={HHH,HHT,HTH,THH} then A is called the event in which at least two head are obtained.
Exercise
1.
Calculate the theoretical probability , when rolling an ordinary ,fair dice ,of getting each of the
following :
a.
A score of 1
b.
A score of 2,3,4,5 or 6
c.
An odd number
d.
A score less than 6
e.
A score of 7
f.
A score less than 7
1
5
1
5
Ans: 6 ; 6 ; 2 ; 6 ; 0 ; 1
2.
250 balls are numbered from 1 to 250 and placed in a box . a ball is picked at random . find the
probability of picking a ball with :
i.
The number 1
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 102
ii.
An even number
iii.
A three –digit number
iv.
A number less than 300
1
1
151
Ans: 250 ; 2 ; 250 ; 1
3.
A boy was late for school 5 times in the previous 30 school days. If tomorrow is a school day,
1
calculate the probability that he will arrive
4.
Ans: 6
Two fair tetrahedral dice are rolled .if each is numbered 1-4 , Draw a two way table to show all
the possible outcomes.
i.
What is the probability that both dice shows the same number?
ii.
What is the probability that the number on one dice is double the number on the other?
iii.
What is the probability that the sum of both numbers is prime?
1
1
9
Ans: 4 ; 4 ; 16
5.
A particular board game involves players rolling a dice. However, before a player can start, he or
she needs to roll a 6.
a.
Complete the tree diagram below showing all the possible combinations for the first two roll of
the dice.
Outcomes
Roll 2
Roll 1
six
six
Six , six
Probability
1 1
1
× =
6 6 36
Not six
1
6
six
5
6
Not six
b.
Calculate the probability of the following :
i.
Getting a six on the first roll,
ii.
Stating within the first two rolls,
iii.
Starting on the second roll
iv.
Not starting within the first three rolls,
v.
Starting within the first three rolls ,
Not six
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 103
.....................................................................................................................................................................................................................
.....................................................................................................................................................................................................................
.....................................................................................................................................................................................................................
.....................................................................................................................................................................................................................
.....................................................................................................................................................................................................................
.....................................................................................................................................................................................................................
.....................................................................................................................................................................................................................
.....................................................................................................................................................................................................................
.....................................................................................................................................................................................................................
.....................................................................................................................................................................................................................
.....................................................................................................................................................................................................................
.....................................................................................................................................................................................................................
.....................................................................................................................................................................................................................
.....................................................................................................................................................................................................................
.....................................................................................................................................................................................................................
.....................................................................................................................................................................................................................
.....................................................................................................................................................................................................................
.....................................................................................................................................................................................................................
.....................................................................................................................................................................................................................
...............................................................................................................................................................................................................
c.
If you add the answer to 𝑏) 𝑖𝑣) 𝑎𝑛𝑑 𝑣) what do you notice. Explain.
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
...............................................................................................................................................................................................................
Ans:
1
6
;
11
36
;
5
36
;
125
216
;
91
216
; 𝑡ℎ𝑒𝑦 𝑎𝑟𝑒 𝑐𝑜𝑚𝑝𝑙𝑒𝑚𝑒𝑛𝑡 𝑒𝑣𝑒𝑛𝑡𝑠
6.
The probability that a morning bus arrive on time is 65% .
a.
Draw a tree diagram showing all the possible outcomes for three consecutive mornings.
b.
Label your tree diagram and use it to calculate the probability that :
i.
The bus is on time on all three mornings.
ii.
The bus is late the first two mornings.
iii.
The bus is on time two out of the three mornings.
iv.
The bus is on time at least twice.
Ans: 0.275 ; 0.123 ; 0.444 ; 0.719
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 104
7.
Samir has a bag of sweets containing 3 chocolates and 4 toffees. Mohit has a bag of sweets
containing 3 chocolates, 2 toffees and 3 boiled sweets. A sweet is taken at random from Samir’s bag
and put in Mohit’s bag. A sweet is then taken at random from Mohit’s bag.
i. Find the probability that the two sweets taken are a toffee from Samir’s bag and a boiled sweet from
Mohit’s bag.
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
..............................................................................................................................................................................
Find the probability that the sweet taken from Mohit’s bag is a chocolate.
ii.
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
iii.
Find the probability that at least one toffee taken.
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
Ans:
4 8
;
21 21
;
2
3
Mutually exclusive events
A and B are mutually exclusive event if it satisfies the following condition:
𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 0
𝑜𝑟 𝑃(𝐴 𝑜𝑟 𝐵) = 𝑃(𝐴) + 𝑃(𝐵)
Exercise:
1. Events A and B are such that P(A)=0.3 ,P(B)=0.6 ,and P(A and B)=0.72 . State, giving a reason , whether
event A and B are Mutually exclusive
.............................................................................................................................................................................................................
.............................................................................................................................................................................................................
.................................................................. ..........................................................................................................................................
..........................................................................................................................................................................................................
Ans: not exclusive
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 105
2. Two fair dice are thrown
i. Event A is the scores differ by 3 or more. Find the probability of event A.
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
... ................................................................................................................................................................................................................
ii.
Event B is the product of the scores is greater than 8. Find the probability of event B.
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
... ................................................................................................................................................................................................................
iii.
State with a reason whether events A and B are mutually exclusive.
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
... ................................................................................................................................................................................................................
Ans : 1/3 ;5/9 ;not
3. A fair dice is thrown. Events are defined as follows :
A: the score is at most 3
B: the score is at least 3
C: the score is fewer than 3
D: the score higher than 3
i. Show that events A and D are mutually exclusive.
..................................................................................................................................................................................................................
..................................................................................................................................................................................................................
..................................................................................................................................................................................................................
..................................................................................................................................................................................................................
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 106
ii.
Show that events B and C are mutually exclusive
.....................................................................................................................................................................................................................
.....................................................................................................................................................................................................................
.....................................................................................................................................................................................................................
.....................................................................................................................................................................................................................
iii.
Find P(A or B)
.....................................................................................................................................................................................................................
.....................................................................................................................................................................................................................
.....................................................................................................................................................................................................................
.....................................................................................................................................................................................................................
iv.
Find P(A and C )
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
................................................................................................................................................................................................................
Ans : 1 , 1/3
4. Two fair cubical dice are thrown
Event A: the score on the dice are the same
Event B : the product of the score is a multiple of 3 .
Event C: the sum of the score is 7.
State with a reason whether the following pair of event are mutually exclusive?
i.
A and B
ii. A and C
iii.
B and C
5. Two ordinary fair dice, one red and one blue are thrown.
Events A, B, and C are defined as follows:
Event A: the number showing on the red die is 5 or 6.
Event B: the total of the number showing on the two die is 7.
Event C : the total of the numbers showing on the two die is 8.
Find the probability of each events.
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 107
i.
Are A and B mutually exclusive events?
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
..........................................................................................................................................................................................................
ii.
Show that event B and C are mutually exclusive events.
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
Ans : 12/36 , 1/6 , 1/18 , no
Law of addition of mutually exclusive event:
P(A or B) = P(A) + P(B)
i. e P(A and B) = 0
P(A or B or C ) = P(A) + P(B) + P(C)
Law of addition of none mutually exclusive events:
P(A or Bor both ) = P(A) + P(B) − P(A and B)
i. e P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
1. The probability that a randomly chosen boy in AS class is in the football team is 0.4, the
probability that he is in the chess team is 0.5 and the probability that he is in both teams is 0.2.
Find the probability that a boy chosen at random from the class
i>
Is in the football team, but not in the chess team.
................................................................................................................................................................................................................
................................................................................................................................................................................................................
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 108
................................................................................................................................................................................................................
................................................................................................................................................................................................................
ii>
Is in the football team or the chess team.
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
iii>
Is not in either team.
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
Ans : 0.2 , 0.7 ,
Independent Events
Two events independent are if the occurrence of one has no effect on the occurrence of the other. For
instance, if a coin is tossed twice, the outcome of the first toss (heads or tails) has no effect on the
outcome of the second toss.
For independent event A and B
P(A/B) = P(A)
P(B/A) = P(B)
P(A and B) = P(A) × P(B) ′and ′ rule for independent event
In set notation p(A ∩ B) = P(A) × P(B)
Exercise:
1. Events A and B are independent. Find the indicated probability.
i.
P(A) = 0.3
P(B) = 0.9
P(A and B) = ?
ii.
P(A) = ?
P(B) = 0.3
P(A and B) = 0.06
iii.
P(A) = 0.75
P(B) = ?
P(A and B) = 0.15
..................................................................................................................................................................................................................
..................................................................................................................................................................................................................
..................................................................................................................................................................................................................
..................................................................................................................................................................................................................
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 109
..................................................................................................................................................................................................................
..................................................................................................................................................................................................................
..................................................................................................................................................................................................................
..................................................................................................................................................................................................................
Ans : 0.27 , 0.2 , 0.2
2. event A and B are such that P(A)=0.3 P(B)=0.6 and P(A or B)=0.72 state ,giving a reason in each
case ,whether event A and B are
i.
mutually exclusive event
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
ii.
Independent event
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
Ans: Not mutually exclave event , independent
3. Data about employment for males and females in a small rural area are shown in the table.
Unemployed Employed
Total
Male
206
412
618
Female
358
305
663
Total
564
717
1281
A person from this area is chosen at random. Let M be the event that the person is male and let E be
the event that the person is employed.
i.
Find P(M).
................................................................................................................................................................................................................
................................................................................................................................................................................................................
ii.
Find P(M and E).
................................................................................................................................................................................................................
................................................................................................................................................................................................................
iii.
Are M and E independent events? Justify your answer.
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 110
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
Ans : 618/1281 , 412/1281 , not
4. Two ordinary fair dice , one red and one blue are thrown .
Event A ,B ,and C are defined as follows
Event A : the number shown on the red die is 5 or 6
Event B : the total of the numbers showing on the two die is 7
Event C : the total of the numbers showing on the two dice is 8
i. State with a reason, whether the event A and B are mutually exclusive.
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
ii. Show that B and event C are mutually exclusive events .
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
iii. Are the event A and B independent?
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
.......................................................................................................................................................
Ans :Not
5. A school has 100 teachers .in a survey on the use of the school car park, the teachers were asked
whether they had driven a car to school on a particular day. of the 70 full –time ,45 had driven a
car to school and of the 30 part time teachers, 12 had driven a car to school.
i. Copy and complete the two-way table, where C denote the event ‘the teacher had driven a car to
school that day ‘
C
C’
Full-time
Part-time
total
100
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 111
ii. Are ‘the teacher had driven a car to school ‘and ‘the teacher is full-time ‘independent? Give a
reason for your answer.
..............................................................................................................................................................................................................
...............................................................................................................................................................................................................
...............................................................................................................................................................................................................
...............................................................................................................................................................................................................
...............................................................................................................................................................................................................
..........................................................................................................................................
iii. Describe two events that are mutually exclusive.
..............................................................................................................................................................................................................
...............................................................................................................................................................................................................
...............................................................................................................................................................................................................
Ans : no , full- time and part time or drive a car and not drive a car
6. Two fair twelve-sided dice with sides marked 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 are thrown, and the
numbers on the sides which land face down are noted. Events Q and R are defined as follows.
Q : the product of the two numbers is 24.
R : both of the numbers are greater than 8.
i.
Find P(Q).
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
...............................................................................................................................................................................................................
ii.
Find P(R)
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
iii.
Are events Q and R exclusive? Justify your answer.
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
iv.
Are events Q and R independent? Justify your answer.
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 112
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
Ans : 6/144 , 16/144, not ,not
7. Rory has 10 cards. Four of the cards have a 3 printed on them and six of the cards have a 4 printed
on them. He takes three cards at random, without replacement, and adds up the numbers on the
cards.
i. Show that P(the sum of the numbers on the three cards is 11) =0.5
............................................................................................................................................................................................................
............................................................................................................................................................................................................
............................................................................................................................................................................................................
............................................................................................................................................................................................................
............................................................................................................................................................................................................
............................................................................................................................................................................................................
............................................................................................................................................................................................................
Event R is ‘the sum of the numbers on the three cards is 11’. Event S is ‘the number on the first
card taken is a 3’.
ii.
Determine whether events R and S are independent. Justify your answer.
.............................................................................................................................................................................................................
.............................................................................................................................................................................................................
.............................................................................................................................................................................................................
.............................................................................................................................................................................................................
.............................................................................................................................................................................................................
.............................................................................................................................................................................................................
.............................................................................................................................................................................................................
.............................................................................................................................................................................................................
.............................................................................................................................................................................................................
.............................................................................................................................................................................................................
.............................................................................................................................................................................................................
............................................................................................................................................................................................................
iii. Determine whether events R and S are exclusive. Justify your answer.
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 113
.............................................................................................................................................................................................................
.............................................................................................................................................................................................................
.............................................................................................................................................................................................................
.............................................................................................................................................................................................................
Ans : Not indep , Not exclusive
8. Manjit throws two fair dice, each with faces numbered 1 to 6. Event A is ‘one of the numbers
obtained is divisible by 3 and the other number is not divisible by 3’. Event B is ‘the product of the
two numbers obtained is even’.
i. Determine whether events A and B are independent, showing your working.
.............................................................................................................................................................................................................
.............................................................................................................................................................................................................
.............................................................................................................................................................................................................
.............................................................................................................................................................................................................
.............................................................................................................................................................................................................
.............................................................................................................................................................................................................
.............................................................................................................................................................................................................
.............................................................................................................................................................................................................
.............................................................................................................................................................................................................
.............................................................................................................................................................................................................
.............................................................................................................................................................................................................
Are events A and B mutually exclusive? Justify your answer.
..................................................................................................................................................................................................................
..................................................................................................................................................................................................................
..................................................................................................................................................................................................................
Ans : Independent , Not mutually exclusive
Additional Questions
1. Fair twelve-sided dice with sides marked 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 are thrown, and the
numbers on the sides which land face down are noted. Events Q and R are defined as follows.
Q : the product of the two numbers is 24.
R : both of the numbers are greater than 8.
i.
Find P(Q).
ii.
Find P(R).
iii.
Are events Q and R exclusive? Justify your answer.
Ans : 6/144 , 16/144, not
2. Two event X and Y are such that
P(XorY) = 0.8 , P(XandY) = 0.35 and P(X) = 0.6 Find P(Y)and P(Y ′ )
Ans:0.55 , 0.45
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 114
3. Events A and Bare such that P(A)=0.6 , P(B)=0.7 P(A or B or both)=0.9 Find
i.
P(A and B)
ii.
P(neither A nor B )
iii.
P(A occurs or B occurs but not both A and B occur)
Ans : 0.4 , 0.1 , 0.5
4. Two fair die are thrown. Find the probability that the sum of the score is
i.
a multiple of 5
ii.
Greater than 9.
iii
A multiple of 5 or greater than 9.
Iv
A multiple of 5 and greater than 9.
Ans : 7/36 , 1/6, 5/18 , 1/12
5. two fair die are thrown.
i.
Event A is ‘the score differ by 3 or more.’ find the probability of event B.
ii.
Event B is ‘the product of the score is greater than 8’ find the probability of event B.
iii.
State with the reason whether event A and B are mutually exclusive.
Ans :1/3 , 5/9 , not mutually exclusive
Conditional probability
Conditional probability is used when the probability that an event will occur depends whether another
event has occurred.
For events A and B , the conditional probability that event B occurs , given that event A has already
occurred i. e P(B given A) or P(A/B).
Formula of conditional probability:.
P(Aand B)
P(A)
P(B/A)=
This gives the multiplication rule .
P(A and B)=P(A).P(B/A)
Exercise :
1. A company is worried about the high turnover of its employees and decides to investigate whether
they are more likely to stay if they are given training. On January one year the company employed
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 115
256 people (excluding those about to retire). During that year a record was kept of who received
training as well as who left the company. The results are summarised in this table.
Given
Still employed(S)
Left company Total
109
43
152
60
44
104
169
87
256
training(T)
Not given
training
Total
i.
Find the probability that a randomly selected employee
a. received training
.....................................................................................................................................................................................................................
.....................................................................................................................................................................................................................
b. Received training and did not leave the company.
.....................................................................................................................................................................................................................
.....................................................................................................................................................................................................................
ii. Are the events T and S independent?
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
iii.
a.
Find the probability that a randomly selected employee
did not leave the company, given that the person had received training
....................................................................................................................................................................................................................
.....................................................................................................................................................................................................................
.....................................................................................................................................................................................................................
.....................................................................................................................................................................................................................
.....................................................................................................................................................................................................................
b.
Did not leave the company, given that the person had not received training.
....................................................................................................................................................................................................................
.....................................................................................................................................................................................................................
.....................................................................................................................................................................................................................
.....................................................................................................................................................................................................................
Ans: 0 59 ; 0 43; 0 66; 0.392; not independent; 0 72; 0 58
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 116
2. Quark hunting is a dangerous occupation. On a quark hunt, there is a probability of
1
4
that the
hunter is killed. The quark is twice as likely to be killed as the hunter. There is a probability of
1
3
that both survive.
i.
Complete this table of probabilities.
Hunter
Hunter
dies
lives
Quark dies
0.5
Quark lives
Total
Total
1/3
1/4
0.5
1
Find the probability that
i.
both the hunter and the quark die
................................................................................................................................................................................................................
................................................................................................................................................................................................................
ii.
the hunter lives and the quark dies
................................................................................................................................................................................................................
................................................................................................................................................................................................................
iii.
the hunter lives, given that the quark dies.
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
ans: 1/12 ; 5/12 ; 5/6
3. A random sample of 100 people was asked whether they ate healthy foods and whether they
exercised three or more times per week. The results are shown in the table.
“B” exercised 3 or more
“A” Ate healthy
Did not eat
total
foods
healthy foods
27
8
35
21
44
65
48
52
100
times per week
Did not exercised 3 or
more times per week
Total
Suppose that one of these individuals was chosen at random. Calculate the Probability of each of the
following possibilities.
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 117
i.
The person ate healthy foods.
................................................................................................................................................................................................................
................................................................................................................................................................................................................
ii. The person exercised three of more times per week.
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
iii. The person ate healthy foods and exercised three or more times per week.
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
..............................................................................................................................................................
iv. The person did not eat healthy foods.
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
v. The person exercised three or more times per week, given that they ate healthy foods.
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
Ans : P(A)=12/25 , P(B)=7/20 , P(A&B)=27/100, P(not A)=13/25 ,
P(A|B)=27/35
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 118
Tree diagram:
1. The probability of a pregnant woman giving birth to a girl is about 0.49.Draw a tree diagram
showing the possible outcomes if she has two babies (not twins).
From the tree diagram, calculate the following probabilities:
a. that the babies are both girls
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
b. that the babies are the same sex
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
c. that the second baby is of different sex to the first.
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
..............................................................................................................................................................................................................
Ans:0.2401 ; 0.5002 ; 0.4998
2. In a certain district of a large city, the probability of a household suffering a break-in in a particular
year is 0.07 and the probability of its car being stolen is 0.12. Assuming these two trials are
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 119
independent of each other, draw a tree diagram showing the possible outcomes for a particular year.
Calculate, for a randomly selected household with one car, the following probabilities:
a.
that the household is a victim of both crimes during that year
................................................................................................................................................................................................................
................................................................................................................................................................................................................
..........................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
b. that the household suffers only one of these misfortunes during that year
...............................................................................................................................................................................................................
...............................................................................................................................................................................................................
.............................................................................................................................................................................................................
c. that the household suffers at least one of these misfortunes during that year.
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
Ans:0.0084; 0.17 32 ; 0.1816
3. Of the companies in a city, 40% are considered to be large and 60% are considered to be small. The
city is divided into three business districts: north, south, and east. Of the large companies, 26% are
in the north, 32% are in the south, and 42% are in the east. Of the small companies, 38% are in the
north, 44% are in the south, and 18% are in the east.
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 120
i. What is the probability that a company selected at random is located in the northern part of the
city?
...............................................................................................................................................................................................................
...............................................................................................................................................................................................................
...............................................................................................................................................................................................................
...............................................................................................................................................................................................................
...............................................................................................................................................................................................................
............................................................................................................................................................................................................
ii.
What is the probability that it is located in the eastern part of the city?
..................................................................................................................................................................................................................
..................................................................................................................................................................................................................
..................................................................................................................................................................................................................
..................................................................................................................................................................................................................
..................................................................................................................................................................................................................
Ans : 0.332 , 0.276
4. Every year two teams, the Ramblers and the Strollers, meet each other for a quiz night. From past
results it seems that in years when the Ramblers win, the probability of them winning the next year
is 0.7 and in years when the Strollers win, the probability of them winning the next year is 0.5. It is
not possible for the quiz to result in the scores being tied. The Ramblers won the quiz in 2009.
a.
Draw a probability tree diagram for the three years up to 2012.
b.
Find the probability that the Strollers will win in 2012.
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
If the Strollers win in 2012, what is the probability that it will be their first win for at least three
years?
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 121
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
c. Assuming that the Strollers win in 2012, find the smallest value of n such that the probability of
the Ramblers winning the quiz for n consecutive years after 2012 is less than 5%.
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
Ans:0.372 ; 0.395 ; 8
5. Around 0.8% of men are red–green colour-blind (the figure is slightly different for women) and
roughly 1 in 5 men is left-handed. Assuming these characteristics are inherited independently,
calculate with the aid of a tree diagram the probability that a man chosen at random will
a. be both colour-blind and left-handed
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 122
b. be colour-blind and not left-handed
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
c. be colour-blind or left-handed
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
d.
be neither colour-blind nor left-handed.
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
Ans:0.0016 ; 0.0064 ; 0.2064 ; 0.7936
6. At a factory, the percent of total output and the defective output rates for three machines are given
in the following table.
% of total
A
B
C
45%
30%
25%
4%
5%
7%
output
% defected
output
i.
What is the probability that an item selected at random is defective?
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
ii.
Given that the selected item is defective, find the probability the item produce by factory B.
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 123
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
Ans : 0.0505 ,0.297
7. A local weather service predicts a 60% chance of rain. There is a 45% chance of high winds
if it rains and a 15% chance of high winds if it does not rain.
i.
What is the probability that there will be high winds?
..................................................................................................................................................................................................................
..................................................................................................................................................................................................................
..................................................................................................................................................................................................................
..................................................................................................................................................................................................................
..................................................................................................................................................................................................................
..................................................................................................................................................................................................................
ii. Find the probability that randomly chosen day is raining, given that there will be high winds.
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 124
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
Ans : 0.33 , 0.8181
8. ln a tea shop 70% of customers order tea with milk, 20% tea with lemon and 10% tea with neither.
Of those taking tea with milk 3/5 take sugar, of those taking tea with lemon 1/4 take sugar, and of
those taking tea with neither milk nor lemon 11/20 take sugar. A customer is chosen at random.
a. Represent the information given on a tree diagram and use it to find the probability that the
customer takes sugar.
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
b. Find the probability that the customer takes milk or sugar or both.
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
..................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
c. Find the probability that the customer takes sugar and milk. Hence find the probability that the
customer takes milk given that the customer takes sugar.
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 125
................................................................................................................................................................................................................
................................................................................................................................................................................................................
Ans:0.525 ; 0.805; 0.42; 0.8
9. Box A contains 8 white balls and 2 yellow balls. Box B contains 5 white balls and x yellow balls. A ball
is chosen at random from box A and placed in box B. A ball is then chosen at random from box B. The
tree diagram below shows the possibilities for the colours of the
balls chosen
White
Box B
Box A
White
𝑥
𝑥+6
yellow
White
Yellow
yellow
i.
Justify the probability
𝑥
𝑥+6
on the tree diagram.
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
ii.
Complete the tree diagram.
iii.
If the ball chosen from box A is white then the probability that the ball chosen from box B is
1
also white is 3. Show that the value of x is 12.
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
................................................................................................................................................................................................................
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 126
iv. Given that the ball chosen from box B is yellow, find the conditional probability that the ball
chosen from box A was yellow.
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
..................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
.............................................................................................................................................................
Additional Questions:
10. One in every 200 people is infected by a virus. A test is used to determine whether a person is
infected. If a person is infected, the test is positive 80% of the time, and if the person is not infected
the test is still positive 5% of the time.
i.
If a person tests positive, what is the probability that the person is infected?
ii.
Given that the person tests negative, what is the probability that the person is not infected?
Ans :0.00798, 0.998
11. Roger and Andy play a tennis match in which the first person to win two sets wins the match. The
probability that Roger wins the first set is 0.6. For sets after the first, the probability that Roger wins
the set is 0.7 if he won the previous set, and is 0.25 if he lost the previous set. No set is drawn.
a. Find the probability that there is a winner of the match after exactly two sets.
b. Find the probability that Andy wins the match given that there is a winner of the match after
exactly two sets.
P(2 sets in match) = 0.72 , 0.417
12. Tom and Ben play a game repeatedly. The probability that Tom wins any game is 0.3. Each game is
won by either Tom or Ben. Tom and Ben stop playing when one of them (to be called the champion)
has won two games.
i.
Find the probability that Ben becomes the champion after playing exactly 2 games.
ii.
Find the probability that Ben becomes the champion.
iii.
Given that Tom becomes the champion, find the probability that he won the 2nd game.
0.49 , 0.784 , 0.708
13. Boxes of sweets contain toffees and chocolates. Box A contains 6 toffees and 4 chocolates, box B
contains 5 toffees and 3 chocolates, and box C contains 3 toffees and 7 chocolates. One of the boxes
is chosen at random and two sweets are taken out, one after the other, and eaten.
i.
Find the probability that they are both toffees.
ii.
Given that they are both toffees, find the probability that they both came from box A.
53/210 (0.252), 70/159 (0.440)
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 127
14. Maria chooses toast for her breakfast with probability 0.85. If she does not choose toast then she has
a bread roll. If she chooses toast then the probability that she will have jam on it is 0.8. If she has a
bread roll then the probability that she will have jam on it is 0.4.
i.
Draw a fully labelled tree diagram to show this information.
ii.
Given that Maria did not have jam for breakfast, find the probability that she had toast.
= 17/26 or (= 0.654)
15. A fair five-sided spinner has sides numbered 1, 2, 3, 4, 5. Raj spins the spinner and throws two fair
dice. He calculates his score as follows.
• If the spinner lands on an even-numbered side, Raj multiplies the two numbers
showing on the dice to get his score.
• If the spinner lands on an odd-numbered side, Raj adds the numbers showing on the dice to
get his score.
Given that Raj’s score is 12, find the probability that the spinner landed on an even-numbered
side.
Prob(E│12) = 8/11 (0.727)
16. It was found that 68% of the passengers on a train used a cell phone during their train journey. Of
those using a cell phone, 70% were under 30 years old, 25% were between 30 and 65 years old and
the rest were over 65 years old. Of those not using a cell phone, 26% were under 30 years old and
64% were over 65 years old.
i.
Draw a tree diagram to represent this information, giving all probabilities as decimals.
ii.
Given that one of the passengers is 45 years old, find the probability of this passenger using a
cell phone during the journey.
= 0.842 (170/202)
17. Ana meets her friends once every day. For each day the probability that she is early is 0.05 and the
probability that she is late is 0.75. Otherwise she is on time. If she is early there is a probability of 0.7
that she will eat a banana. If she is late she does not eat a banana. If she is on time there is a
probability of 0.4 that she will eat a banana. Given that for one particular meeting with friends she
does not eat a banana, find the probability that she is on time.
Ans = 0.136 (8/59)
18. Fabio drinks coffee each morning. He chooses Americano, Cappuccino or Latte with probabilities
0.5, 0.3 and 0.2 respectively. If he chooses Americano he either drinks it immediately with
probability 0.8, or leaves it to drink later. If he chooses Cappucino he either drinks it immediately
with probability 0.6, or leaves it to drink later. If he chooses Latte he either drinks it immediately
with probability 0.1, or leaves it to drink later.
i.
Find the probability that Fabio chooses Americano and leaves it to drink later.
ii.
Fabio drinks his coffee immediately. Find the probability that he chose Latte.
P (A Later) = 0.5 × 0.2 = 0.1, = 0.0333 (1 / 30)
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 128
19. On Saturday afternoons Mohit goes shopping with probability 0.25, or goes to the cinema with
probability 0.35 or stays at home. If he goes shopping the probability that he spends more than $50
is 0.7. If he goes to the cinema the probability that he spends more than $50 is 0.8. If he stays at
home he spends $10 on a pizza.
i.
Find the probability that Mohit will go to the cinema and spend less than $50.
ii.
Given that he spends less than $50, find the probability that he went to the cinema.
P(C ∩ < 50) = 0.35 × 0.2 = 0.07, = 0.128 (14/109)
20. Maria has 3 pre-set stations on her radio. When she switches her radio on, there is a probability of
0.3 that it will be set to station 1, a probability of 0.45 that it will be set to station 2 and a probability
of 0.25 that it will be set to station 3. On station 1 the probability that the presenter is male is 0.1, on
station 2 the probability that the presenter is male is 0.85 and on station 3 the probability that the
presenter is male is p. When Maria switches on the radio, the probability that it is set to station 3
and the presenter is male is 0.075.
i.
Show that the value of p is 0.3.
ii.
Given that Maria switches on and hears a male presenter, find the probability that the
radio was set to station 2.
= 0.3 , = 0.785
22. Last month a consultant saw 60 men and 65 women suspected of having a particular eye condition.
Tests were carried out and the following table shows the result. The total are shown in bold.
Had eye
Did not have eye
condition(C)
condition (C’)
Men (M)
25
35
60
Women (W)
20
45
65
45
80
125
One of these patients was selected at random to take part in survey. Find the probability that the patient
selected.
i.
Was a woman, given that the patient had the eye condition?
ii.
Had the eye condition, given that the patient was a man?
Ans : 4/9 , 7/12
23. Data about employment for males and females in a small rural area are shown in the table.
Unemployed Employed
Total
Male
206
412
618
Female
358
305
663
Total
564
717
1281
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 129
A person from this area is chosen at random. Let M be the event that the person is male and let E be the
event that the person is employed. Given that the person chosen is unemployed, find the probability that
the person is female.
Ans : 0.635
1
5
24. When Joanna cooks, the probability that the meal is served on time is . The probability that the
3
kitchen is left in a mess is 5. The probability that the meal is not served on time and the kitchen is not
3
left in a mess is 10. Some of this information is shown in the following table.
Kitchen left in a mess Kitchen not left in a
mess
Total
Meal served on time
1/5
Meal not served on
3/10
4/5
time
total
1
i.
Complete the table.
ii.
Given that the kitchen is left in a mess, find the probability that the meal is not served on
time.
Ans : 5/6
25.Suzanne has 20 pairs of shoes, some of which have designer labels. She has 6 pairs of high-heeled
shoes, of which 2 pairs have designer labels. She has 4 pairs of low-heeled shoes, of which 1 pair has
designer labels. The rest of her shoes are pairs of sports shoes. Suzanne has 8 pairs of shoes with
designer labels in total.
i.
Copy and complete the table below to show the number of pairs in each category.
Designer
No Designer
total
labels
labels
High-heeled
shoes
Low-heeled
shoes
Sports shoes
Total
Suzanne chooses 1 pair of shoes at random to wear.
ii.
Find the probability that she wears the pair of low-heeled shoes with designer labels.
iii.
Find the probability that she wears a pair of sports shoes.
iv.
Find the probability that she wears a pair of high-heeled shoes, given that she wears a pair
of shoes with designer labels.
v.
State with a reason whether the events ‘Suzanne wears a pair of shoes with designer labels’
and ‘Suzanne wears a pair of sports shoes’ are independent.
Ans 0.05, 0.5 , 0.25, Not independent
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 130
6.Probability Distribution
1. Find which of the following are discrete random variables, giving a reason for any which are not.
a.
X
2
P(X=x) 0.3
3
4
5
6
0.2
0.2
0.4
0.2
-2
1
2
3
0.2
0.4
0.2
0.1
b.
Y
-3
P(Y=y) 0.1
c.
X
1
P(X=x) 0.2
2
3
4
5
0.2
0.4
0.5
-0.3
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
Ans: Not since ∑ 𝑃(𝑥 = 𝑥) ≠ 1 ; A discrete random
variable. ; Not since 𝑃(𝑥 = 5) < 0
2. Three fair coin are tossed .If x=number of heads seen , list the probability distribution for X.
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 131
........................................................................................................................................................................................................................
.......................................................................................................................................................................................................................
Ans:
X
0
P(X=x) 1/8
1
2
3
3/8
3/8
1/8
3. The random variable x with probability distribution given by :
X
-2
P(x=x) 0.1
-1
0
1
2
0.2
0.4
a
0.2
i.
Find the value of a.
ii.
Find 𝑃(𝑥 ≥ 1)
iii.
Find 𝑃(𝑥 ≥ −1/𝑥 ≤ 1)
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
..............................................................................................................................................................................................................
Ans: 𝑎 = 0.1 ; 0.3 ; 0.875
4. The random variable y with probability distribution given by :
y
1
P(y=y) 0.2
2
3
4
5
0.1
0.3
a
0.2
i.
Find the value of a.
ii.
Find 𝑃(𝑦 ≥ 4)
iii.
Find 𝑃(𝑦 ≥ 2/𝑦 ≤ 4)
Ans: 𝑎 = 0.2 ; 0.4 ; 0.75
5. The random variable x with probability distribution given by :
X
-2
P(x=x) k
i.
-1
0
1
2k
4k
k
Find the value of k.
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 132
ii.
Find 𝑃(𝑥 < 1)
iii.
Find 𝑃(𝑥 ≥ −1/𝑥 < 1)
1
7
6
Ans: 8 ; 8 ; 7
6. For each of the following probability function, list the probability distribution.
a. 𝑃(𝑥 = 𝑥) = 𝑘𝑥
𝑓𝑜𝑟 𝑥 = 1, 2, 3, 4, 5
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
𝑘
b. 𝑃(𝑥 = 𝑥) = 𝑥
𝑓𝑜𝑟 𝑥 = 1, 2, 3, 4
𝑘
c. 𝑃(𝑥 = 𝑥) = 𝑥+1
𝑘(𝑥 − 1)
d. 𝑃(𝑥 = 𝑥) = {
𝑘(12 − 𝑥)
𝑓𝑜𝑟 𝑥 = 0, 1, 2, 3, 4
𝑓𝑜𝑟 𝑥 = 1, 2, 3, 4
𝑓𝑜𝑟 𝑥 = 5, 6, 7, 8
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
........................................................................................................................................................................................................................
.......................................................................................................................................................................................................................
e. 𝑃(𝑥 = 𝑥) =
𝑘−1
36
{13−𝑘
36
𝑓𝑜𝑟 𝑥 = 2, 3, 4, 5 , 6,7
𝑓𝑜𝑟 𝑥 = 8, 9, 10, 11,12
7. The random variable x with probability function
𝑥
𝑃(𝑋 = 𝑥) = 21
𝑓𝑜𝑟 𝑥 = 1, 2, 3, 4, 5, 6
A is the event 𝑋 ≥ 4 and B is the event 𝑋 ≤ 5.
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 133
Find
1. 𝑃(𝐴)
2. 𝑃(𝐵)
3. 𝑃(𝐴/𝐵)
4. 𝑃(𝐵/𝐴)
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5 5 3 3
𝐴𝑛𝑠: ; ; ;
7 7 5 5
8. The random variable with probability function
𝑃(𝑍 = 𝑧) =
𝑘
𝑧
𝑓𝑜𝑟 𝑧 = 1, 2, 3, 4.
12
a. Show that 𝑘 = 25.
b. Find
i. 𝑃(𝑍 > 1)
ii. 𝑃(𝑍 = 4/𝑍 > 2)
𝐴𝑛𝑠:
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
13 3
;
25 4
Page 134
Distribution Using Combination
9. A book club sends 8 Mathematics and 3 Economics books to Mr Sagar. He chooses 4 of these books
at random to take with his on holiday. The random variable X represents the number of Economics
books he chooses.
i.
Show that the probability that he chooses exactly 2 Economics books is 14/ 55.
ii.
Draw up the probability distribution table for X.
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Ans:
X
0
1
2
3
P(X=x) 7/33 28/55 14/55 4/165
10. A small farm has 6 Parrots and 3 Penguins. Four of these birds are to be chosen at random. The
random variable X represents the number of Penguins chosen. Draw up the probability
distribution of X.
Ans:
X
0
P(X=x) 5/42
1
2
10/21 5/14
3
1/21
11. A pet shop has 6 Bulldog and 3 hamsters. 5 of these pets are chosen at random. The random
variable X represents the number of hamsters chosen.
i.
10
Show that the probability that exactly 2 hamsters are chosen is 21 .
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 135
ii.
Draw up the probability distribution table for X.
Ans:
X
0
P(X=x) 1/21
1
2
3
5/14
10/21 5/42
12. A fruit basket contains 13 Apples , of which 6 are red, 3 are green and 4 are yellow. Three Apples
are taken, at random and without replacement, from the basket.
i.
Find the probability that the three Apples are all different colours.
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ii.
Show that the probability that exactly 2 of the Apples taken are red is
105
.
286
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iii.
The number of red apples taken is denoted by the discrete random variable X. Draw up a
probability distribution table for X.
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 136
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36
Ans :143
X
0
1
2
3
P(X=x) 35/286 63/143 105/286 10/143
13. A group of 11 people consists of 2 Business men, 4 Teachers and 5 Students. Three peoples are
taken at random and without replacement from the group.
i.
Show that the probability that exactly 2 of the People taken are Business men is
ii.
3
.
55
The number of Businessmen taken is denoted by the discrete random variable X. Draw up a
probability distribution table for X.
Ans:
X
0
1
2
P(X=x) 28/55 24/55 3/55
14. Lawati has 12 cards. Seven of the cards have a 3 printed on them and five of the cards have a 4
printed on them. He takes three cards at random, without replacement, and adds up the numbers
on the cards.
i.
1
Show that P(the sum of the numbers on the three cards is 12)= 22.
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ii.
Draw up a probability distribution table for the sum of the numbers on the three cards.
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 137
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Ans:
X
9
P(X=x) 7/44
10
11
21/44 7/22
12
1/22
15. Anikit has 7 cards. Three of the cards have a 2 printed on them and four of the cards have a 5
printed on them. He takes three cards at random, without replacement, and multiply the numbers
on the cards.
12
iii.
Show that 𝑝(𝑝𝑟𝑜𝑑𝑢𝑐𝑡 = 20) = 35.
iv.
Draw up a probability distribution table for the Product of the numbers on the three cards.
Ans:
X
8
P(X=x) 1/35
20
50
125
12/35 18/35 4/35
16. Alex chooses three digits at random, without replacement, from the 9-digit number 1, 1, 1, 3, 3, 3 ,
5, 5, 5.
i.
Find the probability that the three digits chosen are equal.
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ii.
Find the probability that one digit is a 5 and others digits are not 5.
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 138
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iii.
The random variable X is the number of 3s that Alex chooses. Draw up a table to show the
probability distribution of X.
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1
15
Ans: 28 ; 28
X
0
P(X=x) 5/21
1
2
15/28 3/14
3
1/84
17. Priyam chooses two digits at random, without replacement, from the 7-digit number 1, 1, 3, 3, 4, 4
,4 .
i.
Find the probability that the Two digits chosen are equal.
ii.
Find the probability that one digit is a 3 and others digits is not a 3.
iii.
The random variable X is the number of 4s that Priyam chooses. Draw up a table to show the
probability distribution of X.
Ans: 5/21 ; 10/21
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 139
X
0
P(X=x) 2/7
1
2
4/7
1/7
18. Mr Anikit plans to plant 5 trees along the centre of a main road. The Anikit buys the trees from a
garden centre which has 4 different hibiscus trees, 3 different jacaranda trees and 2 different
oleander trees for sale.
i.
Show that the probability that exactly 2 of the tree taken are hibiscus is
10
.
21
ii. The number of hibiscus trees taken is denoted by the discrete random variable X. Draw up a
probability distribution table for X.
Ans:
X
0
1
2
3
4
P(X=x) 1/126 10/63 10/21 20/63 5/126
Distribution using fair dice or spinner
19. Two ordinary fair dice are thrown. The resulting score is found as follows.
➢ If the two dice show different numbers, the score is the smaller of the two numbers.
➢ If the two dice show equal numbers, the score is 0.
i.
Draw up the probability distribution table for the score.
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ii.
Find the probability that the score is more than 2, given that the score is less than 5.
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 140
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Ans:
X
0
1
2
3
4
;5/17
5
P(X=x) 6/36 10/36 8/36 6/36 4/36 2/36
20. A fair spinner A has edges numbered 1, 2, 3, 4. A fair spinner B has edges numbered −3, −2, −1, 2.
Each spinner is spun. The number on the edge that the spinner comes to rest on is noted. Let X be
the sum of the numbers for the two spinners.
i.
Draw up a table showing the probability distribution of X.
ii.
Find the probability that X is even, given that X is positive.
Ans:
X
-2
-1
0
1
2
3
4
5
6
; 3/10
X=x) 1/16 2/16 3/16 3/16 2/16 2/16 1/16 1/16 1/16
21. A fair cubical die has two faces numbered 1, three faces numbered 2, and one face numbered 3.
The die is thrown twice. The discrete random variable X is the sum of the two scores.
i.
Complete the possible space showing the possible value of X.
1
1
1
2
2
2
3
3
1
2
2
4
3
2
5
3
ii.
Draw up a table showing the probability distribution of X.
iii.
Find the probability that X is even, given that X is Prime.
Ans:
X
2
3
4
5
6
P(X=x)
4/36
12/36
13/36
6/36
1/36
;2/9
22. Two ordinary fair dice are thrown. The resulting score is found as follows.
•
If the two dice show different numbers, the score is the sum of the two numbers.
•
If the two dice show equal numbers, the score is 0.
Draw up the probability distribution table for the score.
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 141
Ans:
X
0
3
4
5
6
7
8
9
10
11
P(X=x)
6/36 2/36 2/36 4/36 4/36 6/36 4/36 4/36 2/36 2/36
23. Set A consists of the ten digits 0, 0, 0, 0, 0, 0, 2, 2, 2, 4.
Set B consists of the seven digits 0, 0, 0, 0, 2, 2, 2.
One digit is chosen at random from each set. The random variable X is defined as the sum of these
two digits.
i.
3
Show that 𝑝(𝑥 = 2) = 7.
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ii.
Tabulate the probability distribution of X.
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iii.
Given that X = 2, find the probability that the digit chosen from set A was 2.
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 142
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Ans:
X
0
2
4
6
P(X=x)
24/70 30/70 13/70 3/70
24. A fair tetrahedral die has four
;2/5
triangular
faces, numbered 1, 2, 3 and 4. The score when this die is thrown is the number on the face that the
die lands on. This die is thrown three times. The random variable X is the sum of the three scores.
i.
10
Show that 𝑃(𝑋 = 9) = 64.
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ii.
Tabulate the probability distribution of X.
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 143
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Ans:
X
3
4
5
6
7
8
9
10
11
12
P(X=x) 1/64 3/64 6/64 10/64 12/64 12/64 10/64 6/64 3/64 1/64
25. Ayush tosses a Fair coin and throws two fair tetrahedral dice. Each of the dice has four faces,
numbered 1, 2, 3 and 4. Ayush’s score is calculated from the numbers on the faces that the dice
land on, as follows:
➢ if the coin shows a head, the two numbers from the dice are added together;
if the coin shows a tail, the two numbers from the dice are multiplied together.
➢
Draw up the probability distribution table for the score.
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Ans:
X
1
2
3
4
5
6
7
8
9
12
16
(X=x) 1/32 3/32 4/32 6/32 4/32 5/32 2/32 3/32 1/32 2/32 1/32
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 144
26. Tushar has a fair coin and a fair cubical die with faces numbered 1, 2, 3, 4 ,5 ,6. He tosses the coin
once and the die twice. The random variable X is defined as follows.
➢
If the coin shows a head then X is the sum of the scores on the two throws of the die.
➢
If the coin shows a tail then X is the score on the first throw of the die only.
i.
Find 𝑃(𝑥 = 3)
ii.
Tabulate the probability distribution of X.
Ans:
X
1
2
3
4
5
6
7
8
9
10
11
12
P(x=x) 6/72 7/72 8/72 9/72 10/72 11/72 6/72 5/72 4/72 3/72 2/72 1/72
Biased dice or spinner
1. The random variable X has the probability distribution shown in the table.
X
-1
P(X=x) 0.15
0
1
2
0.25
0.48
0.12
Two independent values of X are chosen at random. The discrete random variable Y is the sum of the
two values of x
i.
72
show that 𝑃(𝑦 = 3) = 625
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ii.
Draw up the probability distribution table for Y.
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 145
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Ans:
Y
-2
-1
0
P(Y=y) 9/400 3/40
1
2
3
4
413/2000 69/250 363/1250 72/625 9/625
2. The random variable X has the probability distribution shown in the table.
X
2
P(X=x) 0.5
4
6
0.4
0.1
Two independent values of X are chosen at random. The random variable Y takes the value 0 if
the two values of X are the same. Otherwise, the value of Y is the larger value of X minus the
smaller value of X.
12
.
25
i.
Show that 𝑃(𝑌 = 2) =
ii.
Draw up the probability distribution table for Y.
Ans:
Y
0
P(Y=y) 21/50
2
4
12/25
1/10
3. The random variables X and Y has the probability distribution shown in the tables.
X
3
4
5
P(X=x)
0.4 0.35 0.25
Y
-1
P(Y=y) 0.2
1
2
3
0.3
0.1
0.4
The random variable Z is the positive difference between random variables x and Y.
i.
4
Show that 𝑃(𝑍 = 0) = 25.
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ii.
Draw up the probability distribution table for Z.
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 146
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Ans:
Z
0
1
2
3
4
5
6
P(Z=z) 4/25 9/50 51/200 13/100 31/200 7/100 1/20
4. A spinner has 5 sides, numbered 1, 2, 3, 4 and 5. When the spinner is spun, the score is the
number of the side on which it lands. The score is denoted by the random variable X, which has
the probability distribution shown in the table.
X
1
2
3
4
5
P(x=x)
0.3
0.15
3p
2p
0.05
i.
Find the value of p.
A second spinner has 3 sides, numbered 1, 2 and 3. The score when this spinner is spun is
denoted by the random variable Y. It is given that
𝑃(𝑌 = 1) = 0.3, 𝑃(𝑌 = 2) = 0.5 𝑎𝑛𝑑 𝑃(𝑌 = 3) = 0.2.
ii.
Find the probability that, when both spinners are spun together,
a) The sum of the scores is 4,
b) The product of the scores is less than 8.
The random variable z is defined by the equation 𝑧 = 2𝑥 − 𝑦
iii.
Copy and complete the following table to show the probability distribution of z.
Z
-1
0
1
P(Z=z)
2
3
0.105
4
5
6
0.1
7
8
9
0.015
Ans: 0.1; 0.225; 0.765
5. Sirish tosses a biased coin and throws two fair tetrahedral dice. The probability that the coin
2
shows a head is 3. Each of the dice has four faces, numbered 1, 2, 3 and 4. The discrete random
variable x is define as follows:
➢ if the coin shows a head, the two numbers from the dice are added together;
➢ if the coin shows a tail, the two numbers from the dice are multiplied together.
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 147
1
.
12
i.
Show that 𝑃(𝑋 = 8) =
ii.
Draw up the probability distribution table for x.
Ans:
6.
X
1
2
3
4
5
6
7
P(x=x)
1/48 1/12 1/8 3/16 1/6 1/6 1/12 1/12 1/48 1/24 1/48
Sujal has a biased coin with probability of getting head is
8
1
3
9
12
16
and a biased tetrahedral die with
probabilities shown in the table.
X
1
P(X=x) 0.2
2
3
4
0.3
0.4
0.1
He tosses the coin once and the die twice. The random variable X is defined as follows.
➢
If the coin shows a head then X is the sum of the scores on the two throws of the die.
➢ If the coin shows a tail then X is the score on the first throw of the die only.
i.
1
Show that (𝑋 = 8) = 300 .
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ii.
Draw up the probability distribution table for x.
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 148
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Ans:
X
1
2
3
4
5
6
7
8
P(X=x) 2/15 16/75 23/75 3/20 7/75 11/150 2/75 1/300
7. Coin A is weighted so that the probability of throwing a head is 2/3 . Coin B is weighted so that
the probability of throwing a head is 1/4. Coin A is thrown twice and coin B is thrown once.
a.
Show that the probability of obtaining exactly 1 head and 2 tails is 13/36.
b.
Draw up the probability distribution table for the number of heads obtained.
Ans:
X
0
P(X=x) 3/36
1
2
3
13/36
16/36
4/36
Distribution using Tree diagram
1. Sirish has a bag of sweets containing 7 chocolates and 5 toffees. Anikit has a bag of sweets
containing 3 chocolates, 4 toffees and 2 boiled sweets. A sweet is taken at random from Sirish’s
bag and put in Anikit’s bag. A sweet is then taken at random from Anikit’s bag.
iv.
Find the probability that the two sweets taken are a toffee from Sirish’s bag and a boiled
sweet from Anikit’s bag.
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v.
Given that the sweet taken from Anikit’s bag is a chocolate, find the probability that the
sweet taken from Sirish’s bag was also a chocolate.
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 149
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vi.
The random variable X is the number of times a chocolate is taken. State the possible values
of X and draw up a table to show the probability distribution of X.
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Ans: 0.0833 ; 0.651 ;
X
0
P(X=x) 7/24
1
2
19/40
7/30
2. Box A contains 5 red paper clips and 1 white paper clip. Box B contains 7 red paper clips and 2
white paper clips. One paper clip is taken at random from box A and transferred to box B. One
paper clip is then taken at random from box B.
i.
Find the probability of taking both a white paper clip from box A and a red paper clip from
box B.
ii.
Find the probability that the paper clip taken from box B is red.
iii.
Find the probability that the paper clip taken from box A was red, given that the paper clip
taken from box B is red.
iv.
The random variable X denotes the number of times that a red paper clip is taken. Draw up a
table to show the probability distribution of X.
Ans : 7/60 (0.117) , 47/60 , 0.851,
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 150
X
0
P(X=x) 3/60
1
2
17/60
40/60
3. Every day Ravi tries to phone his friend. Every time he phones there is a 50% chance that his
friend will answer. If his friend answers Ravi does not phone again on that day. If his friend does
not answer, Ravi tries again in a few minutes’ time. If his friend has not answered after 4
attempts, Ravi does not try again on that day.
a. Draw a tree diagram to illustrate this situation.
b. Let X be the number of unanswered phone calls made by Ravi on a day. Copy and complete
the table showing the probability distribution of X.
X
P(X=x)
0
1
2
3
4
1
4
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Ans:
X
0
1
2
3
4
P(X=x) 1/2 1/4 1/8 1/16 1/16
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 151
4. sakar attempts a multiple choice revision question on-line. There are 3 suggested answers, one
of which is correct. When sakar chooses an answer the computer indicates whether the answer
is right or wrong. sakar first chooses one of the three suggested answers at random. If this
answer is wrong he has a second try, choosing an answer at random from the remaining 2. If this
answer is also wrong sakar then chooses the remaining answer, which must be correct.
i.
Draw a fully labelled tree diagram to illustrate the various choices that sakar can make until
the computer indicates that he has answered the question correctly.
ii.
The random variable X is the number of attempts that sakar makes up to and including the
one that the computer indicates is correct. Draw up the probability distribution table for X .
Ans:
X
1
P(X=x) 1/3
2
3
1/3
1/3
5. Samragyi is very forgetful. Every time she logs in to her online bank she only has a 40% chance
of remembering her password correctly. She is allowed 3 unsuccessful attempts on any one day
and then the bank will not let her try again until the next day.
a. Draw a fully labelled tree diagram to illustrate this situation.
b. Let X be the number of unsuccessful attempts Samragyi makes on any day that she tries
to log in to her bank. Copy and complete the following table to show the probability
distribution of X.
X
P(X=x)
0
1
2
3
0.24
Ans:
X
0
P(X=x) 0.4
1
2
3
0.24
0.144
0.216
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 152
Mean and variance
1. The table shows the probability distribution of a random variable
X
1
P(X=x) 0.1
i.
2
3
4
0.3
0.4
0.2
Find E(X ) and Var(X ).
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ii.
Three values of X are chosen at random. Find the probability that X takes the value 2 at least
twice.
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Ans: 2.7 ;0.81 ; 0.216
2. (a) The probability distribution of a random variable W is shown in the table.
W
0
P(W=w) 0.3
2
4
0.4
0.3
Calculate Var(W ).
(b) The random variable X has probability distribution given by
𝑃(𝑋 = 𝑥) = 𝑘(𝑥 + 1) 𝑓𝑜𝑟 𝑥 = 1, 2, 3, 4
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 153
1
.
14
i.
Show that =
ii.
Calculate E(X ).
Ans: 2.4 ; 2.86
3. When a four-sided spinner is spun, the number on which it lands is denoted by X, where X is a
random variable taking values 2, 4, 6 and 8. The spinner is biased so that
𝑃(𝑋 = 𝑥) = 𝑘𝑥, where k is a constant.
(i)
Show that 𝑃(𝑋 = 6) =
(ii)
Find E(X) and Var(X).
3
10
Ans: 6; 4
4. The score when a spinner is spun is given by the discrete random variable X with the following
probability distribution, where a and b are probabilities.
X
-1
P(X=x) b
0
2
4
5
a
a
a
b
a. Explain why E(X) = 2.
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b.
Find a linear equation in a and b.
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Given that Var(X) = 7.1
c. Find a second equation in a and b and simplify your answer.
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 154
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d.
Solve your two equations to find the value of a and the value of b.
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The discrete random variable Y = 10 – 3X
e. Find
i.
E(Y )
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ii.
Var(Y )
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Ans: 3𝑎 + 2𝑏 = 1; 20𝑎 + 26𝑏 = 11.1 ; 𝑎 = 0.1 ; 𝑏 = 0.35 ; 4 ; 63.9
5. The probability distribution of the discrete random variable X is shown in the table below.
X
-3
P(X=x) p
-1
0
4
q
0.15
0.4
Given that E(X) = 0.75, find the values of p and q.
Ans: 0.2 ; 0.25
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 155
6. The discrete random variable X has the following probability distribution.
X
-3
P(X=x) a
0
2
4
b
c
0.4
Given that E(X) = 2.3 and Var(X) = 3.01, find the values of a, b and c.
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Ans: 1/30 ; 1/6 ; 2/5
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 156
7.Binomial Distribution
Conditions for binomial distribution:
➢ A single trial has exactly two possible outcomes (success and failure) and these are mutually
exclusive.
➢ A fixed numbers, n , of trials takes place .
➢ The outcome of each trial is independent of the outcomes of all the other trials.
➢ The probability of success at each trial is constant.
The random variable X , which represent the number of success in the n trials of this experiment , has a
probability distribution given by .
𝑛
𝑃(𝑋 = 𝑥) = ( ) 𝑝 𝑥 𝑞𝑛−𝑥
𝑥
𝑓𝑜𝑟 𝑥 = 0,1 , 2,
3…………………𝑛
Where p is the probability of success and q= 1-p is the probability of failure.
When the random variable X satisfied these conditions, X~B(n,p)
Exercise
1. Given that 𝑋~𝐵(12,0.35), find the following probabilities correct to 3 significance figures.
a.
𝑃(𝑋 = 5)
b. 𝑃(𝑋 < 4)
c. 𝑃(𝑋 > 9)
d. 𝑃(𝑋 ≥ 9)
e. 𝑃(𝑋 ≤ 2)
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 157
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Ans: 0.204; 0.347; 0.00085; 0.0056; 0.151 ; 0.999;0.958 ; 0.999
2. Given that 𝑋~𝐵(15,0.28), find the following probabilities correct to 3 significance figures.
a. 𝑃(4 < 𝑋 < 6)
b. 𝑃(2 ≤ 𝑋 < 5)
c. 𝑃(1 < 𝑋 ≤ 3)
d. 𝑃(4 ≤ 𝑋 ≤ 8)
e. 𝑃(1 < 𝑋 ≤ 12)
f.
𝑃(3 ≤ 𝑋 < 15)
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 158
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Ans: 0.193; 0.535 ; 0.309; 0.632; 0.950; 0.835
3. At the Campion Academy, all students sit Mathematics examination at the end of their As level.
On average, 80% of the students pass this examination. A random sample of 9 students who will
take this examination is chosen. Find the probability that at most 6 of these students will pass
the examination.
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Ans: 0.262
4. Mani attempts the crossword puzzle in his daily newspaper every day. The probability that he
will complete the puzzle on any given day is 0.75, independently of all other days. Find the
probability that he will complete the puzzle at least three times over a period of five days.
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Ans: 0.896
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 159
5.
The random variable X has a binomial distribution with n=`12 and p=0.7.
i.
Exactly 3.
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ii.
At least 3.
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iii.
More than 2.
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iv. Between 8 and 10
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Ans: 0.00148, 0.999, 0.239
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 160
6.
On a certain road 20% of the vehicles are trucks, 16% are buses and the remainder are cars. A
random sample of 11 vehicles is taken. Find the probability that fewer than 3 are buses.
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Ans :0.748
7. The probability that a particular attending a clinic has a particular health condition is 2/5 find
the probability that in a randomly chosen group of 7 patient attending the clinic .
i. Exactly 3 have the condition.
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ii. More than 5 have the condition.
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iii. Fewer than 2 have the condition .
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 161
........................................................................................................................................................................................................
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iv. At least 2 but no more than 4 have the condition .
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Ans : 0.290, 0.0188 , 0.159, 0.745
8. In a survey it is found that 48% of the pupils travel to the local school by bus. Find the
probability that, in a random sample of 6 pupils, more than half of the pupils travel to school by
bus.
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Ans: 0.307
9. In an experiment 5 fair coin are tossed together .find the probability that they land showing
i.
Exactly 3 tails.
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ii. Fewer than 3 tails.
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 162
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iii. More than 3 heads.
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Ans :5/16 , 0.5, 3/16
10. On a average, three quarters of the patients who have a check –up at a particular dental practice do
not need follow –up treatment. Find the probability that, in a random sample of 9 patients from the
practice at most 7 do not need follow – up treatment.
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Ans: 0.700
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 163
Finding number of elements in given sample:
1. Given that 𝑋~𝐵(𝑛, 0.8). Find the value of n.
a. 𝑃(𝑋 = 0) > 0.0015
b. 𝑃(𝑋 = 0) < 0.00021
c. 𝑃(𝑋 = 𝑛) < 0.005
d. 𝑃(𝑋 = 𝑛) ≥ 0.00032
e. 𝑃(𝑋 ≥ 1) ≥ 0.995
f.
𝑃(𝑋 ≥ 1) > 0.9753
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Ans: n=4 ; n=6; n=24; n=36 ; n=4; n=3
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 164
2. A box contains a large number of pens and for each pen the probability that it is faulty is 0.1.
Rahul selects n pens at random from the box .what is the minimum value of n for which the probability
he select at least one faulty pen is greater than 0.95.
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Ans :29
3.
A coin is biased so that it is twice as likely to show head as tails.
i.
What is the probability that the coin will show head when it is tossed?
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ii. Smith tosses the coin n times. find the least value of n for which the probability that the coin shows
heads each time is less than 0.01
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Ans : 2/3 , 12
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 165
Additional Questions
4.
In Restaurant Bijoux 13% of customers rated the food as ‘poor’, 22% of customers rated the food
as ‘satisfactory’ and 65% rated it as ‘good’. A random sample of 12 customers who went for a meal at
Restaurant Bijoux was taken.
i.
Find the probability that more than 2 and fewer than 12 of them rated the food as ‘good’.
On a separate occasion, a random sample of n customers who went for a meal at the restaurant was
taken.
ii.
Find the smallest value of n for which the probability that at least 1 person will rate the food as
‘poor’ is greater than 0.95.
Ans: 0.993, 22
5.
The probability that Sue completes a Sudoku puzzle correctly is 0.75. Sue attempts n Sudoku
puzzles. Find the least value of n for which the probability that she completes all n puzzles correctly is
less than 0.06.
Ans: 10
6.
Fiona uses her calculator to produce 12 random integers between 7 and 21 inclusive. The
random Variable X is the number of these 12 integers which are multiples of 5.
i.
State the distribution of X and give its parameters.
ii.
Calculate the probability that X is between 3 and 5 inclusive.
Fiona now produces n random integers between 7 and 21 inclusive.
iii.
Find the least possible value of n if the probability that none of these integers is a multiple of 5 is
less than 0.01.
Ans : X ~ Bin (12, 0.2) , 0.422, n = 21
7.
In a certain country, on average one student in five has blue eyes. For a random selection of n
students, the probability that none of the students has blue eyes is less than 0.001. Find the least
possible value of n.
Ans : n = 31
8.
In a large consignment of mangoes, 15% of mangoes are classified as small, 70% as medium and
15% as large.
i.
Dev picks 14 mangoes at random. Find the probability that fewer than 12 of them are medium or
large.
ii.
Dev picks n mangoes at random. The probability that none of these n mangoes is small is at least
0.1. Find the largest possible value of n.
Ans : 0.352, n = 14
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 166
Expectation and variance of binomial distribution
If , X~B(n,p), then
Mean or expectation
𝐸(𝑋) = 𝜇 = 𝑛𝑝
𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 (𝜎 2 ) = 𝑛𝑝𝑞
𝑤ℎ𝑒𝑟𝑒 𝑞 = 1 − 𝑝
Exercise :
1. The random variable X is distributed B(n,p) with mean 5 and standard deviation 2 .
Find the values of n and p.
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Ans :n=25, p=0.2
2. The random variable X has distribution B (20, 0.25).
i. Find the mean and the standard deviation of X.
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ii. Find the percentage of the distribution that lies within one standard deviation of the mean.
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Ans: 5, 1.94, 56.1%
3. Biscuits are sold in packets of 18. There is a constant probability that any biscuit is broken,
independently of other biscuits. The mean number of broken biscuits in a packet has been found to be
2.7. Find the probability that a packet contains between 2 and 4 (inclusive) broken biscuits.
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 167
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Ans : 0.655
4. A biased die was thrown 20 times and the number of 5s was noted. This experiment was repeated
many times and the average number of 5s was found to be 4.8. Find the probability that in the next 20
throws the number of 5s will be less than three.
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Ans : 0.109
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 168
Geometric Distribution
The geometric distribution represents the number of failures before you get a first success.
Condition for geometric distribution
i.
Each trial has only two outcomes.
ii.
The trials are independent.
iii.
The probability of the outcomes does not change.
iv.
The variable of interest is the number of trails unit the first" success".
Difference between binomial and geometric distribution.
Binomial Properties
1. Fixed number of trials, n.
2. Only two mutually exclusive outcomes.
Geometric Properties
1. Only two mutually exclusive
outcomes.
3. Independent trials.
2. Independent trials.
4. Probability of outcome is constant for
3. Probability of outcome is constant
each trial, p.
X is binomial random variable.
𝑛
𝑃(𝑥 = 𝑟) = ( ) × 𝑝𝑟 × (1 − 𝑝)𝑛−𝑟
𝑟
where r is the number of successes (or
failures)
Mean Value (Expected Value)
E ( X ) =  x = np
for all trials, p.
X is geometric random variable.
𝑃(𝑥 = 𝑟) = 𝑝 × (1 − 𝑝)𝑟−1
where r is the number of attempts to a
success (or failure)
Mean Value (Expected Value)
E( X ) = x =
1
p
Standard Deviation
Standard Deviation
x =
1− p
p2
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 169
 x = np(1 − p )
Example: Assume 40% of the market prefers
Example: Assume 40% of the market
Pepsi. What is the probability that 5 out of
prefers Pepsi. What is the probability that
10 randomly selected cola drinkers would
the first cola drinker to randomly select
choice Pepsi in a blind taste test?
Pepsi in a blind taste test would be the fifth
customer?
P( X = 5) =
10!
(.4)5 (1 − .4)10−5
5!(10 − 5)!
P( X = 5) = .4(1 − .4)
5 −1
= .05184
= .20066
Example: Assume 40% of the market prefers
Example: Assume 40% of the market
Pepsi. What is the probability that 5 or 6 out
prefers Pepsi. What is the probability that
of 10 randomly selected cola drinkers would
the first cola drinker to randomly select
choice Pepsi in a blind taste test?
Pepsi in a blind taste test would be the fifth
or sixth customer?
P(X = 5 or 6) = P(X = 5) + P(X = 6) = ?
add .20066 + .11148
P(X = 5 or 6) = P(X = 5) + P(X = 6) = ?
add .05184 + .031104
Exercise : 1
1. Given that 𝑋~𝐺𝑒𝑜(0.3).
i.
𝑃(𝑋 = 2)
ii.
𝑃(𝑋 = 4)
iii.
𝑃(𝑋 > 5)
iv.
𝑃(𝑋 > 3)
v.
𝑃(𝑋 ≥ 5)
vi.
𝑃(𝑋 ≥ 2)
vii.
𝑃(𝑋 < 3)
viii.
𝑃(𝑋 < 4)
ix.
𝑃(𝑋 ≤ 7)
x.
𝑃(𝑋 ≤ 4)
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Ans: 0.21; 0.0189; 0.168 ; 0.343; 0.24; 0.7; 0.51; 0.657; 0.918; 0.760
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 171
2. Given that 𝑋~𝐺𝑒𝑜(0.68).
a. 𝑃(1 ≤ 𝑥 ≤ 3)
b. 𝑃(4 ≤ 𝑥 ≤ 7)
c. 𝑃(3 < 𝑥 ≤ 5)
d. 𝑃(5 ≤ 𝑥 < 8)
e. 𝑃(3 < 𝑥 < 6)
f.
𝑃(2 < 𝑥 < 7)
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Ans: 0.967; 0.0324; 0.0294; 0.0104; 0.0294; 0.101
3. Each year Jack enters a ballot for a concert ticket. The probability that Jack will win a ticket in
any particular year is 0.27. Find the probability that the first time Jack wins a ticket is
i. on his 8th attempt,
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
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ii. after his 8th attempt.
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Ans: 0. 0298 ; 0.0806
4. Each day Prashav makes repeated attempts to light his gas fire. If the fire lights he makes no
more attempts. On each attempt, the probability that the fire will light is 0.3 independent of all
other attempts. Find the probability that
i.
the fire lights on the 5th attempt,
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 173
ii.
Prashav needs more than 1 attempt but fewer than 5 attempts to light the fire.
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If the fire does not light on the 6th attempt, Prashav stops and the fire remains unlit.
iii.
Find the probability that, on a particular day, the fire lights.
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iv.
Prashav’s week starts on Monday. Find the probability that, during a certain week, the first
day on which the fire lights is Wednesday.
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Ans: 0.0720; 0.460; 0.882 ; 0.0122
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 174
5.
Puspa makes repeated, independent attempts to hit a target. On each attempt, the probability
that she succeeds is 0.1.
i.
Find the probability that
a. the first time she succeeds is on her 5th attempt,
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b. the first time she succeeds is after her 5th attempt,
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c. The second time she succeeds is before her 4th attempt.
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 175
Amrusha also makes repeated attempts to hit the target. Each attempt of either Amrusha or Puspa is
independent. Each time that Amrusha attempts to hit the target, the probability that she succeeds is 0.2.
Puspa and Amrusha take turns attempting to hit the target, with Puspa going first.
ii.
Find the probability that the first person to hit the target is Puspa, on her
a. 2nd attempt,
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b. 10th attempt.
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Ans: 0.0656; 0.59; 0.028 ; 0.072 ; 0.0052
6. 60% of the voters at a certain polling station are women. Voters enter the polling station one at a
time. The number of voters who enter, up to and including the first woman, is denoted by X.
i.
State a suitable distribution that can be used as a model for X, giving the value(s) of any
parameter(s). State also any necessary condition(s) for this distribution to be a good model.
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Use the distribution stated in part (i)to find
(ii)
P(X = 4),
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(iii)
P(X> 3).
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Ans: 0.0384 ; 0.064
7. Alex, Bibhu, Dipak and Lawati play a game with a fair cubical die. Starting with Alex they take
turns, in alphabetical order, to throw the die. This process is repeated as many times as
necessary until a player throws a 6. When this happens, the game stops and this player is the
winner. Find the probability that
(i)
Dipak wins on his first throw,
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(ii)
Lawati wins on his second throw,
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
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(iii)
Alex gets a third throw,
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(iv)
Bibhu throws the die exactly three times.
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Ans: 0.116 ; 0.0465 ; 0.233 ; 0.1
8. On the 'Online Mock Test', there are four answer choices (A, B, C, and D).
a. What is the probability that on a 10-question section of the 'Online Mock Test' by complete
random guessing that the first correctly guessed answered is the fourth?
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b. What is the probability that on a 15-question section of the 'Online Mock Test' by complete
random guessing that the first correct answer will be within the first 6 guesses?
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
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Ans: 0.105; 0.763
9.
Major universities claim that 72% of their senior athletes graduate that year. fifteen senior
athletic students attending major universities are randomly selected and recorded in order of
selection.
a) What is the probability that the first senior athletic student to graduate in the group of 15
that year is the 5th selected?
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b) What is the probability that the first senior athletic student to graduate in the group of 15
that year is the 13th selected?
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c) What is the probability that the first senior athletic student to graduate in the group of 15
that year is before the first 10 selected?
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d) What is the expected number of senior athletic students to graduate that year?
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 179
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e) What is the standard deviation of senior athletic students graduating that year?
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Ans: 0.075; 0.0054; 0.999; 1.39 ; 0.624
10. Will Fumble is the only receiver for MHS football team with the likelihood of catching a pass of
0.15.
a. What is the probability that the first pass caught is on the 1st pass?
b. What is the probability that the first pass caught is on the 4th pass?
c. What is the probability that the first pass is caught before the first 3 attempts?
d. What is the probability that the first pass is caught after the first 3 attempts?
e. What is the expect number of attempts for the first pass caught?
Ans: 0.15; 0.00287; 0.386; 0.614; 6.67
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 180
Normal distribution
Continuous random variable:
A continuous random variable is a variable that cannot take precise values but con only be given within
a specified interval .it usually arises from measuring a characteristic such as time, mass, or length.
Example: the time between cars passing a checkpoint, the error made by scales when weighing potatoes,
the length of leaves on a particular type of bush.
A continuous random variable X is defined by its probability density function f , together with the values
between which the function is valid for example ; a<x<b.
Area beneath the graph of y=f(x) represent probability, so 𝑃(𝑥1 ≤ 𝑋 ≤ 𝑥2 )is equal to the area beneath
the graph between 𝑥 = 𝑥1 𝑎𝑛𝑑 𝑥 = 𝑥2 .
The total area beneath the graph of y=f(x) is 1 .
The normal distribution: One of the most important continuous random variables in statistics is the
normal variable. Its distribution is known as the normal distribution.
In normal distribution
➢ Mean =median =mode
➢ Mean divide the whole data in two equal part or 50%
➢ Lower quartile divide the data in 25% to 75%
➢ Upper quartile divide the data in 75% to 25%
➢ If mean =0 variance =1 then normal distribution is said to be standard normal distribution.
Graph of normal distribution
➢ Changes in the mean μ alter the position of the curve along the x-axis
➢ Changes in the standard deviation alter the spread of the curve about the mean
𝜎2
𝜇
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 181
Here are some normal curves drawn to the same scale.
𝑥~𝑁(0 , 12 )
i.
𝜇 = 0 ,𝜎 = 1
𝑖𝑖. 𝑥~𝑁(4 , 0.52 )
𝑖𝑖𝑖. 𝑥~𝑁(8 , 22 )
𝜇 = 4 , 𝜎 = 0.5
0
𝜇 = 8 ,𝜎 = 2
4
8
The Normal Distribution 𝑵(𝝁, 𝝈𝟐 )
The standard normal distribution 𝑵(𝟎, 𝟏𝟐 )
The diagram shows the standard normal distribution
Mean, μ, = 0
Standard deviation, σ, = 1
𝜇=0
The tables give the area, Φ(z), from –∞ up to z
To find other probabilities, sketch the curve and use your head
Sketch a graph and shade in the area to represent the probability
Calculating probabilities from the Normal distribution
For a discrete probability distribution we calculate the probability of being less than some value x, i.e.
P(X < x), by simply summing up the probabilities of the values less than x.
For a continuous probability distribution we calculate the probability of being less than some value x,
i.e. P(X < x), by calculating the area under the curve to the left of x.
Suppose Z ~ N(0, 1), what is P(Z < 0) ?
Symmetry) P(Z < 0) = 0:5
Standard Normal distribution
Calculating this area is not easy and so we use probability tables. Probability tables are tables of
probabilities that have been calculated on a computer. All we have to do is identify the right probability
in the table and copy it down! Only one special Normal distribution, N(0, 1), has
been tabulated.
➢ The N(0, 1) distribution is called the standard Normal distribution.
Properties of normal distribution
Every normal distribution has certain properties. You use these properties to determine the relative
standing of any particular result on the distribution, and to find probabilities. The properties of any
normal distribution are as follows:
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 182
•
Its shape is symmetric (that is, when you cut it in half the two pieces are mirror images of each
other).
•
Its distribution has a bump in the middle, with tails going down and out to the left and right.
•
The mean and the median are the same and lie directly in the middle of the distribution (due to
symmetry).
•
Its standard deviation measures the distance on the distribution from the mean to the inflection
point (the place where the curve changes from an "upside-down-bowl" shape to a "right-side-upbowl" shape).
•
Because of its unique bell shape, probabilities for the normal distribution follow the Empirical Rule,
which says the following:
• About 68 percent of its values lie within one standard deviation of the mean. To find this range, take
the value of the standard deviation, then find the mean plus this amount, and the mean minus this
amount.
• About 95 percent of its values lie within two standard deviations of the mean. (Here you take 2
times the standard deviation, then add it to and subtract it from the mean.)
• Almost all of its values (about 99.7 percent of them) lie within three standard deviations of the
mean. (Take 3 times the standard deviation and add it to and subtract it from the mean.)
Symmetry property
Its shape is symmetric (that is, when you cut it in half the two pieces are mirror images of each
other).
𝜇−𝑎
𝜇
𝜇+𝑎
Exercise :
1. Given that 𝑍~𝑁(0,1), find the following probabilities correct to 3 significance figures.
a. 𝑃(𝑍 ≤ 1.68)
b. 𝑃(𝑍 ≤ 0.589)
c. 𝑃(𝑍 ≥ 2.189)
d. 𝑃(𝑍 > 2.018)
e. 𝑃(𝑍 < −1.652)
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 183
f.
𝑃(𝑍 ≤ −0.632)
g. 𝑃(𝑍 ≥ −0.532)
h. 𝑃(𝑍 > 2.576)
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Ans: 0.9535 ; 0.7221; 0.0143 ; 0.0218; 0.0493; 0.2637; 0.7026; 0.005
2. Given that 𝑍~𝑁(0,1), find the following probabilities correct to 3 significance figures.
a. 𝑃(1.2 ≤ 𝑍 ≤ 2.31)
b. 𝑃(0.589 ≤ 𝑍 ≤ 1.212)
c. 𝑃(−2.085 ≤ 𝑍 ≤ 1.051)
d. 𝑃(−2.132 ≤ 𝑍 ≤ 2.317)
e. 𝑃(−2.351 ≤ 𝑍 ≤ 0)
f.
𝑃(−1.581 ≤ 𝑍 ≤ −0.523)
g. 𝑃(|𝑍| ≤ 1.234)
h. 𝑃(|𝑍| ≤ 2.135)
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 184
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Ans: 0.1047; 0.1651; 0.8349; 0.9732; 0.4906; 0.2436; 0.7828; 0.9672
3. Sketch the following pair of normal curve on a same axis .
a. 𝑋~𝑁(10,16)
𝑎𝑛𝑑
𝑦~𝑁(12,16).
b. 𝑋~𝑁(25,23)
𝑎𝑛𝑑
𝑦~𝑁(25,23).
c. 𝑋~𝑁(18,4)
𝑎𝑛𝑑
d. 𝑋~𝑁(16,36)
𝑎𝑛𝑑
e. 𝑊~𝑁(20,12)
f.
𝑊~𝑁(15,7)
,
,
𝑦~𝑁(18,12).
𝑦~𝑁(16,8).
𝑋~𝑁(20,25)
𝑎𝑛𝑑
𝑦~𝑁(25,12).
𝑋~𝑁(8,25)
𝑎𝑛𝑑
𝑦~𝑁(15,12) .
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 185
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 186
4. Given that 𝑋~𝑁(30,16), find the following probabilities correct to 3 significance figures.
a. 𝑃(𝑋 ≤ 35)
b. 𝑃(𝑋 ≤ 37)
c. 𝑃(𝑋 ≥ 24)
d. 𝑃(𝑋 > 27)
e. 𝑃(𝑋 ≤ 21)
f.
𝑃(𝑋 ≤ 22)
g. 𝑃(𝑋 ≥ 36)
h. 𝑃(𝑋 > 39)
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Ans: 0.894; 0.960; 0.933; 0.227; 0.988; 0.977; 0.067; 0.988
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 187
5. Given that 𝑋~𝑁(145,152 ), find the following probabilities correct to 3 significance figures.
a. 𝑃(130 ≤ 𝑋 ≤ 142)
b. 𝑃(145 ≤ 𝑋 ≤ 158)
c. 𝑃(125 ≤ 𝑋 ≤ 149)
d. 𝑃(148 ≤ 𝑋 ≤ 161)
e. 𝑃(140 ≤ 𝑋 ≤ 150)
f.
𝑃(135 ≤ 𝑋 ≤ 148)
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Ans: 0.262; 0.307; 0.514; 0.277; 0.414; 0.564
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 188
6.
The average height of 17 year old boys is normally distributed with mean 179 cm and standard
deviation 8 cm. Calculate the percentage of 17 year old boys whose heights are:
a.
More than 195 cm
b.
Between 163 cm and 195 cm
c. Between 171 cm and 187 cm.
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Ans: 2.28%; 95.4%; 68.3%
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 189
7.
The contents of cans of a certain brand of soft drink are normally distributed with mean 377 𝑚𝐿
and standard deviation 4.2 𝑚L.
a. Find the percentage of cans with contents:
i.
less than 368.6 𝑚𝐿
ii.
between372.8 𝑚𝐿 𝑎𝑛𝑑 389.6 𝑚𝐿.
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b.
Find the probability that a randomly selected can contains between377 𝑚𝐿 𝑎𝑛𝑑 381.2 𝑚𝐿.
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Ans:2.28%; 84%; 0.3413
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 190
8. A bottle shop sells on average 2500 bottles per day with a standard deviation of 300 bottles.
Assuming that the number of bottles sold per day is normally distributed, calculate the percentage of
days when:
i.
Less than 1900 bottles are sold
ii.
More than 2200 bottles are sold
iii.
Between 2200 and 3100, bottles are sold.
Ans: 2.28%; 84.13%; 85.85% 0.383
9.
A firm’s marketing manager believes that total sales for next year will follow the normal
distribution, with mean of $2.5 million and a standard deviation of $300, 000. What is the probability
that the firm’s sales will fall within $150000 of the mean?
Ans: 0.3830.
10. Metal strips are produce by a machine . The lengths of strips follow a normal distribution with
mean 150 cm and standard deviation 10 cm .
Find the probability that a randomly chosen strip from the production line has a length
i.
Less than 165 cm.
ii.
Between 127.5 cm and 139.2 cm.
iii.
That deviates from the mean by more than 1.645 times standard deviation .
iv.
That deviates from the mean by less than 2 times standard deviation.
v.
That deviates from the mean by more than 1.282 times standard deviation above the mean.
Ans: 0.933; 0.128; 0.1; 0.9; 0.10
11. The masses of a certain type of cabbages are normally distributed with mean 1000 g and
standard deviation 150 g. find the proportion of the cabbages with a mass
a. greater than 850 g
b. between 750g and 1290 g
c. That deviates from the mean by more than 220 g .
d. That deviates from the mean by more than 2.576 times standard deviation.
e. That deviates from the mean by less than 1.96 times standard deviation.
f.
That deviates from the mean by more than 1.891 times standard deviation below the
mean.
Ans: 0.8413; 0.9256; 0.182; 0.01; 0.9500; 0.029
12. A line up for tickets to a local concert had an average (mean) waiting time of 20 minutes with a
standard deviation of 4 minutes.
a. What percentage of the people in line waited for more than 28 minutes?
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Page 191
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b.
If 2000 ticket buyers were in line, how many of them would expect to wait for less than 16
minutes?
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Ans: 2.28%; 317
13. In an Oreo factory, the mean mass of a cookie is given as 40 g. For quality control, the standard
deviation is 2 g.
a. If 10 000 cookies were produced, how many cookies are within 2 g of the mean?
b. Cookies are rejected if they weigh more than 44 g or less than 36 g. How many cookies
would you expect to be rejected in a sample of 10 000 cookies?
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 192
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Ans: 6826; 3108
14. The speeds of cars on the highway have a mean of 95 km/h with a standard deviation of 5 km/h.
a. What percentage of cars averaged less than 85 km/h?
b. If a police car stopped cars that were going more than 105 km/h, how many cars would they
stop if there were 8000 cars on the highway?
Ans: 2.28%; 182
15. The mean life of a battery is 50 hours with a standard deviation of 6 hours. The manufacturer
advertises that they will replace all batteries that last less than 38hours. If 50 000 batteries were
produced, how many would they expect to replace?
Ans: 1140
16. A bottle of fruit punch contains at least 473 ml. The machine that fills the bottles is set so that
the mean volume is 477 ml. The volumes in the bottles are normally distributed.
a. What percent of the bottles are under filled if the standard deviation is 2 ml?
b. What percent of the bottles are under filled if the standard deviation is 4 ml?
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 193
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Ans: 0.977 ; 0.8413
17. A grading scale is set up for 1000 students’ test scores. It is assumed that the scores are
normally distributed with a mean score of 75 and a standard deviation of 15
a. How many students will have scores between 45 and 75?
b. If 60 is the lowest passing score, how many students are expected to pass the test?
Ans: 0.477 ; 159
18. The monthly income of 5 000 workers at the Microsoft plant are distributed normally. Suppose
the mean monthly income is $1250 and the standard deviation is $250.
a. How many workers earn more than $1500 per month?
b. How many workers earn less than $750 per month?
c. What percentage of the workers earn between $750 and $1500 per month?
d. What percentage of the workers earn less than $1750 per month?
Ans: 794; 4885 ; 81.82% ; 15.9%
19. The random variable X is the daily profit, in thousands of dollars, made by a company. X is
normally distributed with mean 6.4 and standard deviation 5.2.
a) Find the probability that, on a randomly chosen day, the company makes a profit between
$10 000 and $12 000.
b) Find the probability that, on a randomly chosen day, the company makes loss.
Ans: 0.1032; 0.1092
20. Metal strips are produce by a machine. The lengths of strips follow a normal distribution with
mean 150 cm and standard deviation 10 cm .
Find the probability that a randomly chosen strip from the production line has a length
i.
Less than 165 cm.
ii.
Between 127.5 cm and 139.2 cm.
iii.
That deviates from the mean by more than 28 cm.
Ans: 0.933 ; 0.128 ; 0.0052
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 194
21. Every day Bruno jogs around the park. The time he takes in minute, follow a normal distribution
with mean 12 and variance 2.
i.
Find the probability that the he take longer than 14 minutes.
ii.
Find the probability that he takes less 9 minutes.
iii.
Estimate the number of the day during a year the he takes between 10 and 13 minutes
Ans: 0.0787; 0.0170; 249
22. The masses of a certain type of cabbages are normally distributed with mean 1000 g and
standard deviation 150 g.
i.
find the proportion of the cabbages with a mass
a. greater than 850 g
b. between 750g and 1290 g
ii.
Estimate the number of cabbages in batch of 800 with a mass less than 900g or greater
than 1375g.
ANS:0.8413 ;0.9257 ; 207
23. the weight of vegetable marrow supplied to retailers by a wholesaler have a normal distribution
with mean1.50 kg and standard deviation 0.02 kg . the wholesaler supplies three size of marrow
size 1 under 1.48 kg
size 2 from 1.48kg to 1.53 kg.
size 3 over 1.53 kg
find to three decimal places , the proportion of marrow in the three sizes.
Ans:0.159 ; 0.7734 ; 0.6678
24. The time in minutes taken by Peter to walk to the shop and buy a newspaper is normally
distributed with mean 9.5 and standard deviation 1.3. Find the probability that on a randomly
chosen day Peter takes longer than 10.2 minutes.
Ans:0.295
25. (a) Amy measured her pulse rate while resting, x beats per minute, at the same time each day
on 30 days. The results are summarised below.
𝛴(𝑥 – 80) = −147
𝛴(𝑥 – 80)2 = 952
Find the mean and standard deviation of Amy’s pulse rate.
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Amy’s friend Marok measured her pulse rate every day after running for half an hour. Marok’s pulse
rate, in beats per minute, was found to have a mean of 148.6 and a standard deviation of 18.5. Assuming
that pulse rates have a normal distribution, find what proportion of Marok’s pulse rates, after running
for half an hour, were above 160 beats per minute.
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Ans: 71.5 ,2.78, 0.0269
26. Gem stones from a certain mine have weights, X grams, which are normally distributed with
mean 1.9 g and standard deviation 0.55 g. These gem stones are sorted into three categories for
sale depending on their weights, as follows.
Small: under 1.2 g
Medium: between 1.2 g and 2.5 g
Large: over 2.5 g
Find the proportion of gem stones in each of these three categories.
Ans:0.101; 0.761 ; 0.138
27. A farmer finds that the weights of sheep on his farm have a normal distribution with mean 66.4
kg and standard deviation 5.6 kg.
i.
250 sheep are chosen at random. Estimate the number of sheep which have a weight of
between 70 kg and 72.5 kg.
ii.
The proportion of sheep weighing less than 59.2 kg is equal to the proportion weighing more
than y kg. Find the value of y.
Ans:31 ;73.6
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 196
28. The random variable Y is normally distributed with mean - and standard deviation 3. Given
2
𝑡ℎ𝑎𝑡 𝜎 = 3 𝜇, find the probability that a random value of Y is less than 2μ.
Ans: 0.933
29. The height of sunflowers follows a normal distribution with mean 112 cm and standard
deviation 17.2 cm. Find the probability that the height of a randomly chosen sunflower is greater
than 120 cm.
Ans:0.321
Inverse of normal distribution
Finding z when ∅(Z)is given :
1. Given that 𝑍~𝑁(0,1), find the value of k.
a. 𝑃(𝑍 ≤ 𝑘) = 0.9861
b. 𝑃(𝑍 ≤ 𝑘) = 0.9236
c. 𝑃(𝑍 ≥ 𝑘) = 0.8780
d. 𝑃(𝑍 > 𝑘) = 0.5682
e. 𝑃(𝑍 < 𝑘) = 0.1536
f.
𝑃(𝑍 ≤ 𝑘) = 0.4839
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Ans: 2.20 ; 1.43; 0.624; -1.17; -0.172; -1.02; -0.0404; 0.446; 0.565
2. Given that 𝑍~𝑁(0,1), find the value of k.
a. 𝑃(−𝑘 < 𝑍 < 𝑘) = 0.9235.
b. 𝑃(−𝑘 < 𝑍 < 𝑘) = 0.8546
c. 𝑃(−𝑘 < 𝑍 < 𝑘) = 0.7589
d. 𝑃(−𝑘 < 𝑍 < 𝑘) = 0.9138
e. 𝑃(−𝑘 < 𝑍 < 𝑘) = 0.6861
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 197
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Ans: 1.771; 1.46; 1.172; 1.72; 1.01
3. Given that 𝑍~𝑁(0,1), find the value of a.
a. 𝑃(𝑎 < 𝑍 < 1.531) = 0.3526
b. 𝑃(𝑎 < 𝑍 < 2.536) = 0.4258
c. 𝑃(−2.351 < 𝑍 < 𝑎) = 0.7245
d. 𝑃(−1.281 < 𝑍 < 𝑎) = 0.7538
e. 𝑃(−2.361 < 𝑍 < 𝑎) = 0.4213
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 198
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Ans: 0.2134; 0.173; 0.625; 1.053; -0.716; -0.503
4. Given that 𝑋~𝑁(44,125), find the value of k.
a. 𝑃(𝑋 ≤ 𝑘) = 0.9861
b. 𝑃(𝑋 ≤ 𝑘) = 0.8692
c. 𝑃(𝑋 ≤ 𝑘) = 0.7509
d. 𝑃(𝑋 > 𝑘) = 0.8856
e. 𝑃(𝑋 > 𝑘) = 0.3569
f.
𝑃(𝑋 ≤ 𝑘) = 0.1536
g. 𝑃(𝑋 ≤ 𝑘) = 0.2638
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Page 199
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Ans: 77; 60.8; 54.4; 62.1; 49.5; 28.7; 34.5
5. Suppose that X follows the normal distribution with mean 𝜇 = 5. 𝐼𝑓 𝑃(𝑋 > 9) = 0.2.
Find the variance of X.
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Ans: 22.6
6. The weight X of water melons is normally distributed with mean 𝜇 = 10 pounds and standard
deviation 𝜎 = 2 pounds. Find c such that𝑃(𝑋 > 𝑐) = 0.60.
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Ans: 9.48
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 200
7. Suppose that the height (X) in inches, of a 25-year-old man is a normal random variable with mean
𝜇 = 70 inches. If 𝑃(𝑋 > 79) = 0.025 what is the standard deviation of this random normal variable?
Ans: 4.59
8. Suppose that the weight (X) in pounds, of a 40-year-old man is a normal random variable with
standard deviation 𝜎 = 20 pounds. If 5% of this population weigh less than 160 pounds what is the
mean 𝜇 of this distribution?
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Ans: 233
9. Find an interval that covers the middle 95% of 𝑋 ~ 𝑁(64, 8).
10. Suppose that X follows the normal distribution with mean 𝜇 and standard deviation 𝜎 if
𝑃(𝑋 ≥ 59.1) = 0.0218 and 𝑃(𝑋 ≥ 29.2) = 0.9345.Find the mean and standard deviation of the
distribution, correct to 3 significant figures.
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Ans: 42.0; 8.47
11. Suppose that X follows the normal distribution with mean 𝜇 and standard deviation 𝜎 if
𝑃(𝑋 ≥ 9.81) = 0.1587 and 𝑃(𝑋 < 8.82) = 0.0116.Find the mean and standard deviation of the
distribution, correct to 3 significant figures.
Ans: 9.51; 0.303
Additional Questions:
1. The times taken by a garage to fit a tow bar onto a car have a normal distribution with mean m
hours and standard deviation 0.35 hours. It is found that 95% of times taken are longer than 0.9
hours. Find the value of m.
..
Ans: m = 1.48
2. the height of sunflowers follows a normal distribution with mean 115 cm. Given that 80% of the
heights are now greater than 103 cm, find the standard deviation.
Ans: 14.3
3. Melons are sold in three sizes: small, medium and large. The weights follow a normal
distribution with mean 450 grams and standard deviation 120 grams. Melons weighing less than
350 grams are classified as small.
i.
Find the proportion of melons which are classified as small.
ii.
The rest of the melons are divided in equal proportions between medium and large. Find the
weight above which melons are classified as large.
Ans:0.2025 ; 481
4. The lengths of fish of a certain type have a normal distribution with mean 38 cm. It is found that
5% of the fish are longer than 50 cm.
i.
Find the standard deviation.
ii.
When fish are chosen for sale, those shorter than 30 cm are rejected. Find the proportion of
fish rejected.
Ans: 7.29 ; 0.136
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 202
5. In a certain country the time taken for a common infection to clear up is normally distributed
with mean μ days and standard deviation 2.6 days. 25% of these infections clear up in less than
7 days.
i.
Find the value of μ.
In another country the standard deviation of the time taken for the infection to clear up is the same
as in part (i), but the mean is 6.5 days. The time taken is normally distributed.
ii.
Find the probability that, in a randomly chosen case from this country, the infection takes
longer than 6.2 days to clear up.
Ans: μ = 8.75 ; 0.546
6. (a) The random variable X is normally distributed with mean μand standard deviation 𝜎. It is
given that 3𝜇 = 7𝜎 2 and that 𝑃(𝑋 > 2𝜇) = 0.1016. Find 𝜇 and𝜎.
(b) It is given that Y ~N(33, 21). Find the value of a given that
𝑃(33 − 𝑎 < 𝑌 < 33 + 𝑎) = 0.5.
Ans: σ = 0.545 ;μ = 0.693 ; 3.90
7. The lengths, in centimetres, of drinking straws produced in a factory have a normal distribution
with mean μand variance 0.64. It is given that 10% of the straws are shorter than 20 cm.
i.
Find the value of μ.
ii.
Find the probability that, of 4 straws chosen at random, fewer than 2 will have a length
between 21.5 cm and 22.5 cm.
Ans: μ = 21.0 cm (21.0256) ; 0.2453 ; 0.746
8. The random variable X has the distribution 𝑁(𝜇, 𝜎 2 ). It is given that P(X < 54.1)= 0.5 and P(X >
50.9) = 0.8665. Find the values of μ and σ
Ans: μ = 54.1 ; σ = 2.88
9. Gem stones from a certain mine have weights, X grams, which are normally distributed with mean
1.9 g and standard deviation 0.55 g. These gem stones are sorted into three categories for sale
depending on their weights, as follows.
Small: under 1.2 g
Medium: between 1.2 g and 2.5 g
Large: over 2.5 g
i. Find the proportion of gem stones in each of these three categories.
ii. Find the value of k such that P(k < X < 2.5) = 0.8.
Ans: 0.761; k =1.06
10.Packets of tea are labelled as containing 250 g. The actual weight of tea in a packet has a normal
distribution with mean 260 g and standard deviation σ g. Any packet with a weight less than 250 g is
classed as ‘underweight’. Given that 1% of packets of tea are underweight, find the value of σ.
Ans: σ = 4.30
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 203
Normal and Binomial distribution
1.
The weights of apples sold by a store can be modelled by a normal distribution with mean 120
grams and standard deviation 24 grams. Apples weighing less than 90 grams are graded as ‘small’;
apples weighing more than 140 grams are graded as ‘large’; the remainder are graded as ‘medium’.
i. Show that the probability that an apple chosen at random is graded as medium is 0.692, correct to 3
significant figures.
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ii.Four apples are chosen at random. Find the probability that at least two are graded as medium.
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Ans: 0.910
2. (a) A petrol station finds that its daily sales, in litres, are normally distributed with mean 4520 and
standard deviation 560. Find on how many days of the year (365 days) the daily sales can be expected to
exceed 3900 litres.
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The daily sales at another petrol station are X litres, where X is normally distributed with mean m
and standard deviation 560. It is given that 𝑃(𝑋 > 8000) = 0.122.
c. Find the value of m.
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c. Find the probability that daily sales at this petrol station exceed 8000 litres on fewer than 2 of 6
randomly chosen days.
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Ans:315; 7350; 0.840
3. The amount of fibre in a packet of a certain brand of cereal is normally distributed with mean
160 grams. 19% of packets of cereal contain more than 190 grams of fibre.
a. Find the standard deviation of the amount of fibre in a packet.
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 205
b.
Kate buys 12 packets of cereal. Find the probability that at least 1 of the packets
contains more than 190 grams of fibre.
Ans:34.2 ; 0.920
4. Lengths of a certain type of carrot have a normal distribution with mean 14.2 cm and standard
deviation 3.6 cm.
a.
8% of carrots are shorter than c cm. Find the value of c.
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b.
Rebecca picks 7 carrots at random. Find the probability that at least 2 of them have lengths
between 15 and 16 cm.
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Ans:9.14 ; 0.159
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 206
5. The random variable X is the daily profit, in thousands of dollars, made by a company. X is normally
distributed with mean 6.4 and standard deviation 5.2.
a. Find the probability that, on a randomly chosen day, the company makes a profit between $10 000
and
$12 000.
b. Find the probability that the company makes a loss on exactly 1 of the next 4 consecutive days.
Ans: 0.104 ;0.309
6.
Lengths of rolls of parcel tape have a normal distribution with mean 75m, and 15% of the rolls have
lengths less than 73m.
a.
Find the standard deviation of the lengths.
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Alison buys 8 rolls of parcel tape.
b.
Find the probability that fewer than 3 of these rolls have lengths more than 77m.
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
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Ans:1.93 ; 0.895
7. The random variable X is normally distributed and is such that the mean μ is three times the
standard deviation s. It is given that 𝑃(𝑋 < 25) = 0.648.
a. Find the values of μ and s.
b. Find the probability that, from 6 random values of X, exactly 4 are greater than 25.
Ans:22.2 ; 7.4 ; 0.0967
8. The weights of letters posted by a certain business are normally distributed with mean 20 g. It is
found that the weights of 94% of the letters are within 12 g of the mean.
a.
Find the standard deviation of the weights of the letters.
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b.
Find the probability that a randomly chosen letter weighs more than 13 g.
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c. Find the probability that at least 2 of a random sample of 7 letters have weights which are more
than 12 g above the mean.
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 208
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Ans:6.38 ; 0.864; 0.0171
9. The times taken by students to get up in the morning can be modelled by a normal distribution
with mean 26.4 minutes and standard deviation 3.7 minutes.
a. For a random sample of 350 students, find the number who would be expected to take longer
than 20 minutes to get up in the morning.
b. ‘Very slow’ students are students whose time to get up is more than 1.645 standard
deviations above the mean. Find the probability that fewer than 3 students from a random
sample of 8 students are ‘very slow’.
Ans: 335 ;0.994
NormalApproximation:
1.
Is a normal distribution a reasonable approximation for a binomial distribution with n=50 and
p=0.85? Explain your reasoning.
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Ans: Normal approximation is Applicable since np and nq both greater than 5
2.
Use the normal approximation to find 𝜇 and 𝜎 in a binomial distribution with n = 1000 and p=0.5.
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
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Ans: 500; 15.8
3.
The random variable X has a 𝐵(60, 0.5) distribution. For each of the following binomial
probabilities describe the region under the approximating normal curve whose area gives the best
estimate.
a.
𝑃(𝑋 ≤ 12)
b.
𝑃(𝑋 = 16)
c.
𝑃(𝑋 < 22)
d.
𝑃(𝑋 > 18)
e.
𝑃(12 < 𝑋 ≤ 34)
f.
𝑃(12 ≤ 𝑋 ≤ 21)
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Ans: 𝑃(𝑣 ≤ 12.5); 𝑃(15.5 ≤ 𝑋 ≤ 316.5) ; 𝑃(𝑋 ≤ 21.5) ; 𝑃(𝑋 ≥ 18.5) ; 𝑃(12.5 ≤ 𝑋 ≤ 34.5) ;
𝑃(11.5 ≤ 𝑋 ≤ 21.5)
4.
Given that 𝑋~𝐵(150,0.45), find the following probabilities correct to 3 significance figures.
a.
𝑃(𝑋 ≤ 74)
b.
𝑃(𝑋 ≤ 60)
c.
𝑃(𝑋 < 76)
d.
𝑃(𝑋 < 57)
e.
𝑃(𝑋 ≥ 77)
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CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
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Ans: 0.875; 0.125; 0.905; 0.0355; 0.0698; 0.9302; 0.7442; 0.2059
5.
Given that 𝑋~𝐵(380,0.32), find the following probabilities correct to 3 significance figures.
a.
𝑃(112 ≤ 𝑋 ≤ 128)
b.
𝑃(105 ≤ 𝑋 ≤ 141)
c.
𝑃(102 ≤ 𝑋 < 134)
d.
𝑃(110 ≤ 𝑋 < 138)
e.
𝑃(106 < 𝑋 < 142)
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Ans: 0.6427; 0.9546; 0.8912; 0.8769; 0.7745; 0.889; 0.9373; 0.8615
6.
A School quiz completion has 50 multiple-choice questions, each with three possible answers. Use a
normal approximation to calculate the probability of randomly guessing the correct answers for at least
half of the questions.
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Ans: 0.00941
7.
GIHE Online Mock test has 50 multiple-choice questions, each with four possible answers. Use a
normal approximation to calculate the probability of randomly guessing the correct answers for 10 to
15 (inclusive) of the questions.
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Ans: 0.6728
8.
Himalayan distillery , a hard-drink company, knows that it has a 42% market share in one region of
the province. Himalayan distillery’s marketing department conducts a blind taste test with 100 people
at a mall in the region. Use a normal approximation to calculate the probability that fewer than 40 of
these people will choose Himalayan distillery.
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Ans: 0.3063
9.
Black Chimney, a hard-drink company, knows that it has a 38% market share in one region of
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 213
the province. Black Chimney’s marketing department conducts a blind taste test with 100 people at a
mall in the region. Use a normal approximation to calculate the probability that exactly 37 of these
people will choose Black Chimney.
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Ans: 0.0806
10. The probability of an airline flight arriving on time is 90%. Use the normal approximation to find
the probability that at least 300 of a random sample of 350 flights will arrive on time. Explain each step
in the calculation.
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Ans: 0.9982
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
Page 214
Additional Questions:
1. A bank found that 25% of its loans to new small businesses become delinquent. If 500 small
businesses are selected randomly from the bank’s files, what is the probability that at least 130 of
them are delinquent?
Ans: 0.321
2. It is estimated that 10% of the vehicles entering Canada from the United States carry undeclared
goods. Use the normal approximation to calculate the probability that a search of 500 randomly
selected vehicles will find fewer than 50 with undeclared goods
Ans: 0.5297
3.
The owner of a new apartment building must install 25 water heaters. A certain brand is
guaranteed for 5 years, but the probability that it will last 10 years is 0.25. What is the approximate
probability that 8 or more of the hot water heaters will last at least 10 years?
Ans: 0.282
4.
From many years of observation, a biologist knows that the probability is only 0.65 that any given
Arctic tern will survive the migration from its summer nesting area to its winter feeding grounds. A
random sample of 500 Arctic terns were banded at their summer nesting area. What is the approximate
probability that between 310 and 340 of the banded Arctic terns will survive the migration?
Ans: 0.8530
5.
A professor is giving an exam to a class of 200 students. From past semesters, he knows that 60%
of students taking this course receive at least a 70% on this exam. What is the probability that at least
130 of his students will receive a 70% on the test?
Ans: 0.0853
6.
A particular production process used to manufacture ferrite magnets used to operate reed switches
in electronic meters is known to give 10% defective magnets on average. If 200 magnets are randomly
selected, what is the probability that the number of defective magnets is between 24 and 30?
Ans: 0.1321
7.
A triangular spinner has one red side, one blue side and one green side. The red side is weighted so
that the spinner is four times more likely to land on the red side than on the blue side. The green side is
weighted so that the spinner is three times more likely to land on the green side than on the blue side.
i.
Show that the probability that the spinner lands on the blue side 𝑖𝑠 1/8.
ii.
The spinner is spun 3 times. Find the probability that it lands on a different coloured side each time.
iii.
The spinner is spun 136 times. Use a suitable approximation to find the probability that it lands on
the blue side fewer than 20 times.
Ans: 0.141 ;0.742
8. (i) State three conditions that must be satisfied for a situation to be modelled by a binomial
distribution. On any day, there is a probability of 0.3 that Julie’s train is late.
(ii)
Nine days are chosen at random. Find the probability that Julie’s train is late on more than 7
days or fewer than 2 days.
CAIE Examinations Mathematics Workbook and Practice book: N.Sharma and I. Poudel
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(iii)
90 days are chosen at random. Find the probability that Julie’s train is late on more than 35
days or fewer than 27 days.
Ans: 0.196 ;0.48
9.
When a butternut squash seed is sown the probability that it will germinate is 0.86, independently
of any other seeds. A market gardener sows 250 of these seeds. Use a suitable approximation to find the
probability that more than 210 germinate.
Ans: 0.794
10. On average, 2 apples out of 15 are classified as being underweight. Find the probability that in a
random sample of 200 apples, the number of apples which are underweight is more than 21 and less
than 35.
Ans: 0.807
11. A factory makes water pistols, 8% of which do not work properly.
a.
A random sample of 19 water pistols is taken. Find the probability that at most 2 do not work
properly.
b.
In a random sample of n water pistols, the probability that at least one does not work properly is
greater than 0.9. Find the smallest possible value of n.
c.
A random sample of 1800 water pistols is taken. Use an approximation to find the probability that
there are at least 152 that do not work properly.
d.
Justify the use of your approximation in part(c).
Ans: 0.809;28 ; 0.257
12. In Marumbo, three quarters of the adults own a cell phone.
a.
A random sample of 8 adults from Marumbo is taken. Find the probability that the number of
adults who own a cell phone is between 4 and 6 inclusive.
b.
A random sample of 160 adults from Marumbo is taken. Use an approximation to find the
probability that more than 114 of them own a cell phone.
c.
Justify the use of your approximation in part (b).
Ans:0.606 ; 0.842
13. The times spent by people visiting a certain dentist are independent and normally distributed with
a mean of 8.2 minutes. 79% of people who visit this dentist have visits lasting less than 10 minutes.
a.Find the standard deviation of the times spent by people visiting this dentist.
b. Find the probability that the time spent visiting this dentist by a randomly chosen person deviates
from the mean by more than 1 minute.
c.
Find the probability that, of 6 randomly chosen people, more than 2 have visits lasting longer than
10 minutes.
d.
Find the probability that, of 35 randomly chosen people, fewer than 16 have visits lasting less than
8.2 minutes.
Ans:2.23 ; 0.654;0.112 ;0.250
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