Given that P1A ¨ B2 = .4 and P1A  B2 = .8, find P1B2. 3.100 From a production batch with 16 items, 8 items are randomly selected for quality assurance. In how many different ways can the sample be drawn? Suggest an estimate before computing the exact number. 3.101 The Venn diagram below illustrates a sample space containing six sample points and three events, A, B, and C. The probabilities of the sample points are P112 = .3, P122 = .2, P132 = .1, P142 = .1, P152 = .1, and P162 = .2. 1 3 S B C A 4 6 5 2 SUPPLEMENTARY EXERCISES 3.94–3.130 M03_MCCL3396_14_GE_C03.indd 197 29/09/2021 13:53 198 CHAPTER 3 ● Probability

a. Find P1A ¨ B2, P1B ¨ C2, P1A ∪ C2, P1A ∪ B ∪ C2, P1Bc 2, P1Ac ¨ B2, P1B |C2, and P1B|A2.

b. Are A and B independent? Mutually exclusive? Why?

c. Are B and C independent? Mutually exclusive? Why? 3.102 Find the numerical value of

a. 6!

b. a 10 9 b

c. a 10 1 b

d. a 6 3 b

e. 0! Applet Exercise 3.6 Use the applet entitled Random Numbers to generate a list of 50 numbers between 1 and 100, inclusive. Use this list to find each of the probabilities.

a. The probability that a number chosen from the list is less than or equal to 50

b. The probability that a number chosen from the list is even

c. The probability that a number chosen from the list is less than or equal to 50 and even

d. The probability that a number chosen from the list is less than or equal to 50 given that the number is even

e. Do your results from parts a–d support the conclusion that the events less than or equal to 50 and even are independent? Explain. Applying the Concepts—Basic 3.103 Car comparison. Refer to Exercise 2.8 (p. 67). Recall that Jim retrieved the details of 13 cars that he could afford. The cars were classified according to their production year and brand. The results are reproduced in the following table. One of the 13 cars is randomly selected and the production year and car brand are observed. Brand with Car Comparison and Production Year Number of Cars Chevrolet Sonic Sedan LTZ 1.4 (2014) 1 Nissan Almera Turbo 1.0 (2020) 3 Toyota Vios 1.5 (2020) 4 Volkswagen Vento 1.6 Comfortline (2020) 1 Kia Cerato 1.6 (2017) 1 Honda City 1.5 (2020) 3

a. List the sample points for this problem and assign reasonable probabilities to them.

b. Find the probability that Jim has chosen to buy a car produced in 2020.

c. Find the probability that Jim has chosen to buy a Nissan or Toyota or Honda car produced in 2020. 3.104 Workers’ unscheduled absence survey. Each year CCH, Inc., a firm that provides human resources and employment law information, conducts a survey on absenteeism in the workplace. The latest CCH Unscheduled Absence Survey found that of all unscheduled work absences, 34% are due to “personal illness,” 22% for “family issues,” 18% for “personal needs,” 13% for “entitlement mentality,” and 13% due to “stress.” Consider a randomly selected employee who has an unscheduled work absence.

a. List the sample points for this experiment.

b. Assign reasonable probabilities to the sample points.

c. What is the probability that the absence is due to something other than “personal illness”? 3.105 Public health. A research article in the Journal of Environmental and Public Health (Vol. 2020) compares the CAR characteristics of cigarette smoking and e-cigarette and IQOS use among adolescents in Taiwan. The respondents of the survey conducted as part of the study were asked about their smoking habits over the 30 days prior to the survey. The results found that 73.82% of the cigarette smoking adolescents were male and 33.24% of male cigarette smoker adolescents admitted that they were influenced by cigarette advertisements. An adolescent cigarette smoker is randomly selected in Taiwan and the following events are defined: A = 5the adolescent cigarette smoker is a male6, B = 5the adolescent cigarette smoker is a female6, and C = 5the adolescent cigarette smoker is influenced by cigarette advertisements}.

a. Find P1A2. Express this probability in the words of the problem.

b. Find P1B2. Express this probability in the words of the problem.

c. Find P1A ¨ C2. Express this probability in the words of the problem.

d. Find P1C A2. Express this probability in the words of the problem. 3.106 Problems at major companies. The Organization Development Journal (Summer 2006) reported on the results of a survey of human resource officers (HROs) at major employers. The focus of the study was employee behavior, namely, absenteeism and turnover. The study found that 55% of the HROs had problems with employee absenteeism; also, 41% had problems with turnover. Suppose that 22% of the HROs had problems with both absenteeism and turnover. Use this information to find the probability that an HRO selected from the group surveyed had problems with either employee absenteeism or employee turnover. 3.107 New car crash tests. Refer to the National Highway Traffic Safety Administration (NHTSA) crash tests of new car models, Exercise 2.153 (p. 136). Recall that the NHTSA has developed a “star” scoring system, with results ranging from one star (*) to five stars (*****). The more stars in the rating, the better the level of crash protection in a head-on collision. A summary of the driver-side star ratings for 98 cars is reproduced in the accompanying Minitab printout. Assume that one of the 98 cars is selected at random. State whether each of the following is true or false.

a. The probability that the car has a rating of two stars is 4.

b. The probability that the car has a rating of four or five stars is .7857.

c. The probability that the car has a rating of one star is 0.

d. The car has a better chance of having a two-star rating than of having a five-star rating. 3.108 Speeding linked to fatal car crashes. According to a report published by the National Highway Traffic and Safety Administration’s National Center for Statistics and Analysis (NCSA), “Speeding is one of the most prevalent factors contributing to fatal traffic crashes” (NHTSA Technical Report, 2020). The report states that in 2020, CRASH M03_MCCL3396_14_GE_C03.indd 198 29/09/2021 13:53 Supplementary Exercises 3.94–3.130 199 13% of car crashes were fatal and only about 9% of the older American population were involved in the fatal crashes. Suppose a fatal crash occurs. What is the probability that the crash involved an older American? 3.109 Choosing portable grill displays. Consider a study of how people attempt to influence the choices of others by offering undesirable alternatives (Journal of Consumer Research, March 2003). Such a phenomenon typically occurs when family members propose a vacation spot, friends recommend a restaurant for dinner, and realtors show the buyer potential homes. In one phase of the study, the researcher had each of 124 college students select showroom displays for portable grills. Five different displays (representing five different-sized grills) were available, but only three displays would be selected. The students were instructed to select the displays to maximize purchases of Grill #2 (a smaller-sized grill).

a. In how many possible ways can the three grill displays that include Grill #2 be selected from the five displays? List the possibilities.

b. The table shows the grill display combinations and the number of each selected by the 124 students. Use this information to assign reasonable probabilities to the different display combinations.

c. Find the probability that a student who participated in the study selected a display combination involving Grill #1. Grill Display Combination Number of Students 1-2-3 35 1-2-4 8 1-2-5 42 2-3-4 4 2-3-5 1 2-4-5 34 Source: Based on R. W. Hamilton, “Why Do People Suggest What They Do Not Want? Using Context Effects to Influence Others’ Choices,” Journal of Consumer Research, Vol. 29, No. 4, March 2003 (Table 1). 3.110 Road traffic incidents. A paper published in Sustainability in 2020 addresses the issue of road traffic crashes in Indonesia. The data for the districts of Aceh Utara and Lhokseumawe (including the total number of fatalities, accidents that caused major injuries, and accidents that caused minor injuries) are listed in the following table. District Total Fatality Total Major Injury Total Minor Injury Total Aceh Utara 12 4 77 93 Lhokseumawe 10 5 160 175 Source: Data from Satria, R., et al. “A Combined Approach to Address Road Traffic Crashes beyond Cities: Hot Zone Identification and Countermeasures in Indonesia,” Sustainability, 2020 (Table 7).

a. List the simple events for this experiment.

b. Assign reasonable probabilities to the simple events.

c. Find the probability that the crashes happened in Aceh Utara.

d. Find the probability that the crashes caused major injuries.

e. Find the probability that the crashes happened in Lhokseumawe and resulted in fatalities.

f. Find the probability that the crashes either happened in Aceh Utara or caused minor injuries. CRASH g. Find the probability that the crashes did not result in fatalities. 3.111 Is a product “green”? A “green” product (e.g., a product built from recycled materials) is one that has minimal impact on the environment and human health. How do consumers determine if a product is “green”? The 2011 ImagePower Green Brands Survey asked this question of more than 9,000 international consumers. The results are shown in the following table. Reason for Saying a Product Is Green Percentage of Consumers Certification mark on label 45 Packaging 15 Reading information about the product 12 Advertisement 6 Brand Web site 4 Other 18 Total 100 Source: Based on 2011 ImagePower Green Brands Survey.

a. What method is an international consumer most likely to use to identify a green product?

b. Find the probability that an international consumer identifies a green product by a certification mark on the product label or by the product packaging.

c. Find the probability that an international consumer identifies a green product by reading about the product or from information at the brand’s Web site.

d. Find the probability that an international consumer does not use advertisements to identify a green product. 3.112 O2O apps users. O2O commerce is a new business model combining online shopping and offline transactions. A research study published in Electronic Markets (Vol. 30, 2020) applies an expectation confirmation model (ECM) to predict users’ continuance intention to use O2O apps and studies the difference between task-oriented O2O apps users and entertainment-oriented O2O apps users. Refer to the data in the following table: Task-Oriented EntertainmentOriented Satisfied 108 179 Not Satisfied 17 29 Source: Electronic Markets (Vol. 30, 2020) Consider the following events: A = 5Task@oriented O2O apps users6 B = 5Users satisfied with O2O apps6 Are A and B independent events? Applying the Concepts—Intermediate 3.113 Court statistics. On December 22, 2020, the House of Commons Library published a briefing paper on court statistics for England and Wales. In particular, the paper analyzed the data on court performance as seen in the accompanying table. “Defense witness absent” covers all trials that are ineffective due to the absence of the defense witness. “Defense availability” covers all trials that are ineffective because the defense asked for an additional prosecution witness to attend, the defense increased the time O2O TRIAL M03_MCCL3396_14_GE_C03.indd 199 29/09/2021 13:53 200 CHAPTER 3 ● Probability estimate due to insufficient time for the trial to start, the defense advocate was engaged in another trial, the defense advocate failed to attend, and the defendant dismissed the advocate. “Prosecution availability” covers all trials that are ineffective due to the prosecution advocate being engaged in another trial, the prosecution advocate failing to attend, and the prosecution increasing the time estimate due to insufficient time for the trial to start. Suppose one of the 6,903 cases is selected at random, where all combinations of reasons were observed. Reason Magistrate Crown Total Defense witness absent 487 25 512 Defense availability 355 54 409 Prosecution witness absent 324 33 357 Prosecution availability 4,478 1,147 5,625 Total 5,644 1,259 6,903

a. Find P1A2, where A = 5Magistrate trial6.

b. Find P1B2, where B = 5Reason of defense witness absent6.

c. Are A and B mutually exclusive events?

d. Find P1Ac 2.

e. Find P1A ∪ B2.

f. Find P1A ¨ B2. 3.114 Characteristics of a new product. The long-run success of a business depends on its ability to market products with superior characteristics that maximize consumer satisfaction and that give the firm a competitive advantage (Kotler & Keller, Marketing Management, 2015). Ten new products have been developed by a food-products firm. Market research has indicated that the 10 products have the characteristics described by the following Venn diagram: P S A 1 4 5 7 6 3 8 10 9 Superior Product Characteristics Consumer Satisfaction Competitive Advantage 2

a. Write the event that a product possesses all the desired characteristics as an intersection of the events defined in the Venn diagram. Which products are contained in this intersection?

b. If one of the 10 products were selected at random to be marketed, what is the probability that it would possess all the desired characteristics?

c. Write the event that the randomly selected product would give the firm a competitive advantage or would satisfy consumers as a union of the events defined in the Venn diagram. Find the probability of this union.

d. Write the event that the randomly selected product would possess superior product characteristics and satisfy consumers. Find the probability of this intersection.

e. Two of the 10 products will be selected for an ad campaign. How many different pairs of products are possible? 3.115 Testing a watch manufacturer’s claim. A manufacturer of a new SmartWatch claims that the probability of its watch running more than 1 minute slow or 1 minute fast after 1 year of use is .05. A consumer protection agency has purchased four of the manufacturer’s watches with the intention of testing the claim.

a. Assuming that the manufacturer’s claim is correct, what is the probability that none of the watches are as accurate as claimed?

b. Assuming that the manufacturer’s claim is correct, what is the probability that exactly two of the four watches are as accurate as claimed?

c. Suppose that only one of the four tested watches is as accurate as claimed. What inference can be made about the manufacturer’s claim? Explain.

d. Suppose that none of the watches tested are as accurate as claimed. Is it necessarily true that the manufacturer’s claim is false? Explain. 3.116 Car number plate. Danny is going to purchase a new car. For its number plate, he prefers the letters T, Y, and F followed by any combination of digits. If Danny does not have a preference as to which of the three letters is to be put first, what is the probability that

a. the letter T is put first,

b. the letter T is put last,

c. the letter T is put last and the letter F is put second, and

d. the letter F is put first, the letter T is put second, and at the letter Y is put third? 3.117 Global mobile payment user. A research paper published in the Journal of Retailing and Consumer Services (Vol. 57, 2020) investigates the mobile payment and onlineto-offline retail business models. It identifies that China was a global leader in near-end mobile payment applications in 2018 as it accounted for 61.2% of the global mobile payment users.

a. Assuming that this percentage is valid, what is the probability that a person who is a mobile payment user is not Chinese?

b. If five people who are mobile payment users are randomly selected, what is the probability that at least one of them is Chinese? 3.118 Which events are independent? Use your intuitive understanding of independence to form an opinion about whether each of the following scenarios represents independent events.

a. A group of students’ preferences on the type of soda they drink (diet or regular).

b. The stock price movements on different stocks when a financial statement is released.

c. You have two $10 notes and six $5 notes in your purse and you are taking out two notes one at a time without putting them back in.

d. The items purchased by shoppers in a hypermarket.

e. The emails received in a company mailbox and a personal mailbox.

f. Two people who separately buy movie tickets for the same movie and do not ask for a specific seat end up sitting -in the same row. M03_MCCL3396_14_GE_C03.indd 200 29/09/2021 13:53 Supplementary Exercises 3.94–3.130 201 3.119 Stock market participation and IQ. The Journal of Finance (December 2011) published a study of whether the decision to invest in the stock market is dependent on IQ. Information on a sample of 158,044 adults living in Finland formed the database for the study. An IQ score (from a low score of 1 to a high score of 9) was determined for each Finnish citizen as well as whether or not the citizen invested in the stock market. The table below gives the number of Finnish citizens in each IQ score/investment category. Suppose one of the 158,044 citizens is selected at random.

a. What is the probability that the Finnish citizen invests in the stock market?

b. What is the probability that the Finnish citizen has an IQ score of 6 or higher?

c. What is the probability that the Finnish citizen invests in the stock market and has an IQ score of 6 or higher?

d. What is the probability that the Finnish citizen invests in the stock market or has an IQ score of 6 or higher?

e. What is the probability that the Finnish citizen does not invest in the stock market?

f. Are the events {Invest in the stock market} and {IQ score of 1} mutually exclusive? g. Given that the Finnish citizen has an IQ score of 6 or higher, what is the probability that he/she invests in the stock market? h. Given that the Finnish citizen has an IQ score of 5 or lower, what is the probability that he/she invests in the stock market? i. Based on the results, parts g and h, does it appear that investing in the stock market is dependent on IQ? Explain. IQ Score Invest in Market No Investment Totals 1 893 4,659 5,552 2 1,340 9,409 10,749 3 2,009 9,993 12,002 4 5,358 19,682 25,040 5 8,484 24,640 33,124 6 10,270 21,673 31,943 7 6,698 11,260 17,958 8 5,135 7,010 12,145 9 4,464 5,067 9,531 Totals 44,651 113,393 158,044 Source: Based on M. Grinblatt, M. Keloharju, and J. Linnainaa, “IQ and Stock Market Participation,” The Journal of Finance, Vol. 66, No. 6, December 2011 (adapted from Table 1 and Figure 1). 3.120 World Cup soccer match draws. Every 4 years the world’s 32 best national soccer teams compete for the World Cup. Run by FIFA (Fédération Internationale de Football Association), national teams are placed into eight groups of four teams, with the group winners advancing to play for the World Cup. Chance (Spring 2007) investigated the fairness of the World Cup draw. Each of the top 8 seeded teams (teams ranked 1–8, called pot 1) were placed into one of the eight groups (named Group A, B, C, D, E, F, G, and H). The remaining 24 teams were assigned to 3 pots of 8 teams each to achieve the best possible geographic INVIQ distribution between the groups. The teams in pot 2 were assigned to groups as follows: the first team drawn was placed into Group A, the second team drawn was placed in to Group B, etc. Teams in pots 3 and 4 were assigned to the groups in similar fashion. Because teams in pots 2–4 are not necessarily placed there based on their world ranking, this typically leads to a “group of death,” i.e., a group involving at least two highly seeded teams where only one can advance.

a. In one particular World Cup, Germany (as the host country) was assigned as the top seed in Group A. What is the probability that Paraguay (with the highest ranking in pot 2) was assigned to Group A?

b. Many soccer experts viewed the South American teams (Ecuador and Paraguay) as the most dangerous teams in pot 2. What is the probability one of the South American teams was assigned to Group A?

c. Group B was considered the “group of death,” with England (world rank 2), Paraguay (highest rank in pot 2), Sweden (2nd highest rank in pot 3), and Trinidad and Tobago. What is the probability that Group B included the team with the highest rank in pot 2 and the team with one of the top two ranks in pot 3?

d. In drawing teams from pot 2, there was a notable exception. If a South American team (either Ecuador or Paraguay) was drawn into a group with another South American team, it was automatically moved to the next group. This rule impacted Group C (Argentina as the top seed) and Group F (Brazil as the top seed), because they already had South American teams, and groups that followed these groups in the draw. Now Group D included the eventual champion Italy as its top seed. What is the probability that Group D was not assigned one of the dangerous South American teams in pot 2? 3.121 Chance of an Avon sale. The probability that an Avon salesperson sells beauty products to a prospective customer on the first visit to the customer is .4. If the salesperson fails to make the sale on the first visit, the probability that the sale will be made on the second visit is .65. The salesperson never visits a prospective customer more than twice. What is the probability that the salesperson will make a sale to a particular customer? 3.122 Drug testing in athletes. When Olympic athletes are tested for illegal drug use (i.e., doping), the results of a single positive test are used to ban the athlete from competition. In a population of 1,000 athletes, suppose 100 are illegally using testosterone. Of the users, suppose 50 would test positive for testosterone. Of the nonusers, suppose 9 would test positive.

a. Given that the athlete is a user, find the probability that a drug test for testosterone will yield a positive result. (This probability represents the sensitivity of the drug test.)

b. Given the athlete is a nonuser, find the probability that a drug test for testosterone will yield a negative result. (This probability represents the specificity of the drug test.)

c. If an athlete tests positive for testosterone, use Bayes’s Rule to find the probability that the athlete is really doping. (This probability represents the positive predictive value of the drug test.) M03_MCCL3396_14_GE_C03.indd 201 29/09/2021 13:53 202 CHAPTER 3 ● Probability 3.123 Harvest machines. A study was conducted on the effectiveness of machines used for harvesting fruits with shells such as chestnuts, walnuts, and hazelnuts. The four machines that were analyzed were the portable, trailed, mounted, and self-propelled machines. Each machine has its own method of removing the shell from the fruits. The results were recorded, and the machines were classified according to whether they were successful (S) or unsuccessful (U) in removing the shell.

a. List the different sets of classifications that are possible for the four machines (e.g., SUUS).

b. Suppose you are planning to interview factories that have been using these machines in your country to determine the frequency with which they fall into the classifications set by you in part

a. Because no information is available yet, assume initially that there is an equal chance that a factory will classify the machines into any single set. Using this assumption, what is the probability that a factory will classify all four machines as unsuccessful?

c. Using the same assumption as in part

b, what is the probability that a machine will be classified as successful on at least three of the criteria? 3.124 Evaluating the performance of quality inspectors. The performance of quality inspectors affects both the quality of outgoing products and the cost of the products. A product that passes inspection is assumed to meet quality standards; a product that fails inspection may be reworked, scrapped, or reinspected. Quality engineers at an electric company evaluated performances of inspectors in judging the quality of solder joints by comparing each inspector’s classifications of a set of 153 joints with the consensus evaluation of a panel of experts. The results for a particular inspector are shown in the table. One of the 153 solder joints was selected at random. Inspector’s Judgment Committee’s Judgment Joint Acceptable Joint Rejectable Joint acceptable 101 10 Joint rejectable 23 19

a. What is the probability that the inspector judged the joint to be acceptable? That the committee judged the joint to be acceptable?

b. What is the probability that both the inspector and the committee judged the joint to be acceptable? That neither judged the joint to be acceptable?

c. What is the probability that the inspector and the committee disagreed? Agreed? 3.125 Using game simulation to teach a course. In Engineering Management Research (May 2012), a simulation game approach was proposed to teach concepts in a course on production. The proposed game simulation was for color television production. The products are two color television models, A and B. Each model comes in two colors, SJOINT red and black. Also, the quantity ordered for each model can be 1, 2, or 3 televisions. The choice of model, color, and quantity is specified on a purchase order card.

a. Using a tree diagram, list how many different purchase order cards are possible. (These are the sample points for the experiment.)

b. Suppose, from past history, that black color TVs are in higher demand than red TVs. For planning purposes, should the engineer managing the production process assign equal probabilities to the simple events, part a? Why or why not? 3.126 Refer to the Behavioral Research in Accounting (Vol. 32, 2020) study, Exercise