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Chapter 04
1. Probabilities that cannot be estimated from long-run relative frequencies of events are
a
...
Name: Class: Date:
Chapter 04
1. Probabilities that cannot be estimated from long-run relative frequencies of events are
a. objective probabilities b. subjective probabilities
c. complementary probabilities d. joint probabilities
2. The probability of an event and the probability of its complement always sum to
a. 1 b. 0
c. any value between 0 and 1 d. any positive value
3. If events A and B are mutually exclusive, then the probability of both events occurring simultaneously is
equal to
a. 0.0 b. 0.5
c. 1.0 d. any value between 0.5 and 1.0
4. Probabilities that can be estimated from long-run relative frequencies of events are
a. objective probabilities b. subjective probabilities
c. complementary probabilities d. joint probabilities
5. Let A and B be the events of the FDA approving and rejecting a new drug to treat hypertension,
respectively. The events A and B are
a. independent b. conditional
c. unilateral d. mutually exclusive
6. A function that associates a numerical value with each possible outcome of an uncertain event is called a
a. conditional variable b. random variable
c. population variable d. sample variable
7. The formal way to revise probabilities based on new information is to use:
a. complementary probabilities b. conditional probabilities
c. unilateral probabilities d. common sense probabilities
8. is the:
a. addition rule b. commutative rule
c. rule of complements d. rule of opposites
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9. The law of large numbers is relevant to the estimation of
a. objective probabilities b. subjective probabilities
c. both objective and subjective probabilities d. neither objective nor subjective probabilities
10. A discrete probability distribution:
a. lists all of the possible values of the random variable and their corresponding probabilities
b. is a tool that can be used to incorporate uncertainty into models
c. can be estimated from long-run proportions
d. is the distribution of a single random variable
11. Which of the following statements are true?
a. Probabilities must be negative
b. Probabilities must be greater than 1
c. The sum of all probabilities for a random variable must be equal to 1
d. All of these options are true.
12. If P(A) = P(A|B), then events A and B are said to be
a. mutually exclusive b. independent
c. exhaustive d. complementary
13. If A and B are mutually exclusive events with P(A) = 0.70, then P(B):
a. can be any value between 0 and 1
b. can be any value between 0 and 0.70
c. cannot be larger than 0.30
d. Cannot be determined with the information given
14. If two events are collectively exhaustive, what is the probability that one or the other occurs?
a. 0.25
b. 0.50
c. 1.00
d. Cannot be determined from the information given.
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15. If two events are collectively exhaustive, what is the probability that both occur at the same time?
a. 0.00
b. 0.50
c. 1.00
d. Cannot be determined from the information given.
16. The probabilities shown in a table with two rows, and two columns, , are as follows:
, , , and . Then ,
calculated up to two decimals, is
a. .33 b. .35
c. .65 d. .67
17. If two events are mutually exclusive, what is the probability that one or the other occurs?
a. 0.25
b. 0.50
c. 1.00
d. Cannot be determined from the information given.
18. If two events are mutually exclusive, what is the probability that both occur at the same time?
a. 0.00
b. 0.50
c. 1.00
d. Cannot be determined from the information given.
19. If two events are mutually exclusive and collectively exhaustive, what is the probability that both occur?
a. 0.00
b. 0.50
c. 1.00
d. Cannot be determined from the information given.
20. There are two types of random variables, they are
a. discrete and continuous b. exhaustive and mutually exclusive
c. complementary and cumulative d. real and unreal
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21. If P(A) = 0.25 and P(B) = 0.65, then P(A and B) is:
a. 0.25
b. 0.40
c. 0.90
d. Cannot be determined from the information given
22. If two events are independent, what is the probability that they both occur?
a. 0
b. 0.50
c. 1.00
d. Cannot be determined from the information given
23. If A and B are any two events with = .8 and , then is
a. .56 b. .14
c. .24 d. None of these options.
24. Which of the following best describes the concept of probability?
a. It is a measure of the likelihood that a particular event will occur.
b. It is a measure of the likelihood that a particular event will occur, given that another event has
already occurred.
c. It is a measure of the likelihood of the simultaneous occurrence of two or more events.
d. None of these options.
25. The probabilities shown in a table with two rows, and two columns, , are as follows:
, , , and . Then
, calculated up to two decimals, is
a. .33 b. .35
c. .65 d. .67
26. If A and B are mutually exclusive events with P(A) = 0.30 and P(B) = 0.40, then the probability that either
A or B occur is:
a. 0.10 b. 0.12
c. 0.70 d. None of these options.
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27. If A and B are any two events with P(A) = .8 and P(B|A) = .4, then the joint probability of A and B is
a. .80 b. .40
c. .32 d. 1.20
28. If A and B are independent events with P(A) = 0.40 and P(B) = 0.50, then P(A/B) is 0.50.
a. True
b. False
29. A random variable is a function that associates a numerical value with each possible outcome of a
random phenomenon.
a. True
b. False
30. Two or more events are said to be exhaustive if one of them must occur.
a. True
b. False
31. You think you have a 90% chance of passing your statistics class. This is an example of subjective
probability.
a. True
b. False
32. The number of cars produced by GM during a given quarter is a continuous random variable.
a. True
b. False
33. Two events A and B are said to be independent if P(A and B) = P(A) + P(B)
a. True
b. False
34. Probability is a number between 0 and 1, inclusive, which measures the likelihood that some event will
occur.
a. True
b. False
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35. If events A and B have nonzero probabilities, then they can be both independent and mutually exclusive.
a. True
b. False
36. The probability that event A will not occur is denoted as .
a. True
b. False
37. If P(A and B) = 1, then A and B must be collectively exhaustive.
a. True
b. False
38. Conditional probability is the probability that an event will occur, with no other events taken into
consideration.
a. True
b. False
39. When we wish to determine the probability that at least one of several events will occur, we would use the
addition rule.
a. True
b. False
40. The law of large numbers states that subjective probabilities can be estimated based on the long run
relative frequencies of events
a. True
b. False
41. Two events are said to be independent when knowledge of one event is of no value when assessing the
probability of the other.
a. True
b. False
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42. Suppose A and B are mutually exclusive events where P(A) = 0.2 and P(B) = 0.5, then P(A or B) = 0.70.
a. True
b. False
43. If A and B are two independent events with P(A) = 0.20 and P(B) = 0.60, then P(A and B) = 0.80
a. True
b. False
44. The relative frequency of an event is the number of times the event occurs out of the total number of
times the random experiment is run.
a. True
b. False
45. Subjective probability is the probability that a given event will occur, given that another event has already
occurred.
a. True
b. False
46. The temperature of the room in which you are writing this test is a continuous random variable.
a. True
b. False
47. Two events A and B are said to mutually be exclusive if P(A and B) = 0.
a. True
b. False
48. Two or more events are said to be exhaustive if at most one of them can occur.
a. True
b. False
49. When two events are independent, they are also mutually exclusive.
a. True
b. False
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50. Two or more events are said to be mutually exclusive if at most one of them can occur.
a. True
b. False
51. Given that events A and B are independent and that P(A) = 0.8 and P(B/A) = 0.4, then P(A and B) = 0.32.
a. True
b. False
52. The time students spend in a computer lab during one day is an example of a continuous random
variable.
a. True
b. False
53. The multiplication rule for two events A and B is: P(A and B) = P(A|B)P(A).
a. True
b. False
54. The number of car insurance policy holders is an example of a discrete random variable.
a. True
b. False
55. Suppose A and B are mutually exclusive events where P(A) = 0.3 and P(B) = 0.4, then P(A and B) =
0.12.
a. True
b. False
56. Suppose A and B are two events where P(A) = 0.5, P(B) = 0.4, and P(A and B) = 0.2, then P(B/A) = 0.5.
a. True
b. False
57. Suppose that after graduation you will either buy a new car (event A) or take a trip to Europe (event B).
Events A and B are mutually exclusive.
a. True
b. False
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58. If P(A and B) = 0, then A and B must be collectively exhaustive.
a. True
b. False
59. The number of people entering a shopping mall on a given day is an example of a discrete random
variable.
a. True
b. False
60. Football teams toss a coin to see who will get their choice of kicking or receiving to begin a game. The
probability that given team will win the toss three games in a row is 0.125.
a. True
b. False
A manufacturing facility needs to open a new assembly line in four months or there will be significant cost overruns.
The manager of this project believes that there are four possible values for the random variable X (the number of
months from now it will take to complete this project): 3, 3.5, 4, and 4.5. It is currently believed that the probabilities of
these four possibilities are in the ratio 1 to 2 to 3 to 2. That is, X = 3.5 is twice as likely as X = 3 and X = 4 is 1.5
times as likely as X = 3.5.
61. Find the probability distribution of X.
ANSWER:
x 3 3.5 4 4.5
P (X = x) 0.125 0.250 0.375 0.250
62. What is the probability that this project will be completed in less than 4 months from now?
ANSWER:
63. What is the probability that this project will not be completed on time?
ANSWER:
64. (A) What is the expected completion time (in months) from now for this project?
(B) How much variability (in months) exists around the expected value found in (A)?
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The possible annual percentage return of the stocks of Gamma, Inc. and Delta, Inc. share a common probability
distribution, given below.
Probability Return of Gamma, Inc. Return of Delta, Inc.
0.05 36.2 -6.4
0.15 23.4 -1.8
0.30 15.18 6.9
0.20 6.2 12.4
0.15 -2.0 16.8
0.05 -4.2 30.2
65. (A) What is the expected annual return of each stock?
(B) What is the standard deviation of the annual return of each stock?
(C) On the basis of your answers to (A) and (B), which of these stocks would you prefer to buy? Defend
your choice.
(D) Are the annual returns of these two stocks positively or negatively associated with each other? How
might the answer to this question influence your decision to purchase shares?
ANSWER: (A)
Stock Expected Value
Gamma 10.58
Delta 10.10
(B)
Stock Standard Deviation
Gamma 10.16
Delta 8.08
(C) Gamma has a higher expected annual return. However, Gamma is also more volatile with
a higher standard deviation. Each stock also has a 20% chance of a negative return.
Therefore, you could select Gamma based on the expected return or Delta based on the
volatility.
(D) These two stocks are negatively associated with one another. Therefore, if you want a
more balanced portfolio, you may want to invest in both stocks.
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A small grocery store is considering installing an express checkout line. Let X be the number of customers in the
regular checkout line. Note that these numbers include the customers being served, if any. The probability distribution
of X is given in the table below.
x 0 1 2 3
P(X=x) 0.28 0.22 0.26 0.24
66. Find the probability that one customer is in the regular checkout line.
ANSWER:
67. Find the probability that no more than one customer is in line.
ANSWER:
68. Find the probability that at least two people are in line.
ANSWER:
69. Find the probability that three or fewer customers are in line.
ANSWER:
70. What is the probability that no one is waiting or being served in the regular checkout line?
ANSWER: P(Regular line is empty) = P(X=0) = 0.28
71. What is the probability that at most one person is waiting or being served in the express checkout line?
ANSWER:
72. What is the probability that three customers are waiting in line?
ANSWER:
73. On average, how many customers would you expect to see in line?
ANSWER:
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Suppose that the manufacturer of a particular product assesses the distribution of the demand for its product in the
upcoming quarter as presented below. Use this information to answer the following questions.
Demand (in units)
x 2000 2500 3000 3500
P(X=x) 0.28 0.26 0.20 0.26
74. Find the expected demand (in units) for the upcoming quarter.
75. What is the probability that the demand for this product will be above its mean in the upcoming quarter?
ANSWER76. What is the probability that the demand of this product will be below its mean in the upcoming quarter?
ANSWER
77. What is the probability that the demand for this product will be less than 3500 units in the upcoming
quarter?
ANSWER:
78. What is the probability that the demand for this product exceeds 2500 units in the upcoming quarter?
ANSWER79. What is the expected number of bats sold on a typical day?
(x) = 0.298; about 3 bats
80. Calculate the variance of the probability distribution.
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81. What is the standard deviation of the probability distribution?
ANSWER
82. What is probability of observing the sale of no more than 1 bat at this sporting goods store?
ANSWER:
83. What is the probability of observing the sale of at least one bat on a given day at this sporting goods
store?
ANSWER: 84. What is the probability of observing the sale of no more than two bats on a given day at this sporting
goods store?
ANSWER: 85. What is the probability of observing the sale of at least two bats on a given day at this sporting goods
store?
ANSWER86. What is the probability that a household answered no?
ANSWER: 87. How many females were interviewed?
ANSWER
88. What is the probability that a respondent was female?
ANSWER: 89. What is the probability that a respondent chosen at random is a male?
ANSWER
90. What is the probability that a respondent chosen at random enjoys shopping for clothing?
ANSWER:
91. What is the probability that a respondent chosen at random does not enjoy shopping for clothing?
ANSWER92. What is the probability that a respondent chosen at random is a female and enjoys shopping for clothing?
ANSWER
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93. What is the probability that a respondent chosen at random is a male and does not enjoy shopping for
clothing?
ANSWER: 94. What is the probability that a respondent chosen at random is a female and does not enjoy shopping for
clothing?
ANSWER: 95. What is the probability that a respondent chosen at random is a male who enjoys shopping for clothing or
a female who enjoys shopping for clothing?
ANSWER96. What is the probability that a respondent chosen at random is a male or a female?
ANSWER:
97. What is the probability that a respondent chosen at random enjoys or does not enjoy shopping for
clothing?
ANSWER98. Does consumer behavior depend on the gender of consumer? Explain using probabilities.
ANSWER99. What is the probability that a randomly selected patron is not male?
ANSWER: P(not male) = 1 - 0.7 = 0.3
100. What is the probability a randomly selected patron prefers wine?
ANSWER101. What is the probability a randomly selected patron is a female?
ANSWER102. What is the probability a randomly selected patron is a female who prefers wine?
ANSWER103. What is the probability a randomly selected patron is a female who prefers beer?
ANSWER: 104. Suppose a randomly selected patron prefers wine. What is the probability the patron is a male?
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105. Suppose a randomly selected patron prefers beer. What is the probability the patron is a male?
ANSWER
106. Suppose a randomly selected patron is a female. What is the probability the patron prefers beer?
ANSWER107. Suppose a randomly selected patron is a female. What is the probability that the patron prefers wine?
ANSWER
108. Are gender of patrons and drinking preference independent? Explain.
ANSWER: 109. What is the probability that any well will not be successful?
ANSWER110. What is the probability that none of the oil wells will be successful?
ANSWER111. If a new pipeline will be constructed in the event that all three wells are successful, what is the probability
that the pipeline will be constructed?
ANSWER112. How many of the wells can the company expect to be successful?
ANSWER113. If it costs $200,000 to drill each well and a successful well will produce $1,000,000 worth of oil over its
lifetime, what is the expected net value of this three-well program if no wells are successful?
ANSWER114. If it costs $200,000 to drill each well and a successful well will produce $1,000,000 worth of oil over its
lifetime, what is the expected net value of this three-well program if all three wells are successful?
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