138 Chapter 3 Elements of Chance: Probability Methods
Key Words 139 3.81 Given P1A12 = 0.40, P1B1u A12 = 0.60, and
P1B1u A22 = 0.70, what is the probability of P1A2u B22?
3.82 Given P1A12 = 0.60, P1B1u A12 = 0.60, and P1B1u A22 = 0.40, what is the probability of P1A1u B12?
Application Exercises
3.83 A publisher sends advertising materials for an ac-counting text to 80% of all professors teaching the appropriate accounting course. Thirty percent of the professors who received this material adopted the book, as did 10% of the professors who did not receive the material. What is the probability that a professor who adopts the book has received the advertising material?
3.84 A stock market analyst examined the prospects of the shares of a large number of corporations. When the performance of these stocks was investigated one year later, it turned out that 25% performed much better than the market average, 25%, much worse, and the remaining 50%, about the same as the aver-age. Forty percent of the stocks that turned out to do much better than the market were rated good buys by the analyst, as were 20% of those that did about as well as the market and 10% of those that did much worse. What is the probability that a stock rated a good buy by the analyst performed much better than the average?
3.85 The Watts New Lightbulb Corporation ships large consignments of lightbulbs to big industrial users.
When the production process is functioning cor-rectly, which is 90% of the time, 10% of all bulbs produced are defective. However, the process is
susceptible to an occasional malfunction, leading to a defective rate of 50%. If a defective bulb is found, what is the probability that the process is func-tioning correctly? If a nondefective bulb is found, what is the probability that the process is operating correctly?
3.86 You are the meat products manager for Gigantic Foods, a large retail supermarket food distribu-tor who is studying the characteristics of its whole chicken product mix. Chickens are purchased from both Free Range Farms and Big Foods Ltd. Free Range Farms produces chickens that are fed with natural grains and grubs in open feeding areas.
In their product mix, 10% of the processed chick-ens weigh less than 3 pounds. Big Foods Ltd. pro-duces chickens in cages using enriched food grains for rapid growth. They note that 20% of their pro-cessed chickens weigh less than three poounds.
Gigantic Foods purchases 40% of its chickens from Free Range Farms and mixes the products together with no identification of the supplier. Suppose you purchase a chicken that weighs more than three pounds. What is the probability the chicken came from Free Range Farms? If you purchase 5 chickens, what is the probability that at least 3 came from Free Range Farms?
3.87 You and a friend are big soccer fans and are debating the possibility that FC Barcelona will win the final of the UEFA Champions League against Manchester United. You are supporting Manchester United, but your friend tells you that the bookmakers have given the following odds for the game: 2:8 (Manchester United vs. FC Barcelona). What is the probability that Manchester United will win?
K
EYW
ORDS• addition rule of probabilities, 112
• basic outcomes, 95
• Bayes’ theorem, 134
• Bayes’ theorem (alternative statement), 135
• classical probability, 101
• collectively exhaustive, 98
• combinations, 104
• complement, 98
• complement rule, 111
• conditional probability, 113
• event, 96
• independent events, 125
• intersection, 96
• joint probability, 96
• marginal probabilities, 123
• multiplication rule of probabilities, 114
• mutually exclusive, 96
• number of combinations, 102
• odds, 126
• overinvolvement ratio, 127
• permutations, 103
• probability postulates, 107
• random experiment, 94
• relative frequency probability, 106
• sample space, 95
• solution steps for Bayes’
theorem, 134
• statistical independence, 116
• subjective probability, 107
• union, 97
140 Chapter 3 Elements of Chance: Probability Methods
C
HAPTERE
XERCISES ANDA
PPLICATIONS 3.88 Suppose that you have an intelligent friend who hasnot studied probability. How would you explain to your friend the distinction between mutually exclu-sive events and independent events? Illustrate your answer with suitable examples.
3.89 State, with evidence, whether each of the following statements is true or false:
a. The complement of the union of two events is the intersection of their complements.
b. The sum of the probabilities of collectively exhaus-tive events must equal 1.
c. The number of combinations of x objects cho-sen from n is equal to the number of combina-tions of 1n - x2 objects chosen from n, where 1… x … 1n - 12.
d. If A and B are two events, the probability of A, given B, is the same as the probability of B, given A, if the probability of A is the same as the prob-ability of B.
e. If an event and its complement are equally likely to occur, the probability of that event must be 0.5.
f. If A and B are independent, then A and B must be independent.
g. If A and B are mutually exclusive, then A and B must be mutually exclusive.
3.90 Explain carefully the meaning of conditional probabil-ity. Why is this concept important in discussing the chance of an event’s occurrence?
3.91 Bayes’ theorem is important because it provides a rule for moving from a prior probability to a poste-rior probability. Elaborate on this statement so that it would be well understood by a fellow student who has not yet studied probability.
3.92 State, with evidence, whether each of the following statements is true or false:
a. The probability of the union of two events cannot be less than the probability of their intersection.
b. The probability of the union of two events can-not be more than the sum of their individual probabilities.
c. The probability of the intersection of two events cannot be greater than either of their individual probabilities.
d. An event and its complement are mutually exclusive.
e. The individual probabilities of a pair of events can-not sum to more than 1.
f. If two events are mutually exclusive, they must also be collectively exhaustive.
g. If two events are collectively exhaustive, they must also be mutually exclusive.
3.93 Distinguish among joint probability, marginal prob-ability, and conditional probability. Provide some ex-amples to make the distinctions clear.
3.94 State, with evidence, whether each of the following claims is true or false:
a. The conditional probability of A, given B, must be at least as large as the probability of A.
b. An event must be independent of its complement.
c. The probability of A, given B, must be at least as large as the probability of the intersection of A and B.
d. The probability of the intersection of two events cannot exceed the product of their individual probabilities.
e. The posterior probability of any event must be at least as large as its prior probability.
3.95 Show that the probability of the union of events A and B can be written as follows:
P1A < B2 = P1A2 + P1B2 31 - P1A u B2 4
3.96 An insurance company estimated that 30% of all au-tomobile accidents were partly caused by weather conditions and that 20% of all automobile accidents involved bodily injury. Further, of those accidents that involved bodily injury, 40% were partly caused by weather conditions.
a. What is the probability that a randomly chosen accident both was partly caused by weather condi-tions and involved bodily injury?
b. Are the events “partly caused by weather condi-tions” and “involved bodily injury” independent?
c. If a randomly chosen accident was partly caused by weather conditions, what is the probability that it involved bodily injury?
d. What is the probability that a randomly chosen accident both was not partly caused by weather conditions and did not involve bodily injury?
3.97 A company places a rush order for wire of two thick-nesses. Consignments of each thickness are to be sent immediately when they are available. Previous expe-rience suggests that the probability is 0.8 that at least one of these consignments will arrive within a week. It is also estimated that, if the thinner wire arrives within a week, the probability is 0.4 that the thicker wire will also arrive within a week. Further, it is estimated that, if the thicker wire arrives within a week, the probabil-ity is 0.6 that the thinner wire will also arrive within a week.
a. What is the probability that the thicker wire will arrive within a week?
b. What is the probability that the thinner wire will arrive within a week?
c. What is the probability that both consignments will arrive within a week?
3.98 Staff, Inc., a management consulting company, is sur-veying the personnel of Acme Ltd. It determined that 35% of the analysts have an MBA and that 40% of all analysts are over age 35. Further, of those who have an MBA, 30% are over age 35.
a. What is the probability that a randomly chosen analyst both has an MBA and also is over age 35?
b. What is the probability that a randomly chosen analyst who is over age 35 has an MBA?
Chapter Exercises and Applications 141 c. What is the probability that a randomly chosen
analyst has an MBA or is over age 35?
d. What is the probability that a randomly chosen analyst who is over age 35 does not have an MBA?
e. Are the events MBA and over age 35 independent?
f. Are the events MBA and over age 35 mutually exclusive?
g. Are the events MBA and over age 35 collectively exhaustive?
3.99 In a campus restaurant it was found that 35% of all customers order vegetarian meals and that 50% of all customers are students. Further, 25% of all customers who are students order vegetarian meals.
a. What is the probability that a randomly chosen customer both is a student and orders a vegetarian meal?
b. If a randomly chosen customer orders a vegetarian meal, what is the probability that the customer is a student?
c. What is the probability that a randomly chosen customer both does not order a vegetarian meal and is not a student?
d. Are the events “customer orders a vegetarian meal” and “customer is a student” independent?
e. Are the events “customer orders a vegetarian meal” and “customer is a student” mutually exclusive?
f. Are the events “customer orders a vegetarian meal” and “customer is a student” collectively exhaustive?
3.100 It is known that 20% of all farms in a state exceed 160 acres and that 60% of all farms in that state are owned by persons over 50 years old. Of all farms in the state exceeding 160 acres, 55% are owned by persons over 50 years old.
a. What is the probability that a randomly chosen farm in this state both exceeds 160 acres and is owned by a person over 50 years old?
b. What is the probability that a farm in this state either is bigger than 160 acres or is owned by a person over 50 years old (or both)?
c. What is the probability that a farm in this state, owned by a person over 50 years old, exceeds 160 acres?
d. Are size of farm and age of owner in this state sta-tistically independent?
3.101 In a large corporation, 80% of the employees are men and 20% are women. The highest levels of education obtained by the employees are graduate training for 10% of the men, undergraduate training for 30% of the men, and high school training for 60% of the men. The highest levels of education obtained are also graduate training for 15% of the women, undergraduate train-ing for 40% of the women, and high school traintrain-ing for 45% of the women.
a. What is the probability that a randomly chosen employee will be a man with only a high school education?
b. What is the probability that a randomly chosen employee will have graduate training?
c. What is the probability that a randomly chosen employee who has graduate training is a man?
d. Are gender and level of education of employees in this corporation statistically independent?
e. What is the probability that a randomly chosen employee who has not had graduate training is a woman?
3.102 A large corporation organized a ballot for all its work-ers on a new bonus plan. It was found that 65% of all night-shift workers favored the plan and that 40%
of all female workers favored the plan. Also, 50% of all employees are night-shift workers and 30% of all employees are women. Finally, 20% of all night-shift workers are women.
a. What is the probability that a randomly chosen employee is a woman in favor of the plan?
b. What is the probability that a randomly chosen employee is either a woman or a night-shift worker (or both)?
c. Is employee gender independent of whether the night shift is worked?
d. What is the probability that a female employee is a night-shift worker?
e. If 50% of all male employees favor the plan, what is the probability that a randomly chosen em-ployee both does not work the night shift and does not favor the plan?
3.103 A jury of 12 members is to be selected from a panel consisting of 8 men and 8 women.
a. How many different jury selections are possible?
b. If the choice is made randomly, what is the prob-ability that a majority of the jury members will be men?
3.104 A consignment of 12 electronic components contains 1 component that is faulty. Two components are chosen randomly from this consignment for testing.
a. How many different combinations of 2 compo-nents could be chosen?
b. What is the probability that the faulty component will be chosen for testing?
3.105 Tiger Funds Ltd. operates a number of mutual funds in high technology and in financial sectors. Hussein Roberts is a fund manager who runs a major fund that includes a wide variety of technology stocks. As fund manager he decides which stocks should be pur-chased for the mutual fund. The compensation plan for fund managers includes a first-year bonus for each stock purchased by the manager that gains more than 10% in the first six months it is held. Of those stocks that the company holds, 40% are up in value after be-ing held for two years. In reviewbe-ing the performance of Mr. Roberts, they found that he received a first-year bonus for 60% of the stocks that he purchased that were up after two years. He also received a first-year bonus for 40% of the stocks he purchased that were not up after two years.
142 Chapter 3 Elements of Chance: Probability Methods What is the probability that a stock will be up after two years given that Mr. Roberts received a first-year bonus?
3.106 Of 100 patients with a certain disease, 10 were chosen at random to undergo a drug treatment that increases the cure rate from 50% for those not given the treat-ment to 75% for those given the drug treattreat-ment.
a. What is the probability that a randomly chosen patient both was cured and was given the drug treatment?
b. What is the probability that a patient who was cured had been given the drug treatment?
c. What is the probability that a specific group of 10 patients was chosen to undergo the drug treatment? (Leave your answer in terms of factorials.)
3.107 Subscriptions to a particular magazine are classified as gift, previous renewal, direct mail, and subscription service. In January 8% of expiring subscriptions were gifts; 41%, previous renewal; 6%, direct mail; and 45%, subscription service. The percentages of renew-als in these four categories were 81%, 79%, 60%, and 21%, respectively. In February of the same year, 10%
of expiring subscriptions were gift; 57%, previous re-newal; 24%, direct mail; and 9%, subscription service.
The percentages of renewals were 80%, 76%, 51%, and 14%, respectively.
a. Find the probability that a randomly chosen sub-scription expiring in January was renewed.
b. Find the probability that a randomly chosen sub-scription expiring in February was renewed.
c. Verify that the probability in part (b) that is higher than that in part (a). Do you believe that the edi-tors of this magazine should view the change from January to February as a positive or negative development?
3.108 The Customs Inspection agency at international airports has developed a traveler profiling sys-tem (TPS) to detect passengers who are trying to bring more liquor into the country than is allowed by present regulations. Long-term studies indicate that 20% of the passengers are carrying more li-quor than is allowed. Tests on the new TPS scheme has shown that of those carrying illegal amounts of liquor, 80% will be identified and subject to com-plete luggage search. In addition 20% of those not carrying illegal amounts of liquor will also be iden-tified by TPS and subject to a complete luggage search.
If a passenger is identified by TPS, what is the prob-ability that the passenger is carrying an illegal amount of liquor? Comment on the value of this system.
3.109 In a large city, 8% of the inhabitants have contracted a particular disease. A test for this disease is positive in 80% of people who have the disease and is negative in 80% of people who do not have the disease. What is the probability that a person for whom the test result is positive has the disease?
3.110 A life insurance salesman finds that, of all the sales he makes, 70% are to people who already own policies.
He also finds that, of all contacts for which no sale is made, 50% already own life insurance policies. Fur-thermore, 40% of all contacts result in sales. What is the probability that a sale will be made to a contact who already owns a policy?
3.111 A professor finds that she awards a final grade of A to 20% of her students. Of those who obtain a final grade of A, 70% obtained an A on the midterm ex-amination. Also, 10% of the students who failed to obtain a final grade of A earned an A on the midterm exam. What is the probability that a student with an A on the midterm examination will obtain a final grade of A?
3.112 The accompanying table shows, for 1,000 forecasts of earnings per share made by financial analysts, the numbers of forecasts and outcomes in particular cat-egories (compared with the previous year).
Forecast
Outcome Improvement
About the
Same Worse
Improvement 210 82 66
About the same 106 153 75
Worse 75 84 149
a. Find the probability that if the forecast is for a worse performance in earnings, this outcome will result.
b. If the forecast is for an improvement in earnings, find the probability that this outcome fails to result.
3.113 A dean has found that 62% of entering freshmen and 78% of community college transfers eventually graduate. Of all entering students, 73% are entering freshmen and the remainder are community college transfers.
a. What is the probability that a randomly chosen entering student is an entering freshman who will eventually graduate?
b. Find the probability that a randomly chosen enter-ing student will eventually graduate.
c. What is the probability that a randomly chosen entering student either is an entering freshman or will eventually graduate (or both)?
d. Are the events “eventually graduates” and “en-ters as community college transfer” statistically independent?
3.114 A market-research group specializes in providing as-sessments of the prospects of sites for new children’s toy stores in shopping centers. The group assesses prospects as good, fair, or poor. The records of assess-ments made by this group were examined, and it was found that for all stores that had annual sales over
$1,000,000, the assessments were good for 70%, fair for 20%, and poor for 10%. For all stores that turned out to be unsuccessful, the assessments were good for
Chapter Exercises and Applications 143 20%, fair for 30%, and poor for 50%. It is known that
60% of new clothing stores are successful and 40% are unsuccessful.
a. For a randomly chosen store, what is the probabil-ity that prospects will be assessed as good?
b. If prospects for a store are assessed as good, what is the probability that it will be successful?
c. Are the events “prospects assessed as good”
and “store is successful” statistically independent?
d. Suppose that five stores are chosen at random.
What is the probability that at least one of them will be successful?
3.115 A restaurant manager classifies customers as regu-lar, occasional, or new, and finds that of all custom-ers 50%, 40%, and 10%, respectively, fall into these categories. The manager found that wine was or-dered by 70% of the regular customers, by 50% of the occasional customers, and by 30% of the new customers.
a. What is the probability that a randomly chosen customer orders wine?
b. If wine is ordered, what is the probability that the person ordering is a regular customer?
c. If wine is ordered, what is the probability that the person ordering is an occasional customer?
3.116 A record-store owner assesses customers entering the store as high school age, college age, or older, and finds that of all customers 30%, 50%, and 20%, respec-tively, fall into these categories. The owner also found that purchases were made by 20% of high school age customers, by 60% of college age customers, and by 80% of older customers.
a. What is the probability that a randomly chosen customer entering the store will make a purchase?
b. If a randomly chosen customer makes a purchase, what is the probability that this customer is high school age?
3.117 Note that this exercise represents a completely imagi-nary situation. Suppose that a statistics class contained exactly 8 men and 8 women. You have discovered that the teacher decided to assign 5 Fs on an exam by ran-domly selecting names from a hat. He concluded that this would be easier than actually grading all those papers and that his students are all equally skilled in statistics—but someone has to get an F. What is the probability that all 5 Fs were given to male students?
3.118 A survey on the best Asian tourist destinations showed that, out of 70 people, 23 ranked Singapore as first, whereas 15 put Hong Kong in first place, 11 put Shanghai first, 7 put Beijing first, and the rest of them chose Tokyo. On the basis of this data, calcu-late the following.
a. The probability of the preferred destination being a city in China. (In this specific case, Hong Kong is not considered part of China.)
b. The probability of the preferred destination not being a Chinese city. (In this case, Hong Kong is considered a Chinese city, even if outside China.) c. The probability of the preferred destination
being Tokyo.
d. The probability of the preferred destination not being Singapore.
3.119 You are responsible for detecting the source of the er-ror when a computer system fails. From your analysis you know that the source of error is the disk drive, the computer memory, or the operating system. You know that 50% of the errors are disk drive errors, 30%
are computer memory errors, and the remainder are operating system errors. From the component perfor-mance standards, you know that when a disk drive error occurs, the probability of failure is 0.60; when a computer memory error occurs, the probability of fail-ure is 0.7; and when an operating system error occurs, the probability of failure is 0.3. Given the information from the component performance standards, what is the probability of a disk drive error, given that a failure occurred?
3.120 After meeting with the regional sales managers, Lauretta Anderson, president of Cowpie Computers, Inc., you find that she believes that the probability that sales will grow by 10% in the next year is 0.70.
After coming to this conclusion, she receives a report that John Cadariu of Minihard Software, Inc., has just announced a new operating system that will be avail-able for customers in 8 months. From past history she knows that in situations where growth has even-tually occurred, new operating systems have been announced 30% of the time. However, in situations where growth has not eventually occurred, new oper-ating systems have been announced 10% of the time.
Based on all these facts, what is the probability that sales will grow by 10%?
3.121 Sally Firefly purchases hardwood lumber for a cus-tom furniture-building shop. She uses three suppliers, Northern Hardwoods, Mountain Top, and Spring Val-ley. Lumber is classified as either clear or has defects, which includes 20% of the pile. A recent analysis of the defect lumber pile showed that 30% came from Northern Hardwoods and 50% came from Mountain Top. Analysis of the clear pile indicates that 40% came from Northern and 40% came from Spring Valley.
What is the percent of clear lumber from each of the three suppliers? What is the percent of lumber from each of the three suppliers?
3.122 Robert Smith uses either regular plowing or minimal plowing to prepare the cornfields on his Minnesota farm. Regular plowing was used for 40% of the field acreage. Analysis after the crop was harvested showed that 50% of the high-yield acres were from minimal-plowing fields and 40% of the low yield fields were from fields with regular plowing. What is the prob-ability of a high yield if regular plowing is used? What is the probability that a field with high yield had been prepared using regular plowing?