American Community School at Beirut
IB Math SL
Unit Title: Statistics and Probability
April-May 2013 Assignments
Objectives
Statistics
H/W 1. # 1 to 10
1- Define Statistics, Data, Population,
Sample. Recognize continuous and
discrete data.
Define mid-interval value, upper and
lower interval boundaries.
Represent grouped data in frequency
tables and histograms.
H/W 2. # 11 to 22
H/W 3. # 23, 25, 27, 28, 30
H/W 4. # 31, 33, 35, 36, 40
Optional Practice p. 324 # 1 to 26
2- Find mean, median and mode of
grouped and ungrouped data.
3- Find variance and standard deviation of
a given set of data. Study the effect of
constant changes to the original data.
4- Find quartiles, inter-quartile range and
percentiles.
Use Box Plots to represent data.
Use cumulative frequency graphs (or
technology to find median, quartiles
and percentiles. (Answers produced by
tech. may be different than those
obtained from cumulative freq. graphs.
5- Study linear correlation of bivariate
data. Calculate Pearson’s correlation
coefficient r.
Create scatter diagrams and find
equations of best fit (line of regression).
Use best fit line to interpolate and
extrapolate.
H/W 5. P. 323 # 3, 4, 5.
Probability
6- Use the product principle to count the
number of possible outcomes of an
experiment. Use permutation to
enumerate possible outcomes.
7- Define number of trials, outcome,
equally likely outcomes, sample space
and event.
Find the probability of an event.
H/W 6. P. 347 # 1, 3, 4, 6, 7, 8, 10
H/W 7. P. 358 # 1, 2, 3, 4, 5, 6
H/W 8. P. 359 # 7, 9.
Practice p. 360 # 1, 3, 4, 5
H/W 9. P. 362 # 7, 9, 10, 12, 13, 14, 16
H/W 10. P. 364 # 18, 19, 22, 25, 27, 28
8- Find the complementary of an event,
the union and the intersection of two
events.
Use Venn Diagram to illustrate
probability rules.
H/W 11. P. 532 # 3, 5, 6, 7, 8, 10, 11, 12, 13.
H/W 12. Additional Worksheet (Random variables) # 1 to 5
9- Find the conditional probability of
event A given event B.
Define independent events.
H/W 13. P. 539 # 2, 6, 7, 9, 11, 12
H/W 14. P. 553 # 1, 2, 3, 5, 6, 8, 9
10- To define discrete random variable and
discrete probability distribution. To
display probability distribution in
tabular form, graphical representation
and function.
11- To find the expected value (mean) for
discrete data.
12- To define the binomial distribution. To
find the expectation value (mean) for
the normal distribution.
13- To find the mean and variance of the
Binomial distribution.
14- To study and use normal distribution as
a model to represent a given set of data.
To use GDC in order to find the
probability of a given standard or nonstandard variable z.
15- To convert a given normal probability
to standard normal and use tables /GDC
to calculate probabilities of a normal
distribution.
H/W 15. P. 554 # 10, 11, 12, 13, 14, 15
H/W 16. P. 556 Practice # 3, 5, 9, 11, 13, 15, 18.
1.
The following frequency distribution of marks has mean 4.5.
(a)
Mark
1
2
3
4
5
6
7
Frequency
2
4
6
9
x
9
4
Find the value of x.
(4)
(b)
Write down the standard deviation.
(2)
(Total 6 marks)
2.
The following table gives the examination grades for 120 students.
(a)
Grade
Number of students
Cumulative frequency
1
9
9
2
25
34
3
35
p
4
q
109
5
11
120
Find the value of
(i)
p;
(ii)
q.
(4)
(b)
Find the mean grade.
(2)
(c)
Write down the standard deviation.
(1)
(Total 7 marks)
3.
The following diagram is a box and whisker plot for a set of data.
The interquartile range is 20 and the range is 40.
(a)
Write down the median value.
(1)
(b)
Find the value of
(i)
a;
(ii)
b.
(4)
(Total 5 marks)
4.
A standard die is rolled 36 times. The results are shown in the following table.
Score
1
2
3
4
5
6
Frequency
3
5
4
6
10
8
(a)
Write down the standard deviation.
(2)
(b)
Write down the median score.
(1)
(c)
Find the interquartile range.
(3)
(Total 6 marks)
5.
In a school with 125 girls, each student is tested to see how many sit-up exercises (sit-ups) she can do in one
minute. The results are given in the table below.
(a)
(i)
Number of sit-ups
Number of students
Cumulative
number of students
15
11
11
16
21
32
17
33
p
18
q
99
19
18
117
20
8
125
Write down the value of p.
(ii)
Find the value of q.
(3)
(b)
Find the median number of sit-ups.
(c)
Find the mean number of sit-ups.
(2)
(2)
(Total 7 marks)
6.
A set of data is
18, 18, 19, 19, 20, 22, 22, 23, 27, 28, 28, 31, 34, 34, 36.
The box and whisker plot for this data is shown below.
(a)
Write down the values of A, B, C, D and E.
A = ......
(b)
B = ......
C= ......
D = ......
E = ......
Find the interquartile range.
(Total 6 marks)
7.
Consider the four numbers a, b, c, d with a  b  c ≤ d, where a, b, c, d  .
The mean of the four numbers is 4.
The mode is 3, the median is 3, and the range is 6. Find the value of a, of b, of c and of d.
(Total 6 marks)
8.
The population below is listed in ascending order.
5, 6, 7, 7, 9, 9, r, 10, s, 13, 13, t
The median of the population is 9.5. The upper quartile Q3 is 13.
(a)
Write down the value of
(i)
r;
(ii)
(b)
The mean of the population is 10. Find the value of t.
s.
(Total 6 marks)
9.
The box and whisker diagram shown below represents the marks received by 32 students.
(a)
Write down the value of the median mark.
(b)
Write down the value of the upper quartile.
(c)
Estimate the number of students who received a mark greater than 6.
(Total 6 marks)
10.
The 45 students in a class each recorded the number of whole minutes, x, spent doing experiments on
Monday. The results are x = 2230.
(a)
Find the mean number of minutes the students spent doing experiments on Monday.
Two new students joined the class and reported that they spent 37 minutes and 30 minutes respectively.
(b)
11.
Calculate the new mean including these two students.
(Total 6 marks)
The following table shows the mathematics marks scored by students.
Mark
1
2
3
4
5
6
7
Frequency
0
4
6
k
8
6
6
(b)
Write down the mode.
The mean mark is 4.6.
(a)
Find the value of k.
(Total 6 marks)
12.
The table below shows the marks gained in a test by a group of students.
Mark
1
2
3
4
5
Number of students
5
10
p
6
2
The median is 3 and the mode is 2. Find the two possible values of p.
13.
Let a, b, c and d be integers such that a < b, b < c and c = d.
The mode of these four numbers is 11. The range of these four numbers is 8.
(Total 6 marks)
The mean of these four numbers is 8.
Calculate the value of each of the integers a, b, c, d.
(Total 6 marks)
14.
The number of hours of sleep of 21 students are shown in the frequency table below.
Hours of sleep
Number of students
4
2
5
5
6
4
7
3
8
4
10
2
12
1
Find
(a)
the median;
(b)
the lower quartile;
(c)
the interquartile range.
(Total 6 marks)
15.
From January to September, the mean number of car accidents per month was 630. From October to
December, the mean was 810 accidents per month.
What was the mean number of car accidents per month for the whole year?
(Total 6 marks)
16.
Three positive integers a, b, and c, where a < b < c, are such that their median is 11, their mean is 9 and their
range is 10. Find the value of a.
17.
Given the following frequency distribution, find
(a)
the median;
Number (x)
(b)
Frequency (f )
1
2
3
4
5
5
9
16
18
20
the mean.
6
7
(Total 4 marks)
18.
The table shows the scores of competitors in a competition.
Score
10
20
30
40
50
Number of competitors
with this score
1
2
5
k
3
The mean score is 34. Find the value of k.
19.
(Total 4 marks)
At a conference of 100 mathematicians there are 72 men and 28 women. The men have a mean height of 1.79
m and the women have a mean height of 1.62 m. Find the mean height of the 100 mathematicians.
(Total 4 marks)
25
20.
The mean of the population x1, x2, ........ , x25 is m. Given that
x
i
= 300 and
i 1
25
 (x
i
– m) 2 = 625, find
i 1
(a)
the value of m;
(b)
the standard deviation of the population.
(Total 4 marks)
21.
A scientist has 100 female fish and 100 male fish. She measures their lengths to the nearest cm. These are
shown in the following box and whisker diagrams.
(a)
Find the range of the lengths of all 200 fish.
(3)
(b)
Four cumulative frequency graphs are shown below.
Which graph is the best representation of the lengths of the female fish?
(2)
(Total 5 marks)
22.
The four populations A, B, C and D are the same size and have the same range.
Frequency histograms for the four populations are given below.
(a)
Each of the three box and whisker plots below corresponds to one of the four populations. Write the
letter of the correct population under each plot.
......
(b)
......
......
Each of the three cumulative frequency diagrams below corresponds to one of the four populations.
Write the letter of the correct population under each diagram.
(Total 6 marks)
A student measured the diameters of 80 snail shells. His results are shown in the following cumulative
frequency graph. The lower quartile (LQ) is 14 mm and is marked clearly on the graph.
90
80
Cumulative frequency
23.
70
60
50
40
30
20
10
0
0
5
10
15
LQ = 14
20
25
Diameter (mm)
30
35
40
45
(a)
On the graph, mark clearly in the same way and write down the value of
(i)
(b)
24.
the median;
(ii)
the upper quartile.
Write down the interquartile range.
(Total 6 marks)
The following is a cumulative frequency diagram for the time t, in minutes, taken by 80 students to complete
a task.
(a)
Write down the median.
(1)
(b)
Find the interquartile range.
(3)
(c)
Complete the frequency table below.
Time
(minutes)
Number of
students
0 ≤ t < 10
5
10 ≤ t < 20
20 ≤ t < 30
20
30 ≤ t < 40
24
40 ≤ t < 50
50 ≤ t < 60
6
(2)
(Total 6 marks)
25.
There are 50 boxes in a factory. Their weights, w kg, are divided into 5 classes, as shown in the following
table.
Class
Weight (kg)
Number of boxes
A
9.5  w 18.5
7
B
18.5  w  27.5
12
C
27.5  w  36.5
13
D
36.5  w  45.5
10
E
45.5  w  54.5
8
(a)
Show that the estimated mean weight of the boxes is 32 kg.
(3)
(b)
There are x boxes in the factory marked “Fragile”. They are all in class E. The estimated mean weight of all
the other boxes in the factory is 30 kg. Calculate the value of x.
(4)
(c)
An additional y boxes, all with a weight in class D, are delivered to the factory. The total estimated mean
weight of all of the boxes in the factory is less than 33 kg. Find the largest possible value of y.
(5)
(Total 12 marks)
26.
The histogram below represents the ages of 270 people in a village.
(a)
Use the histogram to complete the table below.
Age range
Frequency
Mid-interval
value
0  age  20
40
10
20 ≤ age  40
40 ≤ age  60
60 ≤ age  80
80 ≤ age ≤100
(2)
(b)
Hence, calculate an estimate of the mean age.
(4)
(Total 6 marks)
27.
The following diagram represents the lengths, in cm, of 80 plants grown in a laboratory.
20
15
frequency
10
5
0
(a)
0
10
20
30
40
50
60
length (cm)
70
80
90
100
How many plants have lengths in cm between
(i)
50 and 60?
(ii)
70 and 90?
(2)
(b)
Calculate estimates for the mean and the standard deviation of the lengths of the plants.
(4)
(c)
Explain what feature of the diagram suggests that the median is different from the mean.
(1)
(d)
The following is an extract from the cumulative frequency table.
length in cm
cumulative
less than
frequency
.
.
50
22
60
32
70
48
80
62
.
.
Use the information in the table to estimate the median. Give your answer to two significant figures.
(3)
(Total 10 marks)
28.
In a suburb of a large city, 100 houses were sold in a three-month period. The following cumulative
frequency table shows the distribution of selling prices (in thousands of dollars).
Selling price P
($1000)
P  100
P  200
P  300
P  400
P  500
Total number
of houses
12
58
87
94
100
(a)
Represent this information on a cumulative frequency curve, using a scale of 1 cm to represent $50000
on the horizontal axis and 1 cm to represent 5 houses on the vertical axis.
(4)
(b)
Use your curve to find the interquartile range.
(3)
The information above is represented in the following frequency distribution.
Selling price P
($1000)
0 < P  100
Number of
houses
12
100 < P  200 200 < P  300 300 < P  400 400 < P  500
46
29
a
b
(c)
Find the value of a and of b.
(2)
(d)
Use mid-interval values to calculate an estimate for the mean selling price.
(2)
(e)
Houses which sell for more than $350000 are described as De Luxe.
(i)
Use your graph to estimate the number of De Luxe houses sold.
Give your answer to the nearest integer.
(ii)
Two De Luxe houses are selected at random. Find the probability
that both have a selling price of more than $400000.
(4)
(Total 15 marks)
29.
A survey is carried out to find the waiting times for 100 customers at a supermarket.
(a)
waiting time
(seconds)
number of
customers
0–30
5
30– 60
15
60– 90
33
90 –120
21
120–150
11
150–180
7
180–210
5
210–240
3
Calculate an estimate for the mean of the waiting times, by using an appropriate approximation to
represent each interval.
(2)
(b)
Construct a cumulative frequency table for these data.
(1)
(c)
Use the cumulative frequency table to draw, on graph paper, a cumulative frequency graph, using a scale of 1
cm per 20 seconds waiting time for the horizontal axis and 1 cm per 10 customers for the vertical axis.
(4)
(d)
Use the cumulative frequency graph to find estimates for the median and the lower and upper quartiles.
(3)
(Total 10 marks)
30.
The speeds in km h–1 of cars passing a point on a highway are recorded in the following table.
(a)
Speed v
Number of cars
v  60
0
60 < v  70
7
70 < v  80
25
80 < v  90
63
90 < v  100
70
100 < v  110
71
110 < v  120
39
120 < v  130
20
130 < v  140
5
v > 140
0
Calculate an estimate of the mean speed of the cars.
(2)
(b)
The following table gives some of the cumulative frequencies for the information above.
Speed v
Cumulative frequency
v  60
0
v  70
7
v  80
32
v  90
95
v  100
a
v  110
236
v  120
b
v  130
295
v  140
300
(i)
Write down the values of a and b.
(ii)
On graph paper, construct a cumulative frequency curve to represent this information. Use a
scale of 1 cm for 10 km h–1 on the horizontal axis and a scale of 1 cm for 20 cars on the vertical
axis.
(5)
(c)
Use your graph to determine
(i)
the percentage of cars travelling at a speed in excess of 105 km h–1;
(ii)
the speed which is exceeded by 15% of the cars.
(4)
(Total 11 marks)
31.
A supermarket records the amount of money d spent by customers in their store during a busy period. The
results are as follows:
Money in $ (d)
Number of customers (n)
0–20
20–40
40–60
60–80
24
16
22
40
80–100 100–120 120–140
18
10
4
(a)
Find an estimate for the mean amount of money spent by the customers, giving your answer to the nearest
dollar ($).
(2)
(b)
Copy and complete the following cumulative frequency table and use it to draw a cumulative
frequency graph. Use a scale of 2 cm to represent $20 on the horizontal axis, and 2 cm to represent 20
customers on the vertical axis.
(5)
Money in $ (d)
<20
<40
Number of customers (n)
24
40
<60
<80
< 100
< 120
< 140
2
(c)
The time t (minutes), spent by customers in the store may be represented by the equation t = 2d 3 + 3.
(i)
Use this equation and your answer to part (a) to estimate the mean time in minutes spent by customers
in the store.
(3)
(ii)
Use the equation and the cumulative frequency graph to estimate the number of customers who spent
more than 37 minutes in the store.
(5)
(Total 15 marks)
32.
A fisherman catches 200 fish to sell. He measures the lengths, l cm of these fish, and the results are shown in
the frequency table below.
Length l cm
0 ≤ l < 10
10 ≤ l < 20
20 ≤ l < 30
30 ≤ l < 40
40 ≤ l < 60
60 ≤ l < 75
75 ≤ l < 100
Frequency
30
40
50
30
33
11
6
(a)
Calculate an estimate for the standard deviation of the lengths of the fish.
(b)
A cumulative frequency diagram is given below for the lengths of the fish.
Use the graph to answer the following.
(3)
(i)
Estimate the interquartile range.
(ii)
Given that 40 % of the fish have a length more than k cm, find the value of k.
(6)
In order to sell the fish, the fisherman classifies them as small, medium or large.
Small fish have a length less than 20 cm.
Medium fish have a length greater than or equal to 20 cm but less than 60 cm.
Large fish have a length greater than or equal to 60 cm.
(c)
Write down the probability that a fish is small.
(2)
The cost of a small fish is $4, a medium fish $10, and a large fish $12.
(d)
Copy and complete the following table, which gives a probability distribution for the cost $X.
Cost $X
P(X = x)
4
10
12
0.565
(2)
(e)
Find E(X).
(2)
(Total 15 marks)
33.
A test marked out of 100 is written by 800 students. The cumulative frequency graph for the marks is given
below.
(a)
Write down the number of students who scored 40 marks or less on the test.
(2)
(b)
The middle 50 % of test results lie between marks a and b, where a < b. Find a and b.
(4)
(Total 6 marks)
34.
The table below represents the weights, W, in grams, of 80 packets of roasted
peanuts.
Weight (W)
80 < W  85
85 < W  90
90 < W  95
Number of
packets
5
10
15
(a)
95 < W  100 100 < W  105 105 < W  110 110 < W  115
26
13
7
4
Use the midpoint of each interval to find an estimate for the standard deviation of the weights.
(3)
(b)
Copy and complete the following cumulative frequency table for the above data.
Weight (W)
W  85
W  90
Number of
packets
5
15
W  95
W  100
W  105
W 110
W  115
80
(1)
(c)
A cumulative frequency graph of the distribution is shown below, with a scale 2 cm for 10 packets on
the vertical axis and 2 cm for 5 grams on the horizontal axis.
80
70
60
50
Number
of
packets
40
30
20
10
80
85
90
95
100
Weight (grams)
105
110
115
Use the graph to estimate
(i)
the median;
(ii)
the upper quartile (that is, the third quartile).
Give your answers to the nearest gram.
(4)
(d)
Let W1, W2, ..., W80 be the individual weights of the packets, and let W be their mean. What is the
value of the sum
(W1 – W )  (W2 – W )  (W3 – W )  ...  (W79 – W )  (W80 – W ) ?
(2)
(e)
One of the 80 packets is selected at random. Given that its weight satisfies
85 < W  110, find the probability that its weight is greater than 100 grams.
(4)
(Total 14 marks)
The cumulative frequency graph below shows the heights of 120 girls in a school.
130
120
110
100
Cumulative frequency
35.
90
80
70
60
50
40
30
20
10
0
150 155 160 165 170 175 180 185
Height in centimetres
(a)
Using the graph
(i)
(b)
write down the median;
(ii)
find the interquartile range.
Given that 60 of the girls are taller than a cm, find the value of a.
(Total 6 marks)
The following is the cumulative frequency curve for the time, t minutes, spent by 150 people in a store on a
particular day.
150
140
130
120
110
100
cumulative frequency
36.
90
80
70
60
50
40
30
20
10
0
1
2
3
4
5
7
6
8
9
10
11
12
time (t)
(a)
(i)
How many people spent less than 5 minutes in the store?
(ii)
Find the number of people who spent between 5 and 7 minutes in the store.
(iii)
Find the median time spent in the store.
(6)
(b)
Given that 40 of the people spent longer than k minutes, find the value of k.
(3)
(c)
(i)
On your answer sheet, copy and complete the following frequency table.
t (minutes)
0t2
2t4
Frequency
10
23
(ii)
4t6
6t8
8  t  10
10  t  12
15
Hence, calculate an estimate for the mean time spent in the store.
(5)
(Total 14 marks)
37.
In the research department of a university, 300 mice were timed as they each ran through a maze. The results
are shown in the cumulative frequency diagram opposite.
(a)
How many mice complete the maze in less than 10 seconds?
(1)
(b)
Estimate the median time.
(c)
Another way of showing the results is the frequency table below.
(i)
(1)
Time t (seconds)
Number of mice
t7
0
7t8
16
8t9
22
9  t  10
p
10  t  11
q
11  t  12
70
12  t  13
44
13  t  14
31
14  t  15
23
Find the value of p and the value of q.
(ii) Calculate an estimate of the mean time.
(4)
(Total 6 marks)
38.
The cumulative frequency curve below shows the marks obtained in an examination by a group of 200
students.
200
190
180
170
160
150
140
Number
of
130
students
120
110
100
90
80
70
60
50
40
30
20
10
0
(a)
20
30
40
50
60
Mark obtained
70
80
90
100
Use the cumulative frequency curve to complete the frequency table below.
Mark (x)
Number of
students
(b)
10
0  x < 20
20  x < 40
40  x < 60
22
Forty percent of the students fail. Find the pass mark.
60  x < 80
80  x < 100
20
(Total 6 marks)
39.
The cumulative frequency curve below shows the heights of 120 basketball players in centimetres.
120
110
100
90
80
70
60
Number of players
50
40
30
20
10
0
160
165
170
175
180
185
190
195
200
Height in centimetres
Use the curve to estimate
(a)
the median height;
(b)
the interquartile range.
(Total 6 marks)
40.
One thousand candidates sit an examination. The distribution of marks is shown in the following grouped
frequency table.
Marks
1–10
Number of
15
candidates
(a)
11–20 21–30 31–40 41–50 51–60 61–70 71–80 81–90 91–100
50
100
170
260
220
90
45
30
20
Copy and complete the following table, which presents the above data as a cumulative frequency
distribution.
(3)
Mark
10
20
Number of
candidates
15
65
30
40
50
60
70
80
90
100
905
(b)
Draw a cumulative frequency graph of the distribution, using a scale of 1 cm for 100 candidates on the
vertical axis and 1 cm for 10 marks on the horizontal axis.
(5)
(c)
Use your graph to answer parts (i)–(iii) below,
(i)
Find an estimate for the median score.
(2)
(ii)
Candidates who scored less than 35 were required to retake the examination. How many candidates
had to retake?
(3)
(iii)
The highest-scoring 15% of candidates were awarded a distinction.
Find the mark above which a distinction was awarded.
(3)
(Total 16 marks)
A taxi company has 200 taxi cabs. The cumulative frequency curve below shows the fares in dollars ($) taken
by the cabs on a particular morning.
200
180
160
140
120
Number of cabs
41.
100
80
60
40
20
10
(a)
20
30
40
50
Fares ($)
60
70
80
Use the curve to estimate
(i)
the median fare;
(ii)
the number of cabs in which the fare taken is $35 or less.
(2)
The company charges 55 cents per kilometre for distance travelled. There are no other charges. Use the curve
to answer the following.
(b)
On that morning, 40% of the cabs travel less than a km. Find the value of a.
(c)
What percentage of the cabs travel more than 90 km on that morning?
(4)
(4)
Random variables
1.
Two fair 4-sided dice, one red and one green, are thrown. For each die, the faces are labelled 1, 2, 3, 4. The
score for each die is the number which lands face down.
(a)
List the pairs of scores that give a sum of 6.
(3)
The probability distribution for the sum of the scores on the two dice is shown below.
(b)
Sum
2
3
4
5
6
7
8
Probability
p
q
3
16
4
16
3
16
r
1
16
Find the value of p, of q, and of r.
(3)
2.
José travels to school on a bus. On any day, the probability that José will miss the bus is
If he misses his bus, the probability that he will be late for school is
1
.
3
7
.
8
3
.
8
Let E be the event “he misses his bus” and F the event “he is late for school”.
The information above is shown on the following tree diagram.
If he does not miss his bus, the probability that he will be late is
(a)
Find
(i) P(E ∩ F);
(ii)
P(F).
(4)
(b)
Find the probability that
(i)
José misses his bus and is not late for school;
(ii)
José missed his bus, given that he is late for school.
(5)
The cost for each day that José catches the bus is 3 euros. When he misses the bus, he walks to school. José
goes to school on Monday and Tuesday of each week.
(c)
Copy and complete the probability distribution table.
X (cost in euros)
0
P (X)
1
9
3
6
(3)
(d)
Find the expected cost for José for both days.
(2)
(Total 14 marks)
3.
A four-sided die has three blue faces and one red face. The die is rolled.
Let B be the event a blue face lands down, and R be the event a red face lands down.
(a)
Write down
(i)
P (B);
(ii)
P (R).
(2)
(b)
If the blue face lands down, the die is not rolled again. If the red face lands down, the die is rolled once
again. This is represented by the following tree diagram, where p, s, t are probabilities.
Find the value of p, of s and of t.
(2)
Guiseppi plays a game where he rolls the die. If a blue face lands down, he scores 2 and is finished. If the red
face lands down, he scores 1 and rolls one more time. Let X be the total score obtained.
3
.
16
(c)
(i)
Show that P (X = 3) =
(ii)
(d)
(i)
Construct a probability distribution table for X.
Find P (X = 2).
(3)
(ii) Calculate the expected value of X.
(5)
(e)
If the total score is 3, Guiseppi wins $10. If the total score is 2, Guiseppi gets nothing.
Guiseppi plays the game twice. Find the probability that he wins exactly $10.
(4)
(Total 16 marks)
4.
A discrete random variable X has a probability distribution as shown in the table below.
x
0
1
2
3
P(X = x)
0.1
a
0.3
b
(a)
Find the value of a + b.
(2)
(b)
Given that E(X) =1.5, find the value of a and of b.
(4)
(Total 6 marks)
5.
Bag A contains 2 red balls and 3 green balls. Two balls are chosen at random from the bag without
replacement. Let X denote the number of red balls chosen. The following table shows the probability
distribution for X
X
0
1
2
P(X = x)
(a)
3
10
6
10
1
10
Calculate E(X), the mean number of red balls chosen.
(3)
Bag B contains 4 red balls and 2 green balls. Two balls are chosen at random from bag B.
(b)
(i)
Draw a tree diagram to represent the above information, including the probability of each event.
(ii)
Hence find the probability distribution for Y, where Y is the number of red balls chosen.
(8)
A standard die with six faces is rolled. If a 1 or 6 is obtained, two balls are chosen from bag A, otherwise two
balls are chosen from bag B.
(c)
Calculate the probability that two red balls are chosen.
(5)
(d)
Given that two red balls are obtained, find the conditional probability that a 1 or 6 was rolled on the
die.
(3)
(Total 19 marks)