True/False
1. A contingency table is a tabular summary of probabilities concerning two sets of
complementary events.
Answer: True
2. An event is a collection of sample space outcomes.
Answer: True
3. Two events are independent if the probability of one event is influenced by whether or
not the other event occurs.
Answer: False
4. Mutually exclusive events have a nonempty intersection.
Answer: False
5. A subjective probability is a probability assessment that is based on experience,
intuitive judgment, or expertise.
Answer: True
6. The probability of an event is the sum of the probabilities of the sample space
outcomes that correspond to the event.
Answer: True
7. If events A and B are mutually exclusive, then P(B / A ) is always equal to zero.
Answer: True
8. If events A and B are independent, then P(A | B) is always equal to zero.
Answer: False
9. If events A and B are mutually exclusive, then P(A  B) is always equal to zero.
Answer: True
10. Events that have no sample space outcomes in common, and, therefore cannot occur
simultaneously are referred to as independent events.
Answer: False
Multiple Choice
11. Two mutually exclusive events having positive probabilities are______________
dependent.
A) Always
B) Sometimes
C) Never
Answer: A
12. ___________________ is a measure of the chance that an uncertain event will occur.
A) Random experiment
B) Sample Space
C) Probability
D) A complement
E) A population
Answer: C
13. A manager has just received the expense checks for six of her employees. She
randomly distributes the checks to the six employees. What is the probability that exactly
five of them will receive the correct checks (checks with the correct names).
A) 1
B) ½
C) 1/6
D) 0 E) 1/3
Answer: D
14. In which of the following are the two events A and B, always independent?
A) A and B are mutually exclusive.
B) The probability of event A is not influenced by the probability of event B.
C) The intersection of A and B is zero.
D) P(A / B) = P(A).
E) B and D.
Answer: E
15. If two events are independent, we can _____ their probabilities to determine the
intersection probability.
A) Divide
B) Add
C) Multiply
D) Subtract
Answer: C
16. Events that have no sample space outcomes in common, and therefore, cannot occur
simultaneously are:
A) Independent
B) Mutually Exclusive
C) Intersections
D) Unions
Answer: B
17. If events A and B are independent, then the probability of simultaneous occurrence
of event A and event B can be found with:
A) P(A) P(B)
B ) P(A) P(A / B)
C) P(B) P(B / A)
D) All of the above are correct
Answer: D
18. The set of all possible experimental outcomes is called a(n):
A) Sample space
B) Event
C) Experiment
D) Probability
Answer: A
19. A(n) ____________ is the probability that one event will occur given that we know
that another event already has occurred.
A) Sample space outcome
B) Subjective Probability
C) Complement of events
D) Long-run relative frequency
E) Conditional probability
Answer: E
20. The _______ of two events X and Y is another event that consists of the sample
space outcomes belonging to either event X or event Y or both event X and Y.
A) Complement
B) Union
C) Intersection
D) Conditional probability
Answer: B
21. If P(A) > 0 and P(B) > 0 and events A and B are independent, then:
A) P(A) = P(B)
B) P(B / A)=P(A)
C) P(A  B) = 0
D) P(A  B) = P(A) P(B  A)
Answer: B
22. P(A  B) = P(A) + P(B) - P(A  B) represents the formula for the
A) conditional probability
B) addition rule
C) addition rule for two mutually exclusive events
D) multiplication rule
Answer: B
23. The management of a company believes that weather conditions significantly affect
the level of demand for its product. 48 monthly sales reports are randomly selected.
These monthly sales reports showed 15 months with high demand, 28 months with
medium demand, and 5 months with low demand. 12 of the 15 months with high demand
had favorable weather conditions. 14 of the 28 months with medium demand had
favorable weather conditions. Only 1 of the 5 months with low demand had favorable
weather conditions. What is the probability that weather conditions are poor, given that
the demand is high?
A) .2
B) .5
C) .8
D) .25
E) .75
Answer: A
24.The management believes that the weather conditions significantly impact the level of
demand and the estimated probabilities of poor weather conditions given different levels
of demand is presented below.
P(Poor I High)  .2, P(Poor I Medium) .5, P( Poor I Low)  .8
What is the probability of high demand given that the weather conditions are poor.
A) .06
B) .44
C) .1364
D) .12
E) .1818
Answer: C
Use the following Information to answer questions 25 -26:
An automobile insurance company is in the process of reviewing its policies. Currently
drivers under the age of 25 have to pay a premium. The company is considering
increasing the value of the premium charged to drivers under 25. According to company
records, 35% of the insured drivers are under the age of 25. The company records also
show that 280 of the 700 insured drivers under the age of 25 had been involved in at least
one automobile accident. On the other hand, only 130 of the 1300 insured drivers 25
years or older had been involved in at least one
automobile accident.
25. An accident has just been reported. What is the probability that the insured driver
is under
the age of 25?
A) 35%
B) 20.5%
C) 14%
D) 68.3%
E) 40%
Answer: D
26. What is the probability that an insured driver of any age will be involved in an
accident?
A) 35%
B) 20.5%
C) 65%
D) 68.3%
E) 79.5%
Answer: B
27. A pharmaceutical company manufacturing pregnancy test kits wants to determine the
probability of a woman not being pregnant when the test results indicate pregnancy. It is
estimated that the probability of pregnancy among potential users of the kit is 10%.
According to the company laboratory test results 1 out of 100 non-pregnant women tested
pregnant (false positive). On the other hand, 1 out of 200 pregnant women tested nonpregnant (false negative). A woman has just used the pregnancy test kit manufactured by
the company and the results showed pregnancy. What is the probability that she is not
pregnant?
A) 90%
B) 0.9%
C) 8.3%
D) 91.7%
E) 10.85%
Answer: C
28. A pharmaceutical company manufacturing pregnancy test kits wants to determine the
probability of a woman actually being pregnant when the test results indicate that she is
not pregnant. It is estimated that the probability of pregnancy among potential users of
the kit is 10%. According to the company laboratory test results 1 out of 100 nonpregnant women tested pregnant (false positive). On the other hand, 1 out of 200
pregnant women tested non-pregnant (false negative). A woman has just used the
pregnancy test kit manufactured by the company and the results showed that she is not
pregnant. What is the probability that she is pregnant?
A) 1%
B) 0.9%
C) 0.05%
D) 8.3%
E) 0.056%
Answer: E
29. A(n) _____ is the set of all of the distinct possible outcomes of an experiment.
Answer: Sample Space
30. The _____ of an event is a number that measures the likelihood that an event will
occur when an experiment is carried out.
Answer: Probability
31. When the probability of one event is influenced by whether or not another event
occurs, the events are said to be _____.
Answer: Dependent
32. A process of observation that has an uncertain outcome is referred to as a(n) _____.
Answer: Experiment
33. When the probability of one event is not influenced by whether or not another event
occurs, the events are said to be _____.
Answer: Independent
34. A probability may be interpreted as a long run _____ frequency.
Answer: Relative
35. If events A and B are independent, then P(A / B) is equal to _____.
Answer: P(A)
36. The simultaneous occurrence of event A and B is represented by the notation:
_______.
Answer: A  B
37. A(n) _______________ probability is a probability assessment that is based on
experience, intuitive judgment, or expertise.
Answer: Subjective
38. A(n) ______________ is a collection of sample space outcomes.
Answer: Event
39. Probabilities must be assigned to experimental outcomes so that the probabilities of
all the experimental outcomes must add up to ___.
Answer: 1
40. Probabilities must be assigned to experimental outcomes so that the probability
assigned to each experimental outcome must be between ____ and ____ inclusive.
Answer: 0,1
41. The __________ of event X consists of all sample space outcomes that do not
correspond to the occurrence of event X.
Answer: Complement
42. The _______ of two events A and B is another event that consists of the sample
space outcomes belonging to either event A or event B or both event A and B.
Answer: Union
43. The _______ of two events A and B is the event that consists of the sample space
outcomes belonging to both event A and event B.
Answer: Intersection
44. __________________ statistics is an area of statistics that uses Bayes' theorem to
update prior belief about a probability or population parameter to a posterior belief.
Answer: Bayesian
45. In the application of Bayes' theorem the sample information is combined with prior
probabilities to obtain ___________________ probabilities.
Answer: posterior
46. What is the probability of rolling a seven with a pair of fair dice?
Answer: 1/6
6
36
47. What is the probability of rolling a value higher than eight with a pair of fair dice?
Answer: .2777
10
36
 2777
48. What is the probability that an even number appears on the toss of a die?
Answer: .5
49. What is the probability that a king appears in drawing a single card form a deck of 52
cards?
Answer: 1/13
50. If we consider the toss of four coins as an experiment, how many outcomes does the
sample space consist of?
Answer: 16
51. What is the probability of at least one tail in the toss of three fair coins?
Answer: 7/8
52. A lot contains 12 items, and 4 are defective. If three items are drawn at random from
the lot, what is the probability they are not defective?
Answer: .2545
8
7
6
(12) (11) (10) = .2545
53. A person is dealt 5 cards from a deck of 52 cards. What is the probability they are all
clubs?
Answer: .0004951
13
12
11
10
9
(52) (51) (50) (49) (48) = 0.0004951
54. A group has 12 men and 4 women. If 3 people are selected at random from the
group, what
is the probability that they are all men?
Answer: .392857
12
11
10
(16) (15) (14) = 392857
Use the following information to answer questions 55-57:Container 1 has 8 items, 3
of which are defective. Container 2 has 5 items, 2 of which are defective. If one item
is drawn from each container:
55. What is the probability that both items are not defective?
Answer: .375
56. What is the probability that the item from container one is defective and the item
from container 2 is not defective?
Answer: .225
57. What is the probability that one of the items is defective?
Answer: .45
58. A coin is tossed 6 times. What is the probability that at least one head occurs?
Answer: 63/64
59. Suppose P(A) = .45, P(B) =.20, P(C) = .35, P (A / E) = .10, P(B / E) = .05, and
P(CE) = 0. What is P(E)?
Answer: .055
P(E) = (.45)(.10) + (.20)(.05) + (.35)(0) = .055
60. Suppose P(A) = .45, P(B) = .20, P(C) = .35, P (AE) = .10, P (BE)= .05, and P
(CE)= 0. What is P(EA)?
Answer: .8182
61. Suppose P(A) = .45, P(B) = .20, P(C) = .35, P (AE) = .10, P (BE)= .05, and
P(CE) = 0. What is P(EB)?
Answer: 1818
62. Suppose P(A) = .45, P(B) = .20, P(C) = .35, P (AE)= .10, P(BE) = .05, and
P(CE) = 0. What is P(EC)?
Answer: 0
63. Given the standard deck of cards, what is the probability of drawing a red card,
given that it is a face card?
Answer: .5
64. Given a standard deck of cards, what is the probability of drawing a face card,
given that it is a red card?
Answer: 3/13
65. A machine is made up of 3 components: an upper part, a mid-part, and a lower
part. The machine is then assembled. 5 percent of the upper parts are defective; 4
percent of the mid parts are defective; 1 percent of the lower parts are defective.
What is the probability that a machine is
non-defective?
Answer: .9029
(.95)(.96)(.99) = .9029
66. A machine is produced by a sequence of operations. Typically one defective
machine is produced per 1000 parts. What is the probability of two non-defective
machines being produced?
Answer: .998
(.999)(.999) = .998
67. A pair of dice is thrown. What is the probability that one of the faces is a 3,
given that the sum of the two faces is 9?
Answer: 1/4
68. A card is drawn from a standard deck. What is the probability the card is an ace,
given that
it is a club?
Answer: 1/13
69. A card is drawn from a standard deck. Given that a face card is drawn, what is
the probability it will be a king?
Answer: 1/3
(4 kings) / (12 face cards)
70. Independently a coin is tossed, a card is drawn from a deck, and a die is thrown.
What is the probability of observing a head on the coin, an ace on the card, and a five
on the die?
Answer: 1/156
71. A family has two children. What is the probability that both are girls, given that
at least one is a girl?
Answer: 1/3
72. What is the probability of winning four games in a row, if the prob
ability of winning each game individually is 1/2?
Answer: 1/16
Use the following to answer questions 73-77:
At a college, 70 percent of the students are women and 50 percent of the students
receive a grade of C. 25 percent of the students are neither female nor C students.
Use this contingency table.
C
C
Women
.45
.25
.70
Men
.05
.25
.30
.50
.50
1.00
73. What is the probability that a student is female and a C student?
Answer: .45
74.What is the probability that a student is male and not a C student?
Answer: .25
75. If the student is male, what is the probability he is a C student?
Answer: .1667
76. If the student has received a grade of C, what is the probability that he is male?
Answer: .10
77. If the student has received a grade of C, what is the probability that she is female?
Answer: .90
Use the following information to answer questions 78-79:
Two percent (2%) of the customers of a store buy cigars. Half of the customers who
buy cigars buy beer. 25 percent who buy beer buy cigars. Determine the probability
that a customer using this contingency table:
Cigars
Cigars
Beer
.01
.03
.04
Beer
.01
.95
.69
78. Buys beer.
Answer: .04
79. Neither buys beer nor buys cigars.
Answer: .95
.02
.95
1.0
Use the following information to answer questions 80-81:
An urn contains five white, three red, and four black balls. Three are drawn at
random without replacement.
80. What is the probability that no ball is red?
Answer: .3818
81. What is the probability that all balls are the same color?
Answer: .0682
82. What is the probability that any two people chosen at random were born on the
same day of the week?
Answer: 1/7
83. A letter is drawn from the alphabet of 26 letters. What is the probability that the
letter drawn is a vowel?
Answer: 5/26
84. How many times must a die be tossed if the expected number of ones is five?
Answer: 30
85. List two properties of a valid discrete probability distribution.
Answer: P(X)  0, for all X and ∑𝑛𝑖 𝑋i
86. If A and B are independent events, P(A) = .2, and P(B) = .7, determine P(A  B)
Answer: .76
87. If events A and B are mutually exclusive, calculate P(B / A).
Answer: Zero
88. What is the probability of rolling a six with a fair die five times in a row?
Answer: 1/7,776
89. If a product is made using five individual components, and P(product meets
specifications) = .98, what is the probability of an individual component meeting
specifications assuming that this probability is the same for all five components?
5
Answer: .9960
√98
90. If P(B / A) = .2 and P(B) = .8, determine the intersection of event A and B.
Answer: .16
(.2)(.8) = .16
91. If P(A  B )= .3 and P(B / A) = .9, find P(B).
Answer: .333