Quantitative Analysis for Management, 11e (Render)
Chapter 2 Probability Concepts and Applications
1) Subjective probability implies that we can measure the relative frequency of the values of the random variable.
2) The use of “expert opinion” is one way to approximate subjective probability values.
3) Mutually exclusive events exist if only one of the events can occur on any one trial.
4) Stating that two events are statistically independent means that the probability of one event occurring is
independent of the probability of the other event having occurred.
5) Saying that a set of events is collectively exhaustive implies that one of the events must occur.
6) Saying that a set of events is mutually exclusive and collectively exhaustive implies that one and only one of
the events can occur on any trial.
7) A posterior probability is a revised probability.
8) Bayes’ theorem enables us to calculate the probability that one event takes place knowing that a second event
has or has not taken place.
9) A probability density function is a mathematical way of describing Bayes’ theorem.
10) The probability, P, of any event or state of nature occurring is greater than or equal to 0 and less than or equal
to 1.
11) A probability is a numerical statement about the chance that an event will occur.
12) If two events are mutually exclusive, the probability of both events occurring is simply the sum of the
individual probabilities.
13) Given two statistically dependent events (A,B), the conditional probability of P(A|B) = P(B)/P(AB).
15) Given three statistically independent events (A,B,C), the joint probability of P(ABC) = P(A) × P(B) × P(C).
16) Given two statistically independent events (A,B), the conditional probability P(A|B) = P(A).
17) Suppose that you enter a drawing by obtaining one of 20 tickets that have been distributed. By using the
classical method, you can determine that the probability of your winning the drawing is 0.05.
18) Assume that you have a box containing five balls: two red and three white. You draw a ball two times, each
time replacing the ball just drawn before drawing the next. The probability of drawing only one white ball is
0.20.
19) If we roll a single die twice, the probability that the sum of the dots showing on the two rolls equals four (4), is
1/6.
20) For two events A and B that are not mutually exclusive, the probability that either A or B will occur is P(A) ×
P(B) – P(A and B).
21) If we flip a coin three times, the probability of getting three heads is 0.125.
22) Consider a standard 52–card deck of cards. The probability of drawing either a seven or a black card is 7/13.
23) If a bucket has three black balls and seven green balls, and we draw balls without replacement, the probability
of drawing a green ball is independent of the number of balls previously drawn.
24) Assume that you have an urn containing 10 balls of the following description:
4 are white (W) and lettered (L)
2 are white (W) and numbered (N)
3 are yellow (Y) and lettered (L)
1 is yellow (Y) and numbered (N)
If you draw a numbered ball (N), the probability that this ball is white (W) is 0.667.
25) Assume that you have an urn containing 10 balls of the following description:
4 are white (W) and lettered (L)
2 are white (W) and numbered (N)
3 are yellow (Y) and lettered (L)
1 is yellow (Y) and numbered (N)
If you draw a numbered ball (N), the probability that this ball is white (W) is 0.60.
26) Assume that you have an urn containing 10 balls of the following description:
4 are white (W) and lettered (L)
2 are white (W) and numbered (N)
3 are yellow (Y) and lettered (L)
1 is yellow (Y) and numbered (N)
If you draw a lettered ball (L), the probability that this ball is white (W) is 0.571.
27) The joint probability of two or more independent events occurring is the sum of their marginal or simple
probabilities.
28) The number of bad checks written at a local store is an example of a discrete random variable.
29) Given the following distribution:
Outcome
Value of
Random Variable
Probability
A
1
.4
B
2
.3
C
3
.2
D
4
.1
The expected value is 3.
30) A new young executive is perplexed at the number of interruptions that occur due to employee relations. She
has decided to track the number of interruptions that occur during each hour of her day. Over the last month, she
has determined that between 0 and 3 interruptions occur during any given hour of her day. The data is shown
below.
Number of Interruptions in 1 hour
Probability
0 interruption
.5
1 interruptions
.3
2 interruptions
.1
3 interruptions
.1
On average, she should expect 0.8 interruptions per hour.
31) A new young executive is perplexed at the number of interruptions that occur due to employee relations. She
has decided to track the number of interruptions that occur during each hour of her day. Over the last month, she
has determined that between 0 and 3 interruptions occur during any given hour of her day. The data is shown
below.
Number of Interruptions in 1 hour
Probability
0 interruption
.4
1 interruptions
.3
2 interruptions
.2
3 interruptions
.1
On average, she should expect 1.0 interruptions per hour.
32) The expected value of a binomial distribution is expressed as np, where n equals the number of trials and p
equals the probability of success of any individual trial.
33) The standard deviation equals the square of the variance.
34) The probability of obtaining specific outcomes in a Bernoulli process is described by the binomial probability
distribution.
35) The variance of a binomial distribution is expressed as np/(1–p), where n equals the number of trials and p
equals the probability of success of any individual trial.
36) The F distribution is a continuous probability distribution that is helpful in testing hypotheses about
variances.
37) The mean and standard deviation of the Poisson distribution are equal.
38) In a normal distribution the Z value represents the number of standard deviations from a value X to the mean.
39) Assume you have a normal distribution representing the likelihood of completion times. The mean of this
distribution is 10, and the standard deviation is 3. The probability of completing the project in 8 or fewer days is
the same as the probability of completing the project in 18 days or more.
40) Assume you have a normal distribution representing the likelihood of completion times. The mean of this
distribution is 10, and the standard deviation is 3. The probability of completing the project in 7 or fewer days is
the same as the probability of completing the project in 13 days or more.
41) The classical method of determining probability is
A) subjective probability.
B) marginal probability.
C) objective probability.
D) joint probability.
E) conditional probability.
42) Subjective probability assessments depend on
A) the total number of trials.
B) the relative frequency of occurrence.
C) the number of occurrences of the event.
D) experience and judgment.
E) None of the above
43) If two events are mutually exclusive, then
A) their probabilities can be added.
B) they may also be collectively exhaustive.
C) the joint probability is equal to 0.
D) if one occurs, the other cannot occur.
E) All of the above
44) A ________ is a numerical statement about the likelihood that an event will occur.
A) mutually exclusive construct
B) collectively exhaustive construct
C) variance
D) probability
E) standard deviation
45) A conditional probability P(B|A) is equal to its marginal probability P(B) if
A) it is a joint probability.
B) statistical dependence exists.
C) statistical independence exists.
D) the events are mutually exclusive.
E) P(A) = P(B).
Topic: STATISTICALLY INDEPENDENT EVENTS
46) The equation P(A|B) = P(AB)/P(B) is
A) the marginal probability.
B) the formula for a conditional probability.
C) the formula for a joint probability.
D) only relevant when events A and B are collectively exhaustive.
E) None of the above
47) Suppose that we determine the probability of a warm winter based on the number of warm winters
experienced over the past 10 years. In this case, we have used ________.
A) relative frequency
B) the classical method
C) the logical method
D) subjective probability
E) None of the above
48) Bayes’ theorem is used to calculate
A) revised probabilities.
B) joint probabilities.
C) prior probabilities.
D) subjective probabilities.
E) marginal probabilities.
49) If the sale of ice cream and pizza are independent, then as ice cream sales decrease by 60 percent during the
winter months, pizza sales will
A) increase by 60 percent.
B) increase by 40 percent.
C) decrease by 60 percent.
D) decrease by 40 percent.
E) be unrelated.
50) If P(A) = 0.3, P(B) = 0.2, P(A and B) = 0.0 , what can be said about events A and B?
A) They are independent.
B) They are mutually exclusive.
C) They are posterior probabilities.
D) None of the above
E) All of the above
51) Suppose that 10 golfers enter a tournament and that their respective skill levels are approximately the same.
What is the probability that one of the first three golfers that registered for the tournament will win?
A) 0.100
B) 0.001
C) 0.300
D) 0.299
E) 0.700
52) Suppose that 10 golfers enter a tournament and that their respective skill levels are approximately the same.
Six of the entrants are female and two of those are older than 40 years old. Three of the men are older than 40
years old. What is the probability that the winner will be either female or older than 40 years old?
A) 0.000
B) 1.100
C) 0.198
D) 0.200
E) 0.900
53) Suppose that 10 golfers enter a tournament and that their respective skill levels are approximately the same.
Six of the entrants are female and two of those are older than 40 years old. Three of the men are older than 40
years old. What is the probability that the winner will be a female who is older than 40 years old?
A) 0.000
B) 1.100
C) 0.198
D) 0.200
E) 0.900
54) “The probability of event B, given that event A has occurred” is known as a ________ probability.
A) continuous
B) marginal
C) simple
D) joint
E) conditional
55) When does P(A|B) = P(A)?
A) when A and B are mutually exclusive
B) when A and B are statistically independent
C) when A and B are statistically dependent
D) when A and B are collectively exhaustive
E) when P(B) = 0
56) A consulting firm has received 2 Super Bowl playoff tickets from one of its clients. To be fair, the firm is
randomly selecting two different employee names to “win” the tickets. There are 6 secretaries, 5 consultants and 4
partners in the firm. Which of the following statements is not true?
A) The probability of a secretary winning a ticket on the first draw is 6/15.
B) The probability of a secretary winning a ticket on the second draw given that a consultant won a ticket on the
first draw is 6/15.
C) The probability of a consultant winning a ticket on the first draw is 1/3.
D) The probability of two secretaries winning both tickets is 1/7.
E) The probability of a partner winning a ticket on the second draw given that a secretary won a ticket on the first
draw is 4/14.
57) A consulting firm has received 2 Super Bowl playoff tickets from one of its clients. To be fair, the firm is
randomly selecting two different employee names to “win” the tickets. There are 6 secretaries, 5 consultants, and 4
partners in the firm. Which of the following statements is true?
A) The probability of a partner winning on the second draw given that a partner won on the first draw is 3/14.
B) The probability of a secretary winning on the second draw given that a secretary won on the first draw is 2/15.
C) The probability of a consultant winning on the second draw given that a consultant won on the first draw is
5/14.
D) The probability of a partner winning on the second draw given that a secretary won on the first draw is 8/30.
E) None of the above are true.
58) A consulting firm has received 2 Super Bowl playoff tickets from one of its clients. To be fair, the firm is
randomly selecting two different employee names to “win” the tickets. There are 6 secretaries, 5 consultants, and 4
partners in the firm. Which of the following statements is true?
A) The probability of two secretaries winning is the same as the probability of a secretary winning on the second
draw given that a consultant won on the first draw.
B) The probability of a secretary and a consultant winning is the same as the probability of a secretary and
secretary winning.
C) The probability of a secretary winning on the second draw given that a consultant won on the first draw is the
same as the probability of a consultant winning on the second draw given that a secretary won on the first draw.
D) The probability that both tickets will be won by partners is the same as the probability that a consultant and
secretary will win.
E) None of the above are true.
59) At a university with 1,000 business majors, there are 200 business students enrolled in an introductory
statistics course. Of these 200 students, 50 are also enrolled in an introductory accounting course. There are an
additional 250 business students enrolled in accounting but not enrolled in statistics. If a business student is
selected at random, what is the probability that the student is either enrolled in accounting or statistics, but not
both?
A) 0.45
B) 0.50
C) 0.40
D) 0.05
E) None of the above
60) At a university with 1,000 business majors, there are 200 business students enrolled in an introductory
statistics course. Of these 200 students, 50 are also enrolled in an introductory accounting course. There are an
additional 250 business students enrolled in accounting but not enrolled in statistics. If a business student is
selected at random, what is the probability that the student is enrolled in accounting?
A) 0.20
B) 0.25
C) 0.30
D) 0.50
E) None of the above
61) At a university with 1,000 business majors, there are 200 business students enrolled in an introductory
statistics course. Of these 200 students, 50 are also enrolled in an introductory accounting course. There are an
additional 250 business students enrolled in accounting but not enrolled in statistics. If a business student is
selected at random, what is the probability that the student is enrolled in statistics?
A) 0.05
B) 0.20
C) 0.25
D) 0.30
E) None of the above
62) At a university with 1,000 business majors, there are 200 business students enrolled in an introductory
statistics course. Of these 200 students, 50 are also enrolled in an introductory accounting course. There are an
additional 250 business students enrolled in accounting but not enrolled in statistics. If a business student is
selected at random, what is the probability that the student is enrolled in both statistics and accounting?
A) 0.05
B) 0.06
C) 0.20
D) 0.25
E) None of the above
63) At a university with 1,000 business majors, there are 200 business students enrolled in an introductory
statistics course. Of these 200 students, 50 are also enrolled in an introductory accounting course. There are an
additional 250 business students enrolled in accounting but not enrolled in statistics. If a business student is
selected at random and found to be enrolled in statistics, what is the probability that the student is also enrolled
in accounting?
A) 0.05
B) 0.30
C) 0.20
D) 0.25
E) None of the above
64) Suppose that when the temperature is between 35 and 50 degrees, it has historically rained 40% of the time.
Also, historically, the month of April has had a temperature between 35 and 50 degrees on 25 days. You have
scheduled a golf tournament for April 12. What is the probability that players will experience rain and a
temperature between 35 and 50 degrees?
A) 0.333
B) 0.400
C) 0.833
D) 1.000
E) 0.480
65) Suppose that, historically, April has experienced rain and a temperature between 35 and 50 degrees on 20
days. Also, historically, the month of April has had a temperature between 35 and 50 degrees on 25 days. You
have scheduled a golf tournament for April 12. If the temperature is between 35 and 50 degrees on that day, what
will be the probability that the players will get wet?
A) 0.333
B) 0.667
C) 0.800
D) 1.000
E) 0.556
66) At a university with 1,000 business majors, there are 200 business students enrolled in an introductory
statistics course. Of these 200, 50 are also enrolled in an introductory accounting course. There are an additional
250 business students enrolled in accounting but not enrolled in statistics. If a business student is selected at
random, what is the probability that the student is enrolled in neither accounting nor statistics?
A) 0.45
B) 0.50
C) 0.55
D) 0.05
E) None of the above
67) At a university with 1,000 business majors, there are 200 business students enrolled in an introductory
statistics course. Of these 200, 50 are also enrolled in an introductory accounting course. There are an additional
250 business students enrolled in accounting but not enrolled in statistics. If a business student is selected at
random, what is the probability that the student is not enrolled in accounting?
A) 0.20
B) 0.25
C) 0.30
D) 0.50
E) None of the above
68) At a university with 1,000 business majors, there are 200 business students enrolled in an introductory
statistics course. Of these 200, 50 are also enrolled in an introductory accounting course. There are an additional
250 business students enrolled in accounting but not enrolled in statistics. If a business student is selected at
random, what is the probability that the student is not enrolled in statistics?
A) 0.05
B) 0.20
C) 0.25
D) 0.80
E) None of the above
69) A production process is known to produce a particular item in such a way that 5 percent of these are
defective. If two items are randomly selected as they come off the production line, what is the probability that the
second item will be defective?
A) 0.05
B) 0.005
C) 0.18
D) 0.20
E) None of the above
70) A production process is known to produce a particular item in such a way that 5 percent of these are
defective. If two items are randomly selected as they come off the production line, what is the probability that
both are defective (assuming that they are independent)?
A) 0.0100
B) 0.1000
C) 0.2000
D) 0.0025
E) 0.0250
71) A company is considering producing some new Gameboy electronic games. Based on past records,
management believes that there is a 70 percent chance that each of these will be successful and a 30 percent
chance of failure. Market research may be used to revise these probabilities. In the past, the successful products
were predicted to be successful based on market research 90 percent of the time. However, for products that
failed, the market research predicted these would be successes 20 percent of the time. If market research is
performed for a new product, what is the probability that the results indicate a successful market for the product
and the product is actually not successful?
A) 0.63
B) 0.06
C) 0.07
D) 0.24
E) 0.27
72) A company is considering producing some new Gameboy electronic games. Based on past records,
management believes that there is a 70 percent chance that each of these will be successful and a 30 percent
chance of failure. Market research may be used to revise these probabilities. In the past, the successful products
were predicted to be successful based on market research 90 percent of the time. However, for products that
failed, the market research predicted these would be successes 20 percent of the time. If market research is
performed for a new product, what is the probability that the results indicate an unsuccessful market for the
product and the product is actually successful?
A) 0.63
B) 0.06
C) 0.07
D) 0.24
E) 0.21
73) A company is considering producing some new Gameboy electronic games. Based on past records,
management believes that there is a 70 percent chance that each of these will be successful and a 30 percent
chance of failure. Market research may be used to revise these probabilities. In the past, the successful products
were predicted to be successful based on market research 90 percent of the time. However, for products that
failed, the market research predicted these would be successes 20 percent of the time. If market research is
performed for a new product, what is the probability that the results indicate an unsuccessful market for the
product and the product is actually unsuccessful?
A) 0.63
B) 0.06
C) 0.07
D) 0.24
E) 0.21
74) A company is considering producing some new Gameboy electronic games. Based on past records,
management believes that there is a 70 percent chance that each of these will be successful, and a 30 percent
chance of failure. Market research may be used to revise these probabilities. In the past, the successful products
were predicted to be successful based on market research 90 percent of the time. However, for products that
failed, the market research predicted these would be successes 20 percent of the time. If market research is
performed for a new product, what is the probability that the product will be successful if the market research
indicates a success?
A) 0.10
B) 0.90
C) 0.91
D) 0.63
E) 0.09
75) A dry cleaning business offers a pick–up and delivery service for a 10 percent surcharge. Management believes
60 percent of customers will take advantage of this service. They are also considering offering customers the
option of opening an account and receiving monthly bills. They believe 60 percent of their customers (regardless
of whether or not they use the pick–up service) will use the account service. If the two services are introduced to
the market, what is the probability a customer uses both services?
A) 0.12
B) 0.60
C) 0.36
D) 0.24
E) None of the above
76) A dry cleaning business offers a pick–up and delivery service for a 10 percent surcharge. Management believes
60 percent of the existing customers will take advantage of this service. They are also considering offering
customers the option of opening an account and receiving monthly bills. They believe 60 percent of customers
(regardless of whether or not they use the pick–up service) will use the account service. If the two services are
introduced to the market, what is the probability that a customer uses only one of these services?
A) 0.40
B) 0.60
C) 0.48
D) 0.24
E) None of the above
77) A dry cleaning business offers a pick–up and delivery service for a 10 percent surcharge. Management believes
60 percent of the existing customers will take advantage of this service. They are also considering offering
customers the option of opening an account and receiving monthly bills. They believe 60 percent of customers
(regardless of whether or not they use the pick–up service) will use the account service. If the two services are
introduced to the market, what is the probability a customer uses neither of these services?
A) 0.16
B) 0.24
C) 0.80
D) 0.36
E) None of the above
78) A company is considering producing some new Gameboy electronic games. Based on past records,
management believes that there is a 70 percent chance that each of these will be successful and a 30 percent
chance of failure. Market research may be used to revise these probabilities. In the past, the successful products
were predicted to be successful based on market research 90 percent of the time. However, for products that
failed, the market research predicted these would be successes 20 percent of the time. If market research is
performed for a new product, what is the probability that the product will be successful if the market research
indicates a failure?
A) 0.20
B) 0.90
C) 0.91
D) 0.63
E) 0.23