63. {Business Majors Narrative} What is the chance that exactly one majors in accounting?
64. {Business Majors Narrative} What is the chance that none of them major in accounting?
Drunk Drivers
Six hundred accidents that occurred on a Saturday night were analyzed. Two items noted were the
number of vehicles involved and whether alcohol played a role in the accident. The numbers are
shown below:
Number of Vehicles Involved
Did alcohol play a role?
1
2
3
Totals
Yes
75
125
50
250
No
50
225
75
350
Totals
125
350
125
600
65. {Drunk Drivers Narrative} What proportion of accidents involved more than one vehicle?
66. {Drunk Drivers Narrative} What proportion of accidents involved alcohol and single vehicle?
67. {Drunk Drivers Narrative} What proportion of accidents involved alcohol or a single vehicle?
68. {Drunk Drivers Narrative} Given alcohol was involved, what proportion of accidents involved a single
vehicle?
69. {Drunk Drivers Narrative} If multiple vehicles were involved, what proportion of accidents involved
alcohol?
70. {Drunk Drivers Narrative} If 3 vehicles were involved, what proportion of accidents involved alcohol?
71. {Drunk Drivers Narrative} If alcohol was not involved, what proportion of the accidents were single
vehicle?
72. {Drunk Drivers Narrative} If alcohol was not involved, what proportion of the accidents were multiple
vehicle?
73. Suppose A and B are two independent events for which P(A) = 0.20 and P(B) = 0.60.
a.
Find P(A|B).
b.
Find P(B|A).
ANS:
0.20
b.
0.60
GPA and Class
A college professor classifies his students according to their grade point average (GPA) and their class
rank. GPA is on a 0.0-4.0 scale, and class rank is defined as the under class (freshmen and
sophomores) and the upper class (juniors and seniors). One student is selected at random.
GPA
Class
Under 2.0
2.0 – 3.0
Over 3.0
Under
0.05
0.25
0.10
Upper
0.10
0.30
0.20
74. {GPA and Class Narrative} If the student selected is in the upper class, what is the probability that her
GPA is between 2.0 and 3.0?
ANS:
75. {GPA and Class Narrative} If the GPA of the student selected is over 3.0, what is the probability that
the student is in the lower class?
ANS:
76. {GPA and Class Narrative} What is the probability that the student is in the upper class?
77. {GPA and Class Narrative} What is the probability that the student has GPA over 3.0?
78. {GPA and Class Narrative} What is the probability that the student is in the lower class?
79. {GPA and Class Narrative} What is the probability that the student is in the lower class and has GPA
over 3.0?
80. {GPA and Class Narrative} What is the probability that the student is in the upper class and has GPA
under 2.0?
81. {GPA and Class Narrative} Are being in the upper class and having a GPA over 3.0 related? Explain.
Marital Status
An insurance company has collected the following data on the gender and marital status of 570
customers.
Marital Status
Gender
Single
Married
Divorced
Male
50
250
30
Female
100
100
40
Suppose that a customer is selected at random.
82. {Marital Status Narrative} Develop the joint probability table.
Gender
Single
Married
Divorced
Male
Female
83. {Marital Status Narrative} Find the probability that the customer selected is a married female.
84. {Marital Status Narrative} Find the probability that the customer selected is
a.
female and single
b.
married if the customer is male.
c.
not single
ANS:
b.
0.757
0.737
Financial Consultants
A Financial Consultant has classified his clients according to their gender and the composition of their
investment portfolio (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.
85. {Financial Consultants Narrative} Find the following probabilities:
a.
P(A)
b.
P(B)
a.
0.63
b.
0.40
86. {Financial Consultants Narrative} Express each of the following events in words:
a.
A or B
b.
A and B
a.
The client selected either is male or has a balanced portfolio or both.
b.
The client selected is male and has a balanced portfolio.
87. {Financial Consultants Narrative} Find P(A and B).
88. {Financial Consultants Narrative} Express each of the following probabilities in words:
a.
P(A|B)
b.
P(B|A)
The probability that the client selected is male, if the client has a balanced portfolio.
b.
The probability that the client selected has a balanced portfolio, if the client is male.
89. {Financial Consultants Narrative} Find the following probabilities:
a.
P(A|B)
b.
P(B|A)
a.
b.
0.3968
90. Julius and Gabe go to a show during their Spring break and toss a balanced coin to see who will pay
for the tickets. The probability that Gabe will pay three days in a row is 0.125.
91. If events A and B have nonzero probabilities, then they can be both independent and mutually
exclusive.
92. If the event of interest is A, the probability that A will not occur is the complement of A.
93. Assume that A and B are independent events with P(A) = 0.30 and P(B) = 0.50. The probability that
both events will occur simultaneously is 0.80.
94. Two events A and B are said to be independent if P(A) = P(A|B).
95. When A and B are mutually exclusive, P(A or B) can be found by adding P(A) and P(B).
96. Two events A and B are said to be independent if P(A|B) = P(B).
97. If A and B are two independent events with P(A) = 0.9 and P(B|A) = 0.5, then P(A and B) = 0.45.
98. Two events A and B are said to be independent if P(A|B) = P(B|A).
99. The probability of the union of two mutually exclusive events A and B is 0.
100. Two events A and B are said to be mutually exclusive if P(A and B) = 1.0.
101. If P(A and B) = 1, then A and B must be mutually exclusive.
102. Events A and B are either independent or mutually exclusive.
103. If P(B) = .7 and P(B|A) = .4, then P(A and B) must be .28.
104. If P(B) = .7 and P(A|B) = .7, then P(A and B) = 0.
105. If the events A and B are independent with P(A) = 0.35 and P(B) = 0.45, then the probability that both
events will occur simultaneously is:
a.
0
b.
0.16
c.
0.80
d.
Not enough information to tell.
106. Two events A and B are said to be mutually exclusive if:
a.
P(A|B) = 1
b.
P(A|B) = P(A)
c.
P(A and B) =1
d.
P(A and B) = 0
107. If P(A) = 0.84, P(B) = 0.76, and P(A or B) = 0.90, then P(A and B) is:
a.
0.06
b.
0.14
c.
0.70
d.
0.83
108. Which of the following statements is always correct?
a.
P(A and B) = P(A) * P(B)
b.
P(A or B) = P(A) + P(B)
c.
P(A) = 1 − P(Ac)
d.
None of these choices.
109. If P(A) = 0.20, P(B) = 0.30, and P(A and B) = 0, then A and B are:
a.
dependent events
b.
independent events
c.
mutually exclusive events
d.
complementary events
110. If P(A) = 0.65, P(B) = 0.58, and P(A and B) = 0.76, then P(A or B) is:
a.
1.23
b.
0.47
c.
0.24
d.
None of these choices.
111. Suppose P(A) = 0.30. The probability of the complement of A is:
a.
0.30
b.
0.70
c.
−0.30
d.
None of these choices.
112. If events A and B are independent then:
a.
P(A and B) = P(A) * P(B)
b.
P(A and B) = P(A) + P(B)
c.
P(B|A) = P(A)
d.
None of these choices.
113. If A and B are mutually exclusive events, with P(A) = 0.20 and P(B) = 0.30, then the probability that
both events will occur simultaneously is:
a.
0.50
b.
0.06
c.
0
d.
None of these choices.
114. If A and B are independent events with P(A) = 0.60 and P(B) = 0.70, then P(A or B) equals:
a.
1.30
b.
0.88
c.
0.42
d.
Cannot tell from the given information.
115. If A and B are mutually exclusive events with P(A) = 0.30 and P(B) = 0.40, then P(A or B) is:
a.
0.10
b.
0.12
c.
0.70
d.
None of these choices
116. If A and B are any two events with P(A) = .8 and P(B|A) = .4, then P(A and B) is:
a.
.40
b.
.32
c.
1.20
d.
None of these choices.
117. If A and B are any two events with P(A) = .8 and P(B|Ac) = .7, then P(Ac and B) is
a.
0.56
b.
0.14
c.
1.50
d.
None of these choices.
118. The ____________________ rule says that P(Ac) = 1 − P(A).
119. The ____________________ rule is used to calculate the joint probability of two events.
120. If A and B are ____________________ events, the joint probability of A and B is the product of the
probabilities of those two events.
121. The ____________________ rule is used to calculate the probability of the union of two events.
122. If A and B are ____________________ then the probability of the union of A and B is the sum of their
individual probabilities.
123. The first set of branches of a probability tree represent ____________________ probabilities.
124. The second set of branches of a probability tree represent ____________________ probabilities.
125. When you multiply a first level branch with a second level branch on a probability tree you get a(n)
____________________ probability.
126. If two events are complements, their probabilities sum to ____________________.
127. If two events are mutually exclusive their joint probability is ____________________.
128. Suppose A and B are two independent events for which P(A) = 0.20 and P(B) = 0.60.
a.
Find P(A and B).
b.
Find P(A or B).
a.
0.12
b.
0.68
College Professorship
A Ph.D. graduate has applied for a job with two colleges: A and B. The graduate feels that she has a
60% chance of receiving an offer from college A and a 50% chance of receiving an offer from college
B. If she receives an offer from college B, she believes that she has an 80% chance of receiving an
offer from college A. Let A = receiving an offer from college A, and let B = receiving an offer from
college B.
129. {College Professorship Narrative} What is the probability that both colleges will make her an offer?
130. {College Professorship Narrative} What is the probability that at least one college will make her an
offer?
131. {College Professorship Narrative} If she receives an offer from college B, what is the probability that
she will not receive an offer from college A?