From the research the conditional probabilities are as follows:
P 1 A1 C 1 2 = 0.324
P 1 A1 C 2 2 = 0.103
Using these results, the overinvolvement ratio is as follows:
P 1 A1 C 1 2
0.324
=
= 3.15
0.103
P 1 A1 C 2 2
Based on this analysis, there is evidence to conclude that alcohol increases the probability of automobile crashes.
The overinvolvement ratio is a good example of how mathematical manipulations of
probabilities can be used to obtain results that are useful for business decisions. The wide
usage of automated methods of data collection, including bar code scanners, audience
segmentation, and census data on tapes and disks, provides the possibility to compute
many different probabilities, conditional probabilities, and overinvolvement ratios. As a
result, analyses similar to those presented in this chapter have become part of the daily
routine for marketing analysts and product managers.
Exercises
Basic Exercises
Basic Exercises 3.52–3.58 refer to Table 3.10.
3.52 What is the joint probability that a Doctoral student
prefers Movies /Series?
3.53 What is the joint probability that a Doctoral student
prefers Documentaries?
3.54 What is the joint probability that a Masters student
prefers watching News?
3.55 What is the joint probability that a Bachelors student
prefers Documentaries?
3.56 What is the conditional probability that a student who
prefers watching News is a Bachelors student?
3.57 What is the conditional probability a student who prefers Movies/Series is a Bachelors student?
3.58 What is the conditional probability that a Masters student prefers Documentaries?
3.59 The probability of a sale is 0.80. What are the odds in
favor of a sale?
3.60 The probability of a sale is 0.50. What are the odds in
favor of a sale?
3.61 Consider two groups of students: B1, students who
received high scores on tests, and B2, students who
received low scores on tests. In group B1, 60% study
more than 25 hours per week, and in group B2, 35%
study more than 25 hours per week. What is the overinvolvement ratio for high study levels in high test
scores over low test scores?
3.62 Consider two groups of athletes: A1, athletes who e­ xcel
at football, and A2, the athletes who do not excel at
football. In A1, 55% of the athletes jog in the m
­ orning
for more than three days a week, and in A2, 23% jog in
the morning for more than three days a week. What
is the over-involvement ratio for an athlete to excel at
football for A1 over A2?
3.63 Consider two groups of students: B1, students who
received high scores on tests, and B2, students who
received low scores on tests. In group B1, 20% study
more than 25 hours per week, and in group B2, 40%
study more than 25 hours per week. What is the overinvolvement ratio for high study levels in high test
scores over low test scores?
Table 3.10 Probabilities for Television Preferences and Degree Programs:
Tv Preferences
bachelors
Masters
Doctoral
Totals
News
0.15
0.21
0.09
0.45
Movies/Series
0.14
0.07
0.05
0.26
Documentaries
0.08
0.15
0.06
0.29
Totals
0.37
0.43
0.20
1.00
Exercises
133
Application Exercises
3.64 A survey carried out for a supermarket classified customers according to whether their visits to the store
are frequent or infrequent and whether they often,
sometimes, or never purchase generic products. The
accompanying table gives the proportions of people
surveyed in each of the six joint classifications.
Frequency of
Visit
Frequent
Purchase of Generic
Products
Often
Sometimes
0.14
0.42
Infrequent
0.08
0.04
Never
0.16
0.09
a. What is the probability that a customer both is
a frequent shopper and often purchases generic
products?
b. What is the probability that a customer who never
buys generic products visits the store frequently?
c. Are the events “never buys generic products” and
“visits the store frequently” independent?
d. What is the probability that a customer who
infrequently visits the store often buys generic
products?
e. Are the events “often buys generic products” and
“visits the store infrequently” independent?
f. What is the probability that a customer frequently
visits the store?
g. What is the probability that a customer never buys
generic products?
h. What is the probability that a customer either frequently visits the store or never buys generic products or both?
3.65 An analyst at an airline company wants to predict
whether the number of passengers for the various cabin classes will decrease, remain the same, or
­increase over the next year. The following table shows
predictions based on the past records.
Prediction
Decrease
Economy class
0.14
Remains
the same
0.30
Business class
0.09
0.12
0.04
First class
0.10
0.02
0.14
Increase
0.05
a. What is the probability that the number of passengers in the economy class remains the same?
b. What is the probability that the number of passengers decrease?
c. If the number of passengers increased, what is the
probability that this would happen for business
class passengers?
d. If the number of passengers for first class is considered, what is the probability that it will decrease?
3.66 Subscribers to a local newspaper were asked whether
they regularly, occasionally, or never read the business section and also whether they had traded common stocks (or shares in a mutual fund) over the last
134
Chapter 3
Probability
year. The table shown here indicates the proportions
of subscribers in six joint classifications.
Traded
Stocks
Yes
Read Business Section
Regularly
0.18
Occasionally
0.10
Never
0.04
0.16
0.31
0.21
No
a. What is the probability that a randomly chosen
subscriber never reads the business section?
b. What is the probability that a randomly chosen
subscriber has traded stocks over the last year?
c. What is the probability that a subscriber who never
reads the business section has traded stocks over
the last year?
d. What is the probability that a subscriber who
traded stocks over the last year never reads the
business section?
e. What is the probability that a subscriber who does
not regularly read the business section traded stocks
over the last year?
3.67 A corporation regularly takes deliveries of a particular sensitive part from three subcontractors. It found
that the proportion of parts that are good or defective
from the total received were as shown in the following
table:
Part
Good
A
0.27
Defective
Subcontractor
B
0.30
0.02
0.05
C
0.33
0.03
a. If a part is chosen randomly from all those received,
what is the probability that it is defective?
b. If a part is chosen randomly from all those
­received, what is the probability it is from
subcontractor B?
c. What is the probability that a part from subcontractor B is defective?
d. What is the probability that a randomly chosen
­defective part is from subcontractor B?
e. Is the quality of a part independent of the source of
supply?
f. In terms of quality, which of the three subcontractors
is most reliable?
3.68 A form was circulated among the employees at the
Sinopec Group, one of the world’s largest oil refining, gas and petrochemical conglomerate, asking
them if they are planning to apply for leaves in August (Yes/No) and their preferred location. The following table gives proportions of employees and
their preferences.
Destination Type
Response
Yes
Coastal
Resort
0.11
Mountain
Resort
0.13
Thermal
Spa
0.07
Rural
Cottage
0.01
No
0.25
0.32
0.08
0.03
a. Find the probability that the preferred destination for a randomly chosen employee is a rural
Cottage.
b. What is the probability that a randomly chosen
employee will apply for leave in August?
c. Find the probability that the preferred destination
for a randomly chosen employee who will not apply for leave in August is a coastal resort.
d. If a randomly chosen employee’s preferred destination is a rural cottage, what is the probability
that they will take apply for leave in August?
e. If a randomly chosen employee plans to apply
for leave in August, what is the probability that
their preferred destination is not a mountain
resort?
f. Are the “vacation time” and “destination type” statistically independent?
3.69 The accompanying table shows proportions of computer salespeople classified according to marital status
and whether they left their jobs or stayed over a period of 1 year.
Time on job
Ú one year
6 one year
0.64
0.13
Marital Status
Married
Single
0.17
0.06
a. What is the probability that a randomly chosen
salesperson was married?
b. What is the probability that a randomly chosen
salesperson left the job within the year?
c. What is the probability that a randomly chosen
single salesperson left the job within the year?
d. What is the probability that a randomly chosen
salesperson who stayed in the job over the year was
married?
3.70 The accompanying table shows proportions of adults
in metropolitan areas, categorized as to whether they
are public-radio contributors and whether or not they
voted in the last election.
Voted
Yes
No
Contributors
0.63
Noncontributors
0.13
0.14
0.10
a. What is the probability that a randomly chosen
adult from this population voted?
b. What is the probability that a randomly chosen
adult from this population contributes to public
radio?
c. What is the probability that a randomly chosen adult
from this population did not contribute and did not
vote?
3.71 At a business school in London, a student club distributes material about membership to new students
­attending an orientation meeting. Of those receiving
this material 66% were women and 42% were men.
Subsequently, it was found that 10% of the women
and 8% of the men who received this material joined
the club.
a. What is the probability that a randomly chosen
new student who receives the membership material will join the club?
b. What is the probability that a randomly chosen new
student who joins the club after receiving the membership material is a woman?
3.72 An analyst attempting to predict a corporation’s earnings next year believes that the corporation’s business
is quite sensitive to the level of interest rates. He believes that, if average rates in the next year are more
than 1% higher than this year, the probability of significant earnings growth is 0.1. If average rates next
year are more than 1% lower than this year, the probability of significant earnings growth is estimated to
be 0.8. Finally, if average interest rates next year are
within 1% of this year’s rates, the probability for significant earnings growth is put at 0.5. The analyst estimates that the probability is 0.25 that rates next year
will be more than 1% higher than this year and 0.15
that they will be more than 1% lower than this year.
a. What is the estimated probability that both interest rates will be 1% higher and significant earnings
growth will result?
b. What is the probability that this corporation will
experience significant earnings growth?
c. If the corporation exhibits significant earnings
growth, what is the probability that interest rates
will have been more than 1% lower than in the current year?
3.73 Fourty-four percent of a corporation’s blue-collar employees were in favor of a modified health care plan,
and 24% of its blue-collar employees favored a proposal to change the work schedule. Thirty percent of
those favoring the health care plan modification, favored the work schedule change.
a. What is the probability that a randomly selected
blue-collar employee is in favor of both the modified
health care plan and the changed work schedule?
b. What is the probability that a randomly chosen
blue-collar employee is in favor of at least one of
the two changes?
c. What is the probability that a blue-collar employee
favoring the work schedule change also favors the
modified health care plan?
3.74 Suppose the students at the Lebanese University,
­Beirut, can attend advanced math (precalculus) classes
during their final year. Records show that among the
students who took precalculus during their final year,
65% of them were at the top of the University’s math
class, and 45% were in the middle quarter, and 35%
were in the bottom of the math class.
a. What is the probability that a randomly chosen
student took the precalculus during the final year?
b. What is the probability that a randomly chosen
student who attended the precalculus classes will
be a toper of the class?
c. What is the probability that a randomly chosen student who did not attend the precalculus classes will
not be a top quarter of the class?
Exercises
135
3.75 Before books aimed at preschool children are marketed, reactions are obtained from a panel of preschool
children. These reactions are categorized as favorable,
neutral, or unfavorable. Subsequently, book sales are
categorized as high, moderate, or low, according to
the norms of this market. Similar panels have evaluated 1,000 books in the past. The accompanying table
shows their reactions and the resulting market performance of the books.
Sales
High
Favorable
173
Panel Reaction
Neutral
Unfavorable
101
61
Moderate
88
211
70
Low
42
113
141
a. If the panel reaction is favorable, what is the probability that sales will be high?
b. If the panel reaction is unfavorable, what is the
probability that sales will be low?
c. If the panel reaction is neutral or better, what is the
probability that sales will be low?
d. If sales are low, what is the probability that the panel
reaction was neutral or better?
3.76 A manufacturer produces boxes of candy, each containing 10 pieces. Two machines are used for this
purpose. After a large batch has been produced, it is
discovered that one of the machines, which produces
40% of the total output, has a fault that has led to the
introduction of an impurity into 10% of the pieces
of candy it makes. The other machine produced no
defective pieces. From a single box of candy, one
piece is selected at random and tested. If that piece
contains no impurity, what is the probability that
the faulty machine produced the box from which it
came?
3.77 A student feels that 70% of her college courses have
been enjoyable and the remainder have been boring. This student has access to student evaluations
of professors and finds out that professors who had
previously received strong positive evaluations
from their students have taught 60% of his enjoyable courses and 25% of his boring courses. Next semester the student decides to take three courses, all
from professors who have received strongly positive
student evaluations. Assume that this student’s reactions to the three courses are independent of one
another.
a. What is the probability that this student will find
all three courses enjoyable?
b. What is the probability that this student will find at
least one of the courses enjoyable?
3.5 B ayes ’ T heorem
In this section we introduce an important result that has many applications to management decision making. Bayes’ theorem provides a way of revising conditional probabilities by using available information. It also provides a procedure for determining how
probability statements should be adjusted, given additional information.
Reverend Thomas Bayes (1702–1761) developed Bayes’ theorem, originally published
in 1763 after his death and again in 1958 (Bayes 1958). Because games of chance—and,
hence, probability—were considered to be works of the devil, the results were not widely
publicized. Since World War II a major area of statistics and a major area of management
decision theory have developed based on the original works of Thomas Bayes. We begin
our development with an example problem, followed by a more formal development.
Example 3.23 Drug Screening (Bayes’ Theorem)
A number of amateur and professional sports organizations use routine screening tests
to determine if athletes are using performance-enhancing drugs. Jennifer Smith, president of an amateur athletic union, has asked you to determine the feasibility of screening athletes to determine if they are using performance-enhancing drugs. Amateur
athletes are increasingly denied participation or deprived of victories if they are found
to be users.
As part of the study, you propose the following scenario for analysis. Suppose that
10% of the athletes seeking participation in the athletic union have used performanceenhancing drugs. In addition, suppose that a test is available that correctly identifies an
athlete’s drug usage 90% of the time. If an athlete is a drug user, the probability is 0.90
that the athlete is correctly identified by the test as a drug user. Similarly, if the athlete
136
Chapter 3
Probability