21. Suppose P(A) = 0.50, P(B) = 0.30, and P(A or B) = 0.80.
a. Find P(A
B).
b. Find P(B | A).
c. Are A and B mutually exclusive events? Explain using probabilities.
22. Suppose P(A) = 0.40, P(B) = 0.50, and P(A
B) = 0.70.
a. Find P(A B).
b. Find P(B | A).
c. Are A and B independent events? Explain using probabilities.
23. An insurance company has recently recruited ten graduates, four men and six women. Two of the
graduates are to be selected at random to work in the firm’s suburban office.
a. What is the probability that two men will be selected?
b. What is the probability that at least one man will be selected?
24. An insurance company has collected the following data on the gender and marital status of 300
customers.
Marital Status
Gender
Single
Married
Divorced
Male
25
125
30
Female
50
50
20
Suppose that a customer is selected at random. Find the probability that the customer selected is:
a. a married female.
b. not single.
c. married, if the customer is male.
d. female or divorced.
e. Are gender and marital status mutually exclusive? Explain using probabilities.
f. Is marital status independent of gender? Explain using probabilities.
An ice cream vendor sells three flavours: chocolate, strawberry and vanilla. 45% of the sales are
chocolate, 30% are strawberry and the rest are vanilla. Sales are by the cone or the cup. The
percentages of cones sales for chocolate, strawberry and vanilla are 75%, 60% and 40%, respectively.
For a randomly selected sale, define the following events:
A1 = chocolate chosen.
A2 = strawberry chosen.
A3 = vanilla chosen.
B = ice cream in a cone.
= ice cream in a cup.
Use this information to answer the following question(s).
25. Find the probability that the ice cream was sold on a cone and the flavour was:
a. chocolate.
b. strawberry.
c. vanilla.
26. Find the probability that the ice cream was sold in a cup and the flavour was:
a. chocolate.
b. strawberry.
c. vanilla.
27. Find the probability that the ice cream was sold on a cone.
28. Find the probability that the ice cream was sold in a cup.
29. Find the probability that the ice cream was chocolate-flavoured, given that it was sold on a cone.
30. Find the probability that the ice cream was strawberry-flavoured, given that it was sold on a cone.
31. Find the probability that the ice cream was vanilla-flavoured, given that it was sold on a cone.
32. Find the probability that the ice cream was chocolate-flavoured, given that it was sold in a cup.
33. Find the probability that the ice cream was strawberry-flavoured, given that it was sold in a cup.
34. Find the probability that the ice cream was vanilla-flavoured, given that it was sold in a cup.
35. One card is randomly selected from a deck of 52 playing cards. Let:
A = event card selected is a nine.
B = event card selected is a 10.
C = event card selected is a queen.
Find P( or or ), using the addition rule.
36. A law firm has submitted bids on two separate state contracts, A and B. The company feels that it has a
40% chance of winning contract A, and a 60% chance of winning contract B. Furthermore, the
company believes that it has a 75% chance of winning contract B, given that it wins contract A.
What is the probability that the firm will win both contracts?
37. A law firm has submitted bids on two separate state contracts, A and B. The company feels that it has a
40% chance of winning contract A, and a 60% chance of winning contract B. Furthermore, the
company believes that it has a 75% chance of winning contract B, given that it wins contract A.
What is the probability that the firm will win at least one of the two contracts?
38. A law firm has submitted bids on two separate state contracts, A and B. The company feels that it has a
40% chance of winning contract A, and a 60% chance of winning contract B. Furthermore, the
company believes that it has a 75% chance of winning contract B, given that it wins contract A.
If the firm wins contract A, what is the probability that it will not win contract B?
39. A law firm has submitted bids on two separate state contracts, A and B. The company feels that it has a
40% chance of winning contract A, and a 60% chance of winning contract B. Furthermore, the
company believes that it has a 75% chance of winning contract B, given that it wins contract A.
What is the probability that the firm will win at most one of the two contracts?
40. A law firm has submitted bids on two separate state contracts, A and B. The company feels that it has a
40% chance of winning contract A, and a 60% chance of winning contract B. Furthermore, the
company believes that it has a 75% chance of winning contract B, given that it wins contract A.
What is the probability that the firm will win neither contract?
41. An investment firm has classified its clients according to their gender and the composition of their
investment portfolios (primarily bonds, primarily stocks, or a balanced mix of bonds and stocks). The
proportions of clients falling into the various categories are shown in the following table:
Portfolio Composition
Gender
Bonds
Stocks
Balanced
Male
0.18
0.20
0.25
Female
0.12
0.10
0.15
One client is selected at random, and two events A and B are defined as follows:
A: The client selected is male.
B: The client selected has a balanced portfolio.
Find the following probabilities.
a. P(A).
b. P(B).
c. P().
42. An investment firm has classified its clients according to their gender and the composition of their
investment portfolios (primarily bonds, primarily stocks, or a balanced mix of bonds and stocks). The
proportions of clients falling into the various categories are shown in the following table:
Portfolio Composition
Gender
Bonds
Stocks
Balanced
Male
0.18
0.20
0.25
Female
0.12
0.10
0.15
One client is selected at random, and two events A and B are defined as follows:
A: The client selected is male.
B: The client selected has a balanced portfolio.
Find the following probabilities.
a. P(A B).
b. P(A B).
c. P(A ).
d. P( ).
43. An investment firm has classified its clients according to their gender and the composition of their
investment portfolios (primarily bonds, primarily stocks, or a balanced mix of bonds and stocks). The
proportions of clients falling into the various categories are shown in the following table:
Portfolio Composition
Gender
Bonds
Stocks
Balanced
Male
0.18
0.20
0.25
Female
0.12
0.10
0.15
One client is selected at random, and two events A and B are defined as follows:
A: The client selected is male.
B: The client selected has a balanced portfolio.
Express each of the following probabilities in words.
a. P(A | B).
b. P(B | A).
c. P(A | ).
d. P( | B).
44. An investment firm has classified its clients according to their gender and the composition of their
investment portfolios (primarily bonds, primarily stocks, or a balanced mix of bonds and stocks). The
proportions of clients falling into the various categories are shown in the following table:
Portfolio Composition
Gender
Bonds
Stocks
Balanced
Male
0.18
0.20
0.25
Female
0.12
0.10
0.15
One client is selected at random, and two events A and B are defined as follows:
A: The client selected is male.
B: The client selected has a balanced portfolio.
Find the following probabilities.
a. P(A | B).
b. P(B | A).
c. P(A | ).
d. P( | B).
45. An investment firm has classified its clients according to their gender and the composition of their
investment portfolios (primarily bonds, primarily stocks, or a balanced mix of bonds and stocks). The
proportions of clients falling into the various categories are shown in the following table:
Portfolio Composition
Gender
Bonds
Stocks
Balanced
Male
0.18
0.20
0.25
Female
0.12
0.10
0.15
One client is selected at random, and two events A and B are defined as follows:
A: The client selected is male.
B: The client selected has a balanced portfolio.
Are A and B independent events? Explain.
46. An investment firm has classified its clients according to their gender and the composition of their
investment portfolios (primarily bonds, primarily stocks, or a balanced mix of bonds and stocks). The
proportions of clients falling into the various categories are shown in the following table:
Portfolio Composition
Gender
Bonds
Stocks
Balanced
Male
0.18
0.20
0.25
Female
0.12
0.10
0.15
One client is selected at random, and two events A and B are defined as follows:
A: The client selected is male.
B: The client selected has a balanced portfolio.
Are A and independent events? Explain.
47. An investment firm has classified its clients according to their gender and the composition of their
investment portfolios (primarily bonds, primarily stocks, or a balanced mix of bonds and stocks). The
proportions of clients falling into the various categories are shown in the following table:
Portfolio Composition
Gender
Bonds
Stocks
Balanced
Male
0.18
0.20
0.25
Female
0.12
0.10
0.15
One client is selected at random, and two events A and B are defined as follows:
A: The client selected is male.
B: The client selected has a balanced portfolio.
Are A and mutually exclusive events? Explain.
48. An investment firm has classified its clients according to their gender and the composition of their
investment portfolios (primarily bonds, primarily stocks, or a balanced mix of bonds and stocks). The
proportions of clients falling into the various categories are shown in the following table:
Portfolio Composition
Gender
Bonds
Stocks
Balanced
Male
0.18
0.20
0.25
Female
0.12
0.10
0.15
One client is selected at random, and two events A and B are defined as follows:
A: The client selected is male.
B: The client selected has a balanced portfolio.
Express each of the following events in words.
a. A B.
b. A B.
c. A .
d. .
49. A table of joint probabilities is shown below.
1
A
2
A
3
A
1
B
0.15
0.25
0.20
0.10
0.15
0.15
Calculate the marginal probabilities of event A.
1
A
3
A
50. A table of joint probabilities is shown below.
1
A
2
A
3
A
1
B
0.15
0.25
0.20
0.10
0.15
0.15
Calculate the marginal probabilities of event B.
1
B
51. A table of joint probabilities is shown below.
1
A
2
A
3
A
1
B
0.15
0.25
0.20
0.10
0.15
0.15
Calculate P(
1
A
|
1
B
).
52. A table of joint probabilities is shown below.
1
A
2
A
3
A
1
B
0.15
0.25
0.20
0.10
0.15
0.15
Calculate P(
2
A
/
1
B
).
2
A
2
A
1
B
53. A table of joint probabilities is shown below.
1
A
2
A
3
A
1
B
0.15
0.25
0.20
0.10
0.15
0.15
Calculate P(
3
A
/
1
B
).
3
A
1
B
3
A
1
B
54. A table of joint probabilities is shown below.
1
A
2
A
3
A
1
B
0.15
0.25
0.20
0.10
0.15
0.15
Calculate P(
1
A
|
).
1
A
2
B
1
A
2
B
55. A table of joint probabilities is shown below.
1
A
1
B
1
A
1
B
1
B
1
A
2
A
3
A
1
B
0.15
0.25
0.20
0.10
0.15
0.15
Calculate P(
2
B
/
1
A
).
56. A table of joint probabilities is shown below.
1
A
2
A
3
A
1
B
0.15
0.25
0.20
0.10
0.15
0.15
Calculate P(
1
A
/
2
A
).
1
A
2
A
1
A
2
A
57. A table of joint probabilities is shown below.
1
A
2
A
3
A
1
B
0.15
0.25
0.20
0.10
0.15
0.15
Are the events A and B independent? Explain.
2
A
1
B
2
A
2
A
58. A table of joint probabilities is shown below.
1
A
2
A
3
A
1
B
0.15
0.25
0.20
0.10
0.15
0.15
Calculate P(
1
A
1
B
).
A
B
B
A
B
2
B
1
A
2
B
1
A
59. A table of joint probabilities is shown below.
1
A
2
A
3
A
1
B
0.15
0.25
0.20
0.10
0.15
0.15
Calculate P(
1
A
or
2
B
).
1
A
2
B
1
A
2
B
1
A