Name : _________________________
Date: _________________________
Secondary 4 Mathematics: Statistics and Probability
1. Measures of Spread
A. Mean, Mode and Median
•
Mean is the average of the statistical results or data.
-
Mean can be calculated with the formula:
∑"
𝑥̅ = #
∑ $"
𝑥̅ = ∑ $
or
where
𝑥 represents each of the data point
𝑛 represents the total number of data points
𝑓 represents the frequency of each data point
•
Mode represents the data with the highest frequency (occurs most frequently) in the set
of data.
•
Median represents the middle number or the data in the middle set of data that is
arranged in increasing order.
-
For data with odd numbers, the median will be the middle number (only 1 value)
-
For data with even numbers, the median will be the sum of the value of % th position
#
#
and the & % + 1)th position divided by 2.
Example: Find the median of the following data set:
1
3
4
7
7
9
Solution:
#
&
Since there are 6 numbers, the median lies between the % th position = % th position
#
&
and the & % + 1)th position = &% + 1) = 4th position.
1
Median =
3
'()
%
4
7
7
9
= 5.5
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2. Variance and Standard Deviation
•
Variance measures the average degree which each point differs from the mean.
•
Standard Deviation measures the spread of the data. It is the square root of the
variance.
-
A smaller value of standard deviation implies a more consistent (less spread) set of
data.
-
The formula to compute standard deviation:
∑ $("+"̅ )!
-
∑$
or -
∑("+"̅ )!
#
where
o 𝑓 = frequency
𝑛 = number of data points
or
o 𝑥 = value of data point
o 𝑥̅ = mean of data set
Example:
20 students each from Class 4A and Class 4B took part in a Mathematics competition.
Every student has to go through a 10-mark quiz and the results are as follows:
Grouped results from Class 4A:
Marks
3
4
5
6
7
8
9
10
Number of students
1
0
4
5
4
2
3
1
Ungrouped results from Class 4B:
2
5
7
8
2
8
9
6
9
5
9
10
8
7
6
7
5
6
10
3
(a) Determine the mean and standard deviation of each class.
(b) Which class has more coherent (consistent) results?
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Solutions:
∑ $"
(a) Mean for Class 4A = ∑ $
=
(.×0)(('×1)((2×')((&×2)(()×')((3×%)((4×.)((01×0)
0(1('(2('(%(.(0
0.'
= %1
= 6.7
∑"
Mean for Class 4B = #
=
%(2()(3(%(3(4(&(4(2(4(01(3()(&()(2(&(01(.
%1
0.%
= %1
= 6.6
∑ $("+"̅ )!
Standard Deviation for Class 4A = =-
∑$
0(.+&.&)! ('(2+&.&)! (2(&+&.&)! ('()+&.&)! (%(3+&.&)! (.(4+&.&)! (0(01+&.&)!
%1
23.'
= - %1
= 1.71 marks (3.s.f)
Standard Deviation for Class 4B = -
∑("+"̅ )!
#
(%+&.&)! ((2+&.&)! (()+&.&)! ((3+&.&)! ((%+&.&)! ((3+&.&)! ((4+&.&)! (
(&+&.&)! ((4+&.&)! ((2+&.&)! ((4+&.&)! ((01+&.&)! ((3+&.&)! (()+&.&)! (
0
(&+&.&)! (()+&.&)! ((2+&.&)! ((&+&.&)! ((01+&.&)! ((.+&.&)!
=
%1
001.3
= - %1
= 2.35 marks (3.s.f)
(b) Class 4A has more coherent results as the standard deviation is lower.
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3. Cumulative Frequency Graph
•
A typical cumulative frequency graph looks like an ‘S-shaped’ curve as shown below:
•
The following information can be derived from the graph:
-
M1 (25%) = lower quartile
-
M2 (50%) = Median
-
M3 (75%) = Upper quartile
-
M4 = Lower limit (does not have to be 0)
-
M5 = Upper limit
-
M3 – M1 = Interquartile range
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4. Box-and-whisker Diagram
•
The following shows a box-and-whisker diagram and the information that can be
derived from the plot:
-
M1 = lower quartile
-
M2 = Median
-
M3 = Upper quartile
-
M4 = Lower limit
-
M5 = Upper limit
-
M3 – M1 = Interquartile range
5. Independent Events and Mutually Exclusive Events
•
Independent events are events that occur in sequence but are independent from one
another.
-
E.g. When you toss a coin twice, what is the probability that on both occasions the
coin will land on ‘Head’? In this case, the outcome of the first event will not
determine the outcome of the second event. Hence, the two events are termed
independent events.
-
The probabilities of the first event and the second event are multiplied together.
P(𝐴 and 𝐵) = P(𝐴) × P(𝐵)
•
Mutually exclusive events are events that cannot occur concurrently. The first event
cannot occur when the second event occurs.
-
The probabilities of the first event and the second event are added together.
P(𝐴 or 𝐵) = P(𝐴) + P(𝐵)
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6. Possibility Diagram
•
Possibility diagrams, which usually comes in the form of a table, helps to list down all
possible outcomes in an experiment that will be useful in computing probabilities.
Example:
The six sides of a fair yellow dice are labelled with numbers: 11, 12, 15, 16, 19 and 22.
The six sides of a fair blue dice are labelled with a different set of numbers: 9, 14, 16, 18,
19 and 21.
An incomplete possibility diagram is shown.
-
If the value on the yellow dice is greater than the value of the blue dice, a letter A is
used to represent this outcome.
-
Otherwise a letter B will be used.
-
Letter X is used if the numbers on both dice are the same.
Yellow dice
11
9
12
15
16
19
22
A
Blue Dice
14
16
18
19
B
X
21
(a) Complete the possibility diagram above.
(b) Find the probability that
(i)
the value on the yellow dice is more than the value of the blue dice,
(ii)
both numbers are the same.
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Solutions:
(a)
Blue Dice
Yellow dice
(b) (i)
11
12
15
16
19
22
9
A
A
A
A
A
A
14
B
B
A
A
A
A
16
B
B
B
X
A
A
18
B
B
B
B
A
A
19
B
B
B
B
X
A
21
B
B
B
B
B
A
#6789: <$ =
P(𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑦𝑒𝑙𝑙𝑜𝑤 𝑑𝑖𝑐𝑒 > 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑏𝑙𝑢𝑒 𝑑𝑖𝑐𝑒) = =>> ?<@@A8A>ABA9@
0&
= .&
'
=4
(ii)
#6789: <$ C
P(𝑠𝑎𝑚𝑒 𝑣𝑎𝑙𝑢𝑒) = =>> ?<@@A8A>ABA9@
%
= .&
0
= 03
7. Tree Diagram
•
A tree diagram is a pictorial method to list down all the different outcomes in an
experiment.
•
The tree diagram is more appropriate than the possibility diagram for solving problems
with multi-step scenarios such as experiment on cards with two or more draws.
Example:
There are 12 cans of soft drink A, 8 cans of soft drink B and 15 cans of soft drink C in an
ice box. Ivan picked two cans from the ice box ay random without replacement.
(a) Construct a tree diagram.
(b) Find the probability that the 2 cans of drinks are
(i)
soft drink A,
(ii)
1 soft drink A and 1 soft drink B,
(iii)
at least 1 soft drink C.
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Solutions:
(a)
Note that the sum of probabilities in any one branch always add up to ‘1’.
-
0%
3
02
E.g. the first main branch = .2 + .2 + .2 = 1
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(b) (i)
0%
00
P(𝐴 and 𝐴) = .2 × .'
&&
= 242
(ii)
P(1𝐴 + 1𝐵) = P(1𝐴 and 1𝐵) or P(1𝐵 and 1𝐴)
0%
3
3
0%
= &.2 × .') + &.2 × .')
4&
= 242
(ii)
P(at least 1 𝐶) = 1 − P(no 𝐶)
= 1 − [P(𝐴𝐴) + P(𝐵𝐵) + P(1𝐴 and 1𝐵) + P(1𝐵 and 1𝐴)]
&&
3
)
4&
= 1 − V242 + &.2 × .') + 242W
30
= 004
OR
P(at least 1 𝐶) = P(𝐴𝐶) + P(𝐵𝐶) + P(𝐶𝐴) + P(𝐶𝐵) + P(𝐶𝐶)
0%
02
3
02
02
0%
02
3
02
0'
= &.2 × .') + &.2 × .') + &.2 × .') + &.2 × .') + &.2 × .')
30
= 004
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Name : _________________________
Date: _________________________
Statistics and Probability (Worksheet 1)
1. You have found the following ages (in years) of 4 bears. Those bears were randomly
selected from the 22 bears at the zoo.
5,
4,
6,
39
Based on your sample,
(a) what is the average age of the bears?
(b) what is the standard deviation?
Answer: (a) ________________
(b) ________________
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2. You have found the following ages (in years) of all 6 lions at the Zoo:
13,
2,
1,
5,
2,
7
Based on your data shown above,
(a) what is the mode?
(b) what is the median age of the lions?
(c) what is the mean age of the lions?
(d) what is the standard deviation?
Answer: (a) mode = ________________
(b) median = ________________
(c) mean = ________________
(d) standard deviation = ________________
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3. You have found the following ages (in years) of 5 zebras at the Zoo:
8,
11,
17,
7,
19
Based on the data shown above,
(a) what is the median age of the zebras?
(b) what is the mean age of the zebras?
(c) what is the standard deviation?
Answer: (a) median = ________________
(b) mean = ________________
(c) standard deviation = ________________
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4. 43 kids aged from 5 to 13 attended a medical check-up in a children’s clinic. The
breakdown of each age group is tabulated as shown:
Age (years)
5
6
7
8
9
10
11
12
13
Number of kids
3
5
7
4
4
2
6
7
5
Calculate
(a) the mean.
(b) the standard deviation.
Answer: (a) mean = ________________
(b) standard deviation = ________________
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5. The ordered stem-and-leaf diagram shows the height (in cm) of 30 beauty contestants.
Find
(a) the probability of selecting two beauty contestants who are above 189 cm tall.
Answer: (a) P(two beauty contestants > 189 cm) = ________________
(b) the mode,
Answer: (b) mode = ________________
(c) the mean,
Answer: (c) mean = ________________
(d) the median,
Answer: (d) median = ________________
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(e) the standard deviation,
Answer: (e) standard deviation = ________________
6. 30 students took a memory test and their scores are recorded as follows:
Marks, x
4
5
6
7
8
9
10
No. of students
3
7
8
4
5
2
1
Calculate
(a) the mode,
Answer: (a) mode = ________________
(b) the mean,
Answer: (b) mean = ________________
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(c) the median,
Answer: (c) median = ________________
(d) the standard deviation,
Answer: (d) standard deviation = ________________
(e) the probability of selecting one student scoring below 5 marks and the other scoring 9
marks and above.
Answer: (e) P(one student < 5 marks and one score ≥ 9) = ________________
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7. The time taken for 250 students to complete an artwork is shown in the table.
Time
40 < x
45 < x
50 < x
55 < x
60 < x
65 < x
70 < x
(x in mins)
≤ 45
≤ 50
≤ 55
≤ 60
≤ 65
≤ 70
≤ 75
14
30
38
68
45
32
23
No.
of
students
Calculate
(a) the mean time,
(b) the standard deviation.
Answer: (a) mean = ________________
(b) standard deviation = ________________
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8. 1000 students took a Mathematics examination which consisted of Paper 1 and Paper 2.
Each paper has a total of 50 marks. The results of these students are tabulated as shown.
Marks(x)
Number of candidates
Paper 1
Paper 2
0 < x ≤ 10
90
100
10 < x ≤ 20
310
270
20 < x ≤ 30
370
295
30 < x ≤ 40
200
295
40 < x ≤ 50
30
40
(a) Determine the mean and standard deviation of Paper 1 and 2.
Paper 1
Answer: (a) mean of paper 1 = ________________
standard deviation of paper 1 = ________________
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Paper 2
Answer: (a) mean of paper 2 = ________________
standard deviation of paper 2 = ________________
(b) Which paper is more difficult? Explain your answer.
_____________________________________________________________________
_____________________________________________________________________
(c) Which paper has more coherent results? Explain your answer.
_____________________________________________________________________
_____________________________________________________________________
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9. During the monsoon season (Nov to Jan), the duration of rain was recorded for Year 2019
and Year 2020.
Time (x in hours)
Number of days (total 90 days)
2019
2020
0<x≤2
8
6
2<x≤4
12
8
4<x≤6
24
28
6<x≤8
27
35
8 < x ≤ 10
12
10
10 < x ≤ 12
7
3
(a) Determine the mean and standard deviation of Year 2019 and Year 2020.
Year 2019
Answer: (a) mean of Year 2019 = ________________
standard deviation of Year 2019 = ________________
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Year 2020
Answer: (a) mean of Year 2020 = ________________
standard deviation of Year 2020 = ________________
(b) Which year has a longer duration of rain? Explain your answer.
_____________________________________________________________________
_____________________________________________________________________
(c) Which year is more consistent in terms of the duration of the rain? Explain your answer.
_____________________________________________________________________
_____________________________________________________________________
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10. The stem-and-leaf diagram shows the masses, in grams, of some oranges.
(a) Find the median of these masses.
Answer: (a) median = ________________ g
(a) Given that the interquartile range is 10, find the value of m.
Answer: (b) m = ________________
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Name : _________________________
Date: _________________________
Statistics and Probability (Worksheet 2)
1. The cumulative frequency graph shows the amount 250 shoppers spent in a doughnut shop.
From the cumulative frequency graph, estimate
(a) the median
: ______________________
(b) the interquartile range
: ______________________
(c) the number of shoppers who spent $6 or less
: ______________________
(d) the number of shoppers who spent $10 or more
: ______________________
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2. The cumulative frequency graph shows the mass of 1000 Secondary One students in sports
school.
From the cumulative frequency graph, estimate
(a) the median
: ______________________
(b) the interquartile range
: ______________________
(c) the number of students weighing 50 kg or less
: ______________________
(d) the fraction of students weighing 56 kg or more
: ______________________
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3. Two thousand candidates sat for a Mathematics examination. The cumulative frequency
graph below shows the results. The grading system is as follows:
Grades
F
E
D
C6
C5
B4
B3
A2
A1
Marks
≤ 39
40 to 44
45 to 49
50 to 55
56 to 59
60 to 64
65 to 69
70 to 74
> 74
From the cumulative frequency graph, estimate
(a) the median
: ______________________
(b) the 30th percentile
: ______________________
(c) the interquartile range
: ______________________
(d) the percentage of candidates who scored A1
: ______________________
(e) If the passing mark is 45, find the number of students who passed the examination
: ______________________
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4. The diagram is a box-and-whisker plot showing the mass of 500 seashells.
From the plot, estimate:
(a) the lower limit
: ______________________
(b) the upper limit
: ______________________
(c) the lower quartile
: ______________________
(d) the upper quartile
: ______________________
(e) the median
: ______________________
(f) the interquartile range
: ______________________
5. The diagram is a box-and-whisker plot showing the height of 250 pupils in Primary One.
From the plot, estimate:
(a) the lower limit
: ______________________
(b) the upper limit
: ______________________
(c) the lower quartile
: ______________________
(d) the upper quartile
: ______________________
(e) the median
: ______________________
(f) the interquartile range
: ______________________
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6. 1000 runners took part in a marathon and the time they took to complete the run were shown
in the cumulative frequency graph.
From the cumulative frequency graph, estimate
(a) the median
: ______________________
(b) the interquartile range
: ______________________
(c) Given that 45% of the runners completed the marathon within t hrs, find the value of t.
t = ______________________
(d) Using the above results, construct a box-and-whisker plot below:
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7. Two thousand students reported the number of hours they spent reading books on
weekends. The data is plotted on a cumulative frequency graph as shown below:
From the cumulative frequency graph, estimate
(a) the median
: ______________________
(b) the interquartile range
: ______________________
(c) the percentage of students who read more than 5 hrs : ______________________
(d) Using the above results, construct a box-and-whisker plot below:
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8. The cumulative frequency below illustrates the weights of 100 students in Senoko High
School.
(a) Use the graph to find
(i)
the median
: ______________________
(ii)
the interquartile range
: ______________________
(b) The weight of 100 students in Changi High School have a higher median but a smaller
interquartile range.
Describe how the cumulative curve for Changi High School will differ from the curve
for Senoko High School.
_____________________________________________________________________
_____________________________________________________________________
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9. The cumulative frequency graph shows the distribution of marks of 150 students in a
Mathematics examination.
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(a) Use the graph to estimate
(i)
the number of students who score more than 36 marks: ___________________
(ii)
the interquartile range
(b) (i)
: ___________________
Complete the grouped frequency table of the marks of the 150 students:
x (marks)
0 < x ≤ 20
20 < x ≤ 40
40 < x ≤ 60
60 < x ≤ 80
80 < x ≤ 100
Number of
students
(ii)
Using your grouped frequency table, calculate an estimate of
(a) the mean mark
Answer: (b)(ii)(a) mean = ___________________
(b) the standard deviation
Answer: (b)(ii)(b) standard deviation = ___________________
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(c) The same group of students took a Science Examination.
The box-and-whisker plot shows the distribution of their marks.
(i)
Which exam was more difficult? Justify your answer.
_____________________________________________________________________
_____________________________________________________________________
(ii)
Compare and comment on the consistency of the performance of the students in
the two examinations.
_____________________________________________________________________
_____________________________________________________________________
10. (a) The following box-and-whisker diagrams shows the distribution of the mass of 300
students from each school, SK Secondary School and HG Secondary School respectively.
(i)
What is the median mass for each school?
Answer: (a)(i) SK Secondary School’s median = _______________
HG Secondary School’s median = _______________
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(ii)
Compare the mass of the students from SK Secondary School and HG
Secondary School in two ways.
_______________________________________________________________
_______________________________________________________________
(iii)
Mary commented that there are more students in HG Secondary School than SK
Secondary School who weigh more than or equal to 70 kg. Do you agree with
Mary? Support with a reason.
_______________________________________________________________
_______________________________________________________________
(b) The cumulative frequency curve shows the height distribution of 80 plants.
(i)
Use your graph to find the value of m, if 32.5% of the plants have heights more
than m cm.
Answer: (b)(i) m = _______________
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(ii)
The height distribution of the 80 plants was also recorded in the following
frequency table.
Height (h cm)
Number of plants
60 < h ≤ 70
2
70 < h ≤ 80
a
80 < h ≤ 90
9
90 < h ≤ 100
27
100 < h ≤ 110
23
110 < h ≤ 120
b
120 < h ≤ 130
4
(a) Find the value of a and of b.
Answer: (b)(ii)(a) a = _______________
b = _______________
(b) Hence, find the mean and standard deviation of the height of the 80 plants.
Answer: (b)(ii)(a) mean = _______________
standard deviation = _______________
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Name : _________________________
Date: _________________________
Statistics and Probability (Worksheet 3)
1. Bag A contains 4 numbered cards labelled 1, 2, 3 and 4. Bag B contains 5 Bingo balls
numbered 5, 6, 7, 8 and 9. Beatrice picked one item from each bag. She recorded the
numbers on the card and ball and added the values of each item.
(a) Construct a possibility diagram to show all possible sums.
(b) Using the possibility diagram, find the probability that the sum of the two items is
(i)
odd,
Answer: (b)(i) P(odd) = _______________
(ii)
even,
Answer: (b)(ii) P(even) = _______________
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(iii)
less than 8,
Answer: (b)(iii) P(less than 8) = _______________
(iv)
more than or equal to 10,
Answer: (b)(iv) P( ≥10 ) = _______________
(v)
divisible by 2.
Answer: (b)(v) P(divisible by 2) = _______________
2. There are seven numbered cards labelled 10, 11, 13, 15, 17, 18 and 21 in Box P. There are
three numbered cards labelled 4, 5 and 7 in another Box Q. One card is drawn randomly
from each of the box and the values on the cards are added together.
(a) Construct a possibility diagram to show all possible sums.
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(b) Using the possibility diagram, find the probability that the sum of the two items is
(i)
odd,
Answer: (b)(i) P(odd) = _______________
(ii)
even,
Answer: (b)(ii) P(even) = _______________
(iii)
less than or equal to 20,
Answer: (b)(iii) P(less than or equal to 20) = _______________
(iv)
more than 25,
Answer: (b)(iv) P( > 20 ) = _______________
(v)
divisible by 3.
Answer: (b)(v) P(divisible by 3) = _______________
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3. Joel threw two fair dice and recorded the numbers on the dice. He then multiplied the two
numbers.
(a) Construct a possibility diagram to show all possible products.
(b) Using the possibility diagram, find the probability that the sum of the two items is
(i)
odd,
Answer: (b)(i) P(odd) = _______________
(ii)
even,
Answer: (b)(ii) P(even) = _______________
(iii)
a perfect square,
Answer: (b)(iii) P(perfect square) = _______________
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(iv)
divisible by 4,
Answer: (b)(iv) P(divisible by 4) = _______________
(v)
divisible by either 2 or 5.
Answer: (b)(v) P(divisible by either 2 or 5) = _______________
4. A bag contains 4 red marbles, 5 green marbles and 6 blue marbles. Two marbles are drawn
successively without replacement.
(a) Construct a tree diagram.
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(b) Using the tree diagram you have drawn in (a), determine the probability that
(i)
the two marbles are red,
Answer: (b)(i) P(both red) = _______________
(ii)
the two marbles are blue,
Answer: (b)(ii) P(both blue) = _______________
(iii)
the two marbles are green,
Answer: (b)(iii) P(both green) = _______________
(iv)
at least 1 marble is red,
Answer: (b)(iv) P(at least 1 red) = _______________
(v)
none of the marbles is green,
Answer: (b)(v) P(no green) = _______________
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(vi)
the two marbles are of different colours.
Answer: (b)(vi) P(different colours) = _______________
5. Three types of coloured pens (10 reds, 4 black pens and 11 blue pens) were kept in a bag.
Helen picked two pens at random without replacement.
(a) Construct a tree diagram.
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(b) Using the tree diagram you have drawn in (a), determine the probability that
(i)
the two pens are of the same colour,
Answer: (b)(i) P(same colour) = _______________
(ii)
the two pens are of different colour,
Answer: (b)(ii) P(different colour) = _______________
(iii)
there is at least one black pen,
Answer: (b)(iii) P(at least one black) = _______________
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(iv)
no red,
Answer: (b)(iv) P(no red) = _______________
(v)
a black pen and a blue pen are picked,
Answer: (b)(v) P(1 black and 1 blue) = _______________
(vi)
a red pen is picked and followed by a blue pen.
Answer: (b)(vi) P(red then blue) = _______________
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6. In a boc of chocolate cookies jar, there were 10 dark chocolates cookies, 12 milk cookies
and 8 white chocolate cookies. Ivan picks two cookies at random.
(a) Construct a tree diagram.
(b) Using the tree diagram you have drawn in (a), determine the probability that
(i)
both cookies are of different flavour,
Answer: (b)(i) P(different flavour) = _______________
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(ii)
there is no milk chocolate cookies,
Answer: (b)(ii) P(no milk chocolate cookies) = _______________
(iii)
there is one milk chocolate cookie and one white cholcate cookie,
Answer: (b)(ii) P(1 milk and 1 white) = _______________
7. 50 students were asked how much pocket monet they received and their answers were
tabulated as follows:
Weekly money pocket ($)
15
16
17
18
19
20
Number of students
3
5
10
21
8
3
Two students are chosen at random one after another. Find the probability that the sum of
their pocket money is
(a) $39 or more,
Answer: (a) P(sum ≥ $39) = _______________
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(b) $31 or less,
Answer: (b) P(sum ≤ $31) = _______________
(c) exactly $36.
Answer: (c) P(sum = $36) = _______________
8. 25 students took a memory test and their marks are recorded as follows:
Marks
4
5
6
7
8
9
10
Number of students
3
5
7
3
4
2
1
Two students are selected at random one after another. Find the probability that the sum of
their marks is
(a) exactly 16 marks,
Answer: (a) P(sum = 16) = _______________
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(b) 19 marks or above,
Answer: (b) P(sum ≥ 19) = _______________
(c) less than 9 marks,
Answer: (c) P(sum < 9) = _______________
9. The ordered stem-and-leaf diagram shows the height (in cm) of 30 beauty contestents.
Two contestents are chosen at random without replacement. Find the proabability that
(a) the heights of both contestents are more than 188 cm,
Answer: (a) P(both > 188 cm) = _______________
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(b) the heights of both contestents are less than or equal to 175 cm,
Answer: (b) P(both ≤ 175 cm) = _______________
(c) the height of one contestent is above 180 cm and the other is below 170 cm,
Answer: (c) P(one is > 180 cm and one is < 170 cm) = _______________
(d) the heights of both contestents are between 175 cm and 180 cm inclusive.
Answer: (d) P(both between 175 cm and 180 cm inclusive) = _______________
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10. A box contains a number of blue, green and red balls. There are 27 blue and green balls
altogether. A ball is selected at random from the box. The probability of drawing a
green ball is twice the probability of drawing a blue ball. The probability of drawing a red
'
ball is ) .
%
(a) Show that the probability of drawing a green ball is ) .
(b) Find the number of red balls in the bag.
Answer: (b) Number of red balls = _______________
(c) Two balls are selected at random from the box, with replacement.
Find the probability that both balls are blue.
Answer: (c) P(both blue) = _______________
(d) Two balls are selected at random from the same box, without replacement.
Find the probability that both balls are different colours.
Answer: (d) P(both different colours) = _______________
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11. A drawer contains 2 blue socks and 6 white socks. Two socks are taken from the drawer at
random without replacement. If the two socks are different colours, then a third sock is
taken from the drawer. Otherwise, no third sock is taken.
(a) Draw a tree diagram to show the probabilities of the possible outcomes.
(b) Find, as a fraction in its simplest form, the probability that
(i) the first two socks taken are white,
Answer: (b)(i) P(first two are white) = _______________
(ii) a third sock is taken and it is the same colour as the first sock.
Answer: (b)(ii) P(3rd is same as 1st) = _______________
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12. A box contains five slips of paper. Each slip has one of the numbers 4, 6, 7, 8 or 9 written
on it. There are two players for the game. The first player reaches into the box and draws
two slips and adds the two numbers. If the sum is even, the player wins. If the sum is odd,
the player loses. What is the probability that the first player wins.
Answer: P(first player wins) = _______________
13. A game is such that a fair die is rolled respectively until a ‘6’ is obtained.
(a) Find the probability that the game ends by the fourth roll.
Answer: (a) P(game ends by 4th roll) = _______________
(b) Suppose now that the game is such that the same die is rolled repeatedly until two ‘6’s
are obtained. Find the probability that
(i) the game ends on the third roll,
Answer: (b)(i) _______________
(ii) the game ends on the third roll and the sum of the scores is odd.
Answer: (b)(ii) _______________
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