Bayes’ Theorem
4.2 The Rules of Probability (pg. 165–188)
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4 Overview
• Probability is the way we measure uncertainty about events. To properly use probability, you need to know the probability rules and the terms associated with probability.
• Important probability concepts and terminology include sample space, dependent and independent events, and mutually exclusive events.
• Probabilities are assessed in three main ways: classical assessment, relative frequency assessment, and subjective assessment.
o u t c o m e 1 Identify situations for which each of the three approaches to assessing probabilities applies.
The Basics of Probability (pg. 153–165)
4.1
Summary
• Section 4.2 introduces nine probability rules, including three addition rules and two multiplication rules.
• Rules for conditional probability and the complement rule are also very useful.
• Bayes’ Theorem is used to calculate conditional probabilities in situations where the probability of the given event is not provided and must be calculated. Bayes’
Theorem is also used to revise prior probabilities after new information affecting the prior becomes available.
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Introduction to Probability(4.12) Probability Rule 9 pg. 179
Multiplication Rule for Independent Events E1 and E2: P1E1 and E22 = P1E12P1E22
(4.13) Bayes’ Theorem pg. 181
P(EiB) = P(Ei)P(BEi)
P(E1)P(BE1) + P(E2)P(BE2) + g + P(Ek)P(BEk) and
P(E2E1) = P(E2) P(E1) 7 0
(4.11) Probability Rule 8 pg. 177
Multiplication Rule for Any Two Events E1 and E2: P1E1 and E22 = P1E12P1E2E12
Key Terms
Classical probability assessment pg. 158 Complement pg. 168
Conditional probability pg. 173 Dependent events pg. 157
Event pg. 155 Experiment pg. 154 Independent events pg. 157 Mutually exclusive events pg. 156
Probability pg. 153
Relative frequency assessment pg. 159 Sample space pg. 154
Subjective probability assessment pg. 161
Chapter Exercises
Conceptual Questions4-59. Discuss what is meant by classical probability
assessment and indicate why classical assessment is not often used in business applications.
4-60. Discuss what is meant by the relative frequency assessment approach to probability assessment. Provide a business-related example, other than the one given in the text, in which this method of probability assessment might be used.
4-61. Discuss what is meant by subjective probability. Provide a business-related example in which subjective
probability assessment would likely be used. Also, provide an example of when you have personally used subjective probability assessment.
4-62. Examine the relationship between independent,
dependent, and mutually exclusive events. Consider two events A and B that are mutually exclusive such that P1A2 ≠ 0.
a. Calculate P1AB2.
b. What does your answer to part a say about whether two mutually exclusive events are dependent or independent?
c. Consider two events C and D such that P1C 2 = 0.4 and P1CD2 = 0.15. (1) Are events C and D mutually exclusive? (2) Are events C and D independent or dependent? Are dependent events necessarily mutually exclusive events?
4-63. Consider the following table:
A A Totals
B 800 200 1,000
B 600 400 1,000
Totals 1,400 600 2,000
Explore the complements of conditional events:
a. Calculate the following probabilities:
P1AB2, P1AB2, P1AB2, P1AB2. b. Now determine which pair of events are
complements of each other. (Hint: Use the probabilities calculated in part a and the Complement Rule.)
4-64. Examine the following table:
A A Totals
B 200 800 1,000
B 300 700 1,000
Totals 500 1,500 2,000
a. Calculate the following probabilities:
P1A2, P1A2, P1AB2, P1AB2, P1AB2, and P1AB2. b. Show that (1) A and B, (2) A and B (3) A and B,
and (4) A and B are dependent events.
Business Applications
4-65. An accounting professor at a state university in Vermont recently gave a three-question multiple-choice quiz.
Each question had four optional answers.
a. What is the probability of getting a perfect score if you were forced to guess at each question?
b. Suppose it takes at least two correct answers out of three to pass the test. What is the probability of passing if you are forced to guess at each question?
What does this indicate about studying for such an exam?
c. Suppose through some late-night studying you are able to correctly eliminate two answers on each question. Now answer parts a and b.
4-66. When customers call into a bank’s customer service phone line, they are asked to complete a survey on
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Chapter Exercises
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Chapter 4 191use credit or debit cards than is true for the nation as a whole? Explain.
4-70. Ponderosa Paint and Glass makes paint at three plants. It then ships the unmarked paint cans to a central
warehouse. Plant A supplies 50% of the paint, and past records indicate that the paint is incorrectly mixed 10%
of the time. Plant B contributes 30%, with paint mixed incorrectly 5% of the time. Plant C supplies 20%, with paint mixed incorrectly 20% of the time. If Ponderosa guarantees its product and spent $10,000 replacing improperly mixed paint last year, how should the cost be distributed among the three plants?
4-71. Recently, several long-time customers at the Sweet Haven Chocolate Company have complained about the quality of the chocolates. It seems there are several partially covered chocolates being found in boxes. The defective chocolates should have been caught when the boxes were packed. The manager is wondering which of the three packers is not doing the job properly. Clerk 1 packs 40% of the boxes and usually has a 2%
defective rate. Clerk 2 packs 30%, with a 2.5%
defective rate. Clerk 3 boxes 30% of the chocolates, and her defective rate is 1.5%. Which clerk is most likely responsible for the boxes that raised the complaints?
4-72. An investment firm is thinking of opening a skiing facility in Colorado. It is trying to decide whether to open an area catering to family skiers or to some other group. To help make its decision, it gathers the following information. Let
A1 = Family will ski A2 = Family will not ski
B1 = Family has children but none in the 8–16 age group
B2 = Family has children in the 8–16 age group B3 = Family has no children
Then, for this location, P1A12 = 0.40 P1B22 = 0.35 P1B12 = 0.25 P1A1B22 = 0.70 P1A1B12 = 0.30
a. Use the probabilities given to construct a joint probability distribution table.
b. What is the probability a family will ski and have children who are not in the 8–16 age group? How do you write this probability?
c. What is the probability a family with children in the 8–16 age group will not ski?
d. Are the categories skiing and family composition independent?
4-73. Fifty chief executive officers of small to medium-sized companies were classified according to their gender and functional background as shown in the table below:
service quality at the end of the call. Six out of ten customers agree to do the survey. Of those who agree, 58% actually respond to all the questions.
a. Calculate the probability that a randomly chosen customer who calls the service phone line will actually complete the survey.
b. Calculate the probability that a randomly chosen customer will not complete the survey.
c. If three customers are selected at random, what is the probability that only two of them will complete the survey?
4-67. An April 14, 2015, article by Kara Brandeisky on Money magazine’s website states that overall the chance of being audited by the IRS in 2014 was 0.86%. If a person’s adjusted gross income was between $50,000 and $74,999, the chance of being audited drops to 0.53%, and for those with incomes higher than $10 million, the chance goes up to 16.22%. (Source: “These are the people who are most likely to get audited,”
http://time.com/money, Apr. 14, 2015.)
a. Considering these probabilities, discuss which probability assessment method you believe was most likely used to arrive at the probabilities.
b. Based on chance alone and not taking income into account, what is the overall probability that you and your best friend will both be audited? (Assume the 2014 chances still apply.)
c. Suppose Harry made $13 million in 2014 and his daughter, Sarah, made $55,000. What is the probability that neither of them was audited by the IRS based on chance alone?
4-68. FM Auto Workshop is conducting a study for its two branches (FM1 and FM2) on oil changes and tire rotations. Two hundred customers were involved in the study. Thirty-six out of 119 oil change customers visited FM2. Thirty-five customers went to FM1 for their tire rotations.
a. Find the probability of customers visiting FM2.
b. Find the probability of customers having their oil change in FM1.
c. Determine whether the two branches are independent from the two types of services.
4-69. A national convenience store chain determines that 70%
of gas purchases are made with a credit or debit card.
a. Indicate the type of probability assessment method that the gas station manager would use to assess this probability.
b. In one local store, 10 randomly chosen customers were observed. All 10 of these customers used a credit or a debit card. If the 70% statistic applies to this area, determine the probability that 10 out of 10 customers would use a credit or debit card.
c. If 90% of gas purchases paid for at the pump were made with a credit or debit card, determine the probability that 10 out of 10 customers would use a credit or debit card.
d. Based on your answers to parts b and c, does it appear that a larger percentage of local individuals
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Introduction to Probabilitythree customers from the financial institution’s database,
a. All three manage their accounts online?
b. Exactly two manage their accounts online?
c. None manage online?
d. At least one manages online?
e. The third sampled customer is the first one to manage online?
4-77. Based on a survey conducted by students in the Recreational Management degree program, 65% of students at the university own a mountain bike, 25%
own a road bike, and 6% own both a mountain bike and a road bike.
a. What is the probability that a randomly selected student at this university has a mountain bike but not a road bike?
b. What is the probability that a randomly selected student at this university has a road bike but not a mountain bike?
c. What is the probability that a randomly selected student at this university has neither a road bike nor a mountain bike?
d. Given that a randomly selected student at this university has a road bike, what is the probability he or she also has a mountain bike?
4-78. After spending some time studying company e-mail accounts, the manager of computer security for a large insurance company has determined that 25% of all e-mail messages sent to employee computers are spam.
If 80% of all spam e-mails sent to personal accounts contain the word guarantee in the Subject line, and only 4% of nonspam e-mails contain the word guarantee, what is the probability that an employee’s next e-mail that contains the word guarantee in the Subject line is spam?
4-79. A large coffeehouse has found that 50% of all customers order food, 80% order coffee, and 40% order both. What is the probability that a customer at the coffeehouse a. Orders food but no coffee?
b. Orders coffee but no food?
c. Orders neither coffee nor food?
4-80. In a large shipment of electronic parts, 0.015 are bad.
The parts are tested using a machine that correctly identifies bad parts as defective with a probability of 0.98, and correctly identifies good parts as nondefective with a probability of 0.95.
a. If a part is randomly sampled from the shipment and tested, what is the probability that the testing machine identifies the part as defective?
b. Given that the test indicates a nondefective part, what is the probability the part is truly good?
4-81. In a large metropolitan area, 65% of commuters ride the train only, 18% ride the bus only, and 6% ride both the train and the bus to get to the central business district.
a. What is the probability that a randomly selected commuter does not ride the train to the central business district?
Functional Background Male Female Total
Marketing 4 10 14
Finance 11 5 16
Operations 17 3 20
Total 32 18 50
a. If a chief executive is randomly selected from this group, what is the probability that the executive is a female?
b. What is the probability that a randomly selected executive from this group is a male whose functional background is marketing?
c. Assume that an executive is selected and you are told that the executive’s functional background was in operations. What is the probability that this executive is a female?
d. Assume that an executive is selected and you are told that the executive is a female. What is the probability the executive’s functional area is marketing?
e. Are gender and functional background independent for this set of executives?
4-74. A manufacturing firm has two suppliers for an
electrical component used in its process: one in Mexico and one in China. The supplier in Mexico ships 82% of all the electrical components used by the firm and has a defect rate of 4%. The Chinese supplier ships 18% of the electrical components used by the firm and has a defect rate of 6%.
a. Calculate the probability that an electrical component is defective.
b. Suppose an electrical component is defective. What is the probability that component was shipped from Mexico? (Hint: Use Bayes’ Theorem.)
4-75. Five hundred smartphone owners were asked if they use their phones to download sports scores. The responses are summarized in the following table:
Yes No Total
Male 195 105 300
Female 70 130 200
Total 265 235 500
a. What is the probability that a randomly sampled respondent from this group has used his or her smartphone to download sports scores?
b. What is the probability that a randomly sampled respondent from this group is female?
c. Given that a randomly sampled respondent has not used the smartphone to download sports scores, what is the probability that he or she is male?
d. Are the responses Yes and No independent of the respondent’s gender?
4-76. Thirty-five percent of all customers of a large, national financial institution manage their accounts online.
What is the probability that in a random sample of
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