Consider the Venn diagram in the next column, whereP(E1) =

Problem 1E An experiment results in one of the following sample points: E1, E2, E3, E4, or E5. a. Find P(E3) if P(E1) = .1, P(E2) = .2, P(E4) = .1, and P(E5) = .1. b. Find P(E3) if P(E1) = P(E3), P(E2) = .1, P(E4) = .2, and P(E5) = .1. c. Find P(E3) if P(E1) = P(E2) = P(E4) = P(E5) = .1.

Problem 3E The sample space for an experiment contains five sample points with probabilities as shown in the table. Find the probability of each of the following events: Sample Points Probabilities 1 0.05 2 0.20 3 0.30 4 0.30 5 0.15 A: {Either 1, 2, or 3 occurs} B: {Either 1, 3, or 5 occurs} C: {4 does not occur}

Problem 4E Compute each of the following:

Problem 130SE Flawed Pentium computer chip. In October 1994, a flaw was discovered in the Pentium microchip installed in personal computers. The chip produced an incorrect result when dividing two numbers. Intel, the manufacturer of the Pentium chip, initially announced that such an error would occur once in 9 billion divisions, or “once in every 27,000 years” for a typical user; consequently, it did not immediately offer to replace the chip. Depending on the procedure, statistical software packages (e.g., Minitab) may perform an extremely large number of divisions to produce the required output. For heavy users of the software, 1 billion divisions over a short time frame is not unusual. Will the flawed chip be a problem for a heavy Minitab user? [Note: Two months after the flaw was discovered, Intel agreed to replace all Pentium chips free of charge.]

Problem 5E Compute the number of ways you can select n elements from N elements for each of the following: a. n = 2, N = 5 b. n = 3, N = 6 c. n = 5, N = 20

Problem 2E The diagram below describes the sample space of a particular experiment and events A and B. a. What is this type of diagram called? b. Suppose the sample points are equally likely. Find P(A) and P(B). c. Suppose P(1) = P(2) = P(3) = P(4) = P(5) = 1/20 and P(6) = P(7) = P(8) = P(9) = P(10) = 3/20. Find P(A) and P(B).

Problem 6E Two fair dice are tossed, and the face on each die is observed. a. Use a tree diagram to find the 36 sample points contained in the sample space. b. Assign probabilities to the sample points in part a. c. Find the probability of each of the following events: A = {3 showing on each die} B = {sum of two numbers showing is 7} C = {sum of two numbers showing is even} Two fair dice are tossed, and the face on each die is observed. a. Use a tree diagram to find the 36 sample points contained in the sample space. b. Assign probabilities to the sample points in part a. c. Find the probability of each of the following events: A = {3 showing on each die} B = {sum of two numbers showing is 7} C = {sum of two numbers showing is even}

Problem 8E Simulate the experiment described in Exercise 3.7 using any five identically shaped objects, two of which are one color and three another color. Mix the objects, draw two, record the results, and then replace the objects. Repeat the experiment a large number of times (at least 100). Calculate the proportion of time events A, B, and C occur. How do these proportions compare with the probabilities you calculated in Exercise 3.7? Should these proportions equal the probabilities? Explain. 3.7 Two marbles are drawn at random and without replacement from a box containing two blue marbles and three red marbles. NW a. List the sample points for this experiment. b. Assign probabilities to the sample points. c. Determine the probability of observing each of the following events: A: {Two blue marbles are drawn.} B: {A red and a blue marble are drawn.} C: {Two red marbles are drawn.}

Colors of M&M’s candies. In 1940, Forrest E. Mars Sr. formed the Mars Corporation to produce chocolate candies with a sugar shell that could be sold throughout the year and wouldn’t melt during the summer. Originally, M&M’s Plain Chocolate Candies came in only a brown color. Today, M&M’s in standard bags come in six colors: brown, yellow, red, blue, orange, and green. According to Mars Corporation, today 24% of all M&M’s produced are blue, 20% are orange, 16% are green, 14% are yellow, 13% are brown, and 13% are red. Suppose you purchase a randomly selected bag of M&M’s Plain Chocolate Candies and randomly select one of the M&M’s from the bag. The color of the selected M&M is of interest. a. Identify the outcomes (sample points) of this experiment. b. Assign reasonable probabilities to the outcomes, part a. c. What is the probability that the selected M&M is brown (the original color)? d. In 1960, the colors red, green, and yellow were added to brown M&Ms. What is the probability that the selected M&M is either red, green, or yellow? e. In 1995, based on voting by American consumers, the color blue was added to the M&M mix. What is the probability that the selected M&M is not blue?

Problem 10E 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”?

Two marbles are drawn at random and without replacement from a box containing two blue marbles and three red marbles. NW a. List the sample points for this experiment. b. Assign probabilities to the sample points. c. Determine the probability of observing each of the following events: A: {Two blue marbles are drawn.} B: {A red and a blue marble are drawn.} C: {Two red marbles are drawn.}

Problem 12E Predicting when a Florida hurricane occurs. Since the early 1900s, the state of Florida has exceeded $450 million in damages due to destructive hurricanes. Consequently, the value of property insured against windstorm damage in Florida is the highest in the nation. Researchers at Florida State University conducted a comprehensive analysis of damages caused by Florida hurricanes and published the results in Southeastern Geographer (Summer 2009). Part of their analysis included estimating the likelihood that a hurricane develops from a tropical storm based on the sequence number of the tropical storm within a season. The researchers discovered that of the 67 Florida hurricanes since 1900, 11 developed from the fifth tropical storm of the season (the sequence with the highest frequency). Also, only 5 hurricanes developed from a tropical storm with a sequence number of 12 or greater. a. Estimate the probability that a Florida hurricane develops from the fifth tropical storm of the season. b. Estimate the probability that a Florida hurricane develops before the 12th tropical storm of the season.

Problem 11E Male nannies. In a survey conducted by the International Nanny Association (INA) and reported at the INA Web site (www.nanny.org), 4,176 nannies were placed in a job in a given year. Only 24 of the nannies placed were men. Find the probability that a randomly selected nanny who was placed during the last year is a male nanny (a “mannie”).

Problem 14E Working on summer vacation. Is summer vacation a break from work? Not according to an Adweek/Harris (July 2011) poll of 3,304 U.S. adults. The poll found that 46% of the respondents work during their summer vacation, 35% do not work at all while on vacation, and 19% were unemployed. Consider the work status during summer vacation of a randomly selected poll respondent. a. List the sample points for this experiment. b. Assign reasonable probabilities to the sample points. c. What is the probability that a randomly selected poll respondent will not work while on summer vacation or is unemployed?

Going online for health information. A cyberchondriac is defined as a person who regularly searches the Web for health care information (see Exercise 1.20, p. 26). A 2011 Harris Poll surveyed 1,019 U.S. adults by telephone and asked each respondent how often (in the past month) he/she looked for health care information online. The results are summarized in the following table. Consider the response category of a randomly selected person who participated in the Harris Poll. a. List the sample points for the experiment. b. Assign reasonable probabilities to the sample points. c. Find the probability that the respondent looks for health care information online more than two times per month.

Problem 15E 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 18 Other 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.

Problem 17E USDA chicken inspection. The U.S. Department of Agriculture (USDA) reports that, under its standard inspection system, one in every 100 slaughtered chickens passes inspection with fecal contamination. a. If a slaughtered chicken is selected at random, what is the probability that it passes inspection with fecal contamination? b. The probability of part a was based on a USDA study that found that 306 of 32,075 chicken carcasses passed inspection with fecal contamination. Do you agree with the USDA’s statement about the likelihood of a slaughtered chicken passing inspection with fecal contamination?

Problem 18E PIN pad shipments. Personal identification number (PIN) pads are devices that connect to point-of-sale electronic cash registers for debit and credit card purchases. Refer to The Nilson Report (Oct. 2008) listing of the volume of PIN pad shipments by manufacturers worldwide, Exercise 2.7 (p. 49). Recall that for the 12 manufacturers listed in the table, a total of 334,039 PIN pads were shipped. Suppose you randomly select one of these PIN pads and identify the manufacturer. Manufacturer Number Shipped (units) Bitel 13,500 CyberNet 16,200 Fujian Landi 119,000 Glintt (ParaRede) 5,990 Intelligent 4,562 KwangWoo 42,000 Omron 20,000 Pax Tech. 10,072 ProvencoCadmus 20,000 SZZT Electronics 67,300 Toshiba TEC 12,415 Urmet 3,000 Source: Data from The Nilson Report, No. 912, October 2008 (p. 9). a. Find the probability that the PIN pad is shipped by either Fujian Landi or SZZT Electronics. b. Suppose that 1,000 of the PIN pads shipped were found to be defective. Find the probability that the PIN pad selected is one of the defectives.

Problem 16E Museum management. Refer to the Museum Management and Curatorship (June 2010) study of the criteria used to evaluate museum performance, Exercise 2.14 (p. 50). Recall that the managers of 30 leading museums of contemporary art were asked to provide the performance measure used most often. A summary of the results is reproduced in the table. Performance Measure Number of Museums Total visitors 8 Paying visitors 5 Big shows 6 Funds raised 7 Members 4 a. If one of the 30 museums is selected at random, what is the probability that the museum uses total visitors or funds raised most often as a performance measure? b. Consider two museums of contemporary art randomly selected from all such museums. Of interest is whether or not the museums use total visitors or funds raised most often as a performance measure. Use a tree diagram to aid in listing the sample points for this problem. c. Assign reasonable probabilities to the sample points of part b. [Hint: Use the probability, part a, to estimate these probabilities.] d. Refer to parts b and c. Find the probability that both museums use total visitors or funds raised most often as a performance measure.

Problem 19E Randomization in a study of TV commercials. Gonzaga University professors conducted a study of more than 1,500 television commercials and published their results in the Journal of Sociology, Social Work and Social Welfare (Vol. 2, 2008). Commercials from eight networks—ABC, FAM, FOX, MTV, ESPN, CBS, CNN, and NBC—were sampled during an 8-day period, with one network randomly selected each day. The table below shows the actual order determined by random draw: a. What is the probability that ESPN was selected on Monday, July 11? b. Consider the four networks chosen for the weekends (Saturday and Sunday). How many ways could the researchers select four networks from the eight for the weekend analysis of commercials? (Assume that the order of assignment for the four weekend days was immaterial to the analysis.) c. Knowing that the networks were selected at random, what is the probability that ESPN was one of the four networks selected for the weekend analysis of commercials?

Problem 21E Investing in stocks. From a list of 15 preferred stocks recommended by your broker, you will select three to invest in. How many different ways can you select the three stocks from the 15 recommended stocks?

Problem 22E Groundwater contamination in wells. Refer to the Environmental Science & Technology (Jan. 2005) study of methyl tert-butyl ether (MTBE) contamination in New Hampshire wells, Exercise 2.17 (p. 51). Data collected for a sample of 223 wells are saved in the accompanying file. Recall that each well was classified according to well class (public or private), aquifier (bedrock or unconsolidated), and detectable level of MTBE (below limit or detect). a. Consider an experiment in which the well class, aquifier, and detectable MTBE level of a well are observed. List the sample points for this experiment. [Hint: One sample point is private/bedrock/below limit.] b. Use statistical software to find the number of the 223 wells in each sample point outcome. Then use this information to compute probabilities for the sample points. c. Find and interpret the probability that a well has a detectable level of MTBE. 2.17 Groundwater contamination in wells. In New Hampshire, about half the counties mandate the use of reformulated gasoline. This has led to an increase in the contamination of groundwater with methyl tert-butyl ether (MTBE). Environmental Science & Technology (Jan. 2005) reported on the factors related to MTBE contamination in public and private New Hampshire wells. Data were collected for a sample of 223 wells. Three of the variables are qualitative in nature: well class (public or private), aquifer (bedrock or unconsolidated), and detectible level of MTBE (below limit or detect). [Note: A detectible level of MTBE occurs if the MTBE value exceeds .2 micrograms per liter.] The data for 11 selected wells are shown in the accompanying table. Well Class Aquifer Detect MTBE Private Bedrock Below Limit Private Bedrock Below Limit Public Unconsolidated Detect Public Unconsolidated Below Limit Public Unconsolidated Below Limit Public Unconsolidated Below Limit Public Unconsolidated Detect Public Unconsolidated Below Limit Public Unconsolidated Below Limit Public Bedrock Detect Public Bedrock Detect Source: Based on Ayotte, J. D., Argue, D. M., and McGarry, F. J. “Methyl tert-Butyl Ether Occurrence and Related Factors in Public and Private Wells in Southeast New Hampshire.” ENVIRONMENTAL SCIENCE & TECHNOLOGY, Vol. 39, No. 1, Jan. 2005, pp. 9–16. a. Use graphical methods to describe each of the three qualitative variables for all 223 wells. b. Use side-by-side bar charts to compare the proportions of contaminated wells for private and public well classes. c. Use side-by-side bar charts to compare the proportions of contaminated wells for bedrock and unconsolidated aquifers. d. What inferences can be made from the bar charts, parts a–c?

Problem 23E Choosing portable grill displays. University of Maryland marketing professor R. W. Hamilton studied how people attempt to influence the choices of others by offering undesirable alternatives (Journal of Consumer Research, Mar. 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 differentsized 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 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 Hamilton, R. W. “Why do people suggest what they do not want? Using context effects to influence others’ choices,” Journal of Consumer Research, Vol. 29, No. 4. Mar. 2003 (Table 1).

Problem 20E Jai-alai bets. The Quinella bet at the paramutual game of jai-alai consists of picking the jai-alai players that will place first and second in a game irrespective of order. In jai-alai, eight players (numbered 1, 2, 3, . . . , 8) compete in every game. a. How many different Quinella bets are possible? b. Suppose you bet the Quinella combination of 2–7. If the players are of equal ability, what is the probability that you win the bet?

Highest-rated new cars. Consumer Reports magazine annually asks readers to evaluate their experiences in buying a new car during the previous year. Analysis of the questionnaires for a recent year revealed that readers were very satisfied with the following three new cars (in no particular order): Hyundai Elantra, Toyota Prius, and Subaru Forrester (Consumer Reports, Apr. 2011). a. List all possible sets of rankings for these top three cars. b. Assuming that each set of rankings in part a is equally likely, what is the probability that readers ranked Toyota Prius first? That readers ranked Hyundai Elantra third? That readers ranked Toyota Prius first and Subaru Forrester second?

Problem 25E Odds of winning a race. Handicappers for greyhound races express their belief about the probabilities that each greyhound will win a race in terms of odds. If the probability of event E is P(E), then the odds in favor of E are P(E) to 1 - P(E). Thus, if a handicapper assesses a probability of .25 that Oxford Shoes will win its next race, the odds in favor of Oxford Shoes are 25/100 to 75/100, or 1 to 3. It follows that the odds against E are 1 - P(E) to P(E), or 3 to 1 against a win by Oxford Shoes. In general, if the odds in favor of event E are a to b, then P(E) = a/(a + b). a. A second handicapper assesses the probability of a win by Oxford Shoes to be 1/3 . According to the second handicapper, what are the odds in favor of Oxford Shoes winning? b. A third handicapper assesses the odds in favor of Oxford Shoes to be 1 to 1. According to the third handicapper, what is the probability of Oxford Shoes winning? c. A fourth handicapper assesses the odds against Oxford Shoes winning to be 3 to 2. Find this handicapper’s assessment of the probability that Oxford Shoes will win.

Lead bullets as forensic evidence. Chance (Summer 2004) published an article on the use of lead bullets as forensic evidence in a federal criminal case. Typically, the Federal Bureau of Investigation (FBI) will use a laboratory method to match the lead in a bullet found at the crime scene with unexpended lead cartridges found in possession of the suspect. The value of this evidence depends on the chance of a false positive (i.e., the probability that the FBI finds a match given that the lead at the crime scene and the lead in possession of the suspect are actually from two different “melts,” or sources). To estimate the false-positive rate, the FBI collected 1,837 bullets that they were confident all came from different melts. The FBI then examined every possible pair of bullets and counted the number of matches using its established criteria. According to Chance, the FBI found 693 matches. Use this information to compute the chance of a false positive. Is this probability small enough for you to have confidence in the FBI’s forensic evidence?

Problem 27E Making your vote count. The recent Democratic and Republican presidential state primary elections were highlighted by the difference in the way winning candidates were awarded delegates. In Republican states, the winner is awarded all the state’s delegates; conversely, the Democratic state winner is awarded delegates in proportion to the percentage of votes. This led to a Chance (Fall 2007) article on making your vote count. Consider the following scenario where you are one of five voters (for example, on a county commission where you are one of the five commissioners voting on an issue). a. Determine the number of ways the five commissioners can vote, where each commissioner votes either for or against. (These outcomes represent the sample points for the experiment.) b. Assume each commissioner is equally likely to vote for or against. Assign reasonable probabilities to the sample points, part a. c. Your vote counts (i.e., is the decisive vote) only if the other four voters split, two in favor and two against. Assuming you are commissioner number 5, how many of the sample points in part a result in a 2-2 split for the other four commissioners? d. Use your answers to parts a–c to find the probability that your vote counts. e. Now suppose you convince two other commissioners to “vote in bloc,” i.e., you all agree to vote among yourselves first, and whatever the majority decides is the way all three will vote. With only five total voters, this guarantees that the bloc vote will determine the outcome. In this scenario, your vote counts only if the other two commissioners in the bloc split votes, one in favor and one against. Find the probability that your vote counts.

Problem 28E Suppose P(A) = .4, P(B) = .7, and P(A B) = .3. Find the following probabilities: a. P(Bc) b. P(Ac) c. P(A ? B)

Problem 30E A pair of fair dice is tossed. Define the following events: A: {You will roll a 7.} (i.e., The sum of the dots on the up faces of the two dice is equal to 7.) B: {At least one of the two dice shows a 4.} a. Identify the sample points in the events A, B, A B, A ? B and Ac. b. Find P(A), P(B), P(A B), P(A ? B), and P(Ac) by summing the probabilities of the appropriate sample points. c. Find P(A ? B) using the additive rule. Compare your answer to that for the same event in part b. d. Are A and B mutually exclusive? Why?

Problem 31E Consider the Venn diagram below, where P(E1) = P(E2) = P(E3) = 1/5 , P(E4) = P(E5) = 1/20, P(E6) = 1/10 , and P(E7) = 1/5 . Find each of the following probabilities: a. P(A) b. P(B) c. P(A ? B) d. P(A B) e. P(Ac) f. P(Bc) g. P(A ? Ac) h. P(Ac B)

Problem 32E Consider the Venn diagram in the next column, where P(E1) = .10, P(E2) = .05, P(E3) = P(E4) = .2, P(E5) = .06, P(E6) = .3, P(E7) = .06, and P(E8) = .03. Find the following probabilities: a. P(Ac) b. P(Bc) c. P(Ac B) d. P(A ? B) e. P(A B) f. P(Ac Bc) g. Are events A and B mutually exclusive? Why?

Problem 33E The outcomes of two variables are (Low, Medium, High) and (On, Off), respectively. An experiment is conducted in which the outcomes of each of the two variables are observed. The probabilities associated with each of the six possible outcome pairs are given in the accompanying two-way table. Low Medium High On .50 .10 .05 Off .25 .07 .03 Consider the following events: A: {On} B: {Medium or On} C: {Off and Low} D: {High} a. Find P(A). b. Find P(B). c. Find P(C). d. Find P(D). e. Find P(Ac). f. Find P(A ? B). g. Find P(A ? C). h. Consider each pair of events (A and B, A and C, A and D, B and C, B and D, C and D). List the pairs of events that are mutually exclusive. Justify your choices.

Problem 34E Refer to Exercise 3.33. Use the same event definitions to do the following exercises. a. Write the event that the outcome is “On” and “High” as an intersection of two events. b. Write the event that the outcome is “Low” or “Medium” as the complement of an event. 3.33 The outcomes of two variables are (Low, Medium, High) and (On, Off), respectively. An experiment is conducted in which the outcomes of each of the two variables are observed. The probabilities associated with each of the six possible outcome pairs are given in the accompanying two-way table. Low Medium High On .50 .10 .05 Off .25 .07 .03 Consider the following events: A: {On} B: {Medium or On} C: {Off and Low} D: {High} a. Find P(A). b. Find P(B). c. Find P(C). d. Find P(D). e. Find P(Ac). f. Find P(A ? B). g. Find P(A ? C). h. Consider each pair of events (A and B, A and C, A and D, B and C, B and D, C and D). List the pairs of events that are mutually exclusive. Justify your choices.

Study of analysts’ forecasts. The Journal of Accounting Research (March 2008) published a study on relationship incentives and degree of optimism among analysts’ forecasts. Participants were analysts at either a large or small brokerage firm who made their forecasts either early or late in the quarter. Also, some analysts were only concerned with making an accurate forecast, while others were also interested in their relationship with management. Suppose one of these analysts is randomly selected. Consider the following events: A = {The analyst is concerned only with making an accurate forecast.} B = {The analyst makes the forecast early in the quarter.} C = {The analyst is from a small brokerage firm.} Describe each of the following events in terms of unions, intersections, and complements (e.g., \(A \cup B,\ A \cap B,\ A^c\), etc.). a. The analyst makes an early forecast and is concerned only with accuracy. b. The analyst is not concerned only with accuracy. c. The analyst is from a small brokerage firm or makes an early forecast. d. The analyst makes a late forecast and is not concerned only with accuracy.

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 located in a southeastern city. The focus of the study was employee behavior, namely absenteeism, promptness to work, 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.

Problem 29E A fair coin is tossed three times, and the events A and B are defined as follows: A: {At least one head is observed.} B: {The number of heads observed is odd.} a. Identify the sample points in the events A, B, A ? B, Ac, and A B. b. Find P(A), P(B), P(A ? B), P(Ac), and P(A B) by summing the probabilities of the appropriate sample points. c. Find P(A ? B) using the additive rule. Compare your answer to the one you obtained in part b. d. Are the events A and B mutually exclusive? Why?

Problem 37E Do social robots walk or roll? Refer to the International Conference on Social Robotics (Vol. 6414, 2010) study of the trend in the design of social robots, Exercise 2.3 (p. 48). Recall that in a random sample of 106 social (or service) robots designed to entertain, educate, and care for human users, 63 were built with legs only, 20 with wheels only, 8 with both legs and wheels, and 15 with neither legs nor wheels. One of the 106 social robots is randomly selected and the design (e.g., wheels only) is noted. a. List the sample points for this study. b. Assign reasonable probabilities to the sample points. c. What is the probability that the selected robot is designed with wheels? d. What is the probability that the selected robot is designed with legs? e. Use the rule of complements to find the probability that the selected robot is designed with either legs or wheels.

Problem 39E Inactive oil and gas structures. U.S. federal regulations require that operating companies clear all inactive offshore oil and gas structures within 1 year after production ceases. Researchers at the Louisiana State University Center for Energy Studies gathered data on both active and inactive oil and gas structures in the Gulf of Mexico (Oil & Gas Journal, Jan. 3, 2005). They discovered that the Gulf of Mexico has 2,175 active and 1,225 idle (inactive) structures. The following table breaks down these structures by type (caisson, well protector, or fixed platform). Consider the structure type and active status of one of these oil/gas structures. Source: Data from Kaiser, M., and Mesyanzhinov, D. “Study tabulates idle Gulf of Mexico structures,” Oil & Gas Journal, Vol. 103, No. 1, Jan. 3, 2005 (Table 2). a. List the simple events for this experiment. b. Assign reasonable probabilities to the simple events. c. Find the probability that the structure is active. d. Find the probability that the structure is a well protector. e. Find the probability that the structure is an inactive caisson. f. Find the probability that the structure is either inactive or a fixed platform. g. Find the probability that the structure is not a caisson.

Problem 38E Scanning errors at Wal-Mart. The National Institute for Standards and Technology (NIST) mandates that for every 100 items scanned through the electronic checkout scanner at a retail store, no more than 2 should have an inaccurate price. A study of the accuracy of checkout scanners at Wal- Mart stores in California was conducted (Tampa Tribune, Nov. 22, 2005). Of the 60 Wal-Mart stores investigated, 52 violated the NIST scanner accuracy standard. If one of the 60 Wal-Mart stores is randomly selected, what is the probability that the store does not violate the NIST scanner accuracy standard?

Problem 40E Social networking Web sites in the United Kingdom. In the United States, MySpace and Facebook are considered the two most popular social networking Web sites. In the United Kingdom (UK), the competition for social networking is between MySpace and Bebo. According to Nielsen/ NetRatings (April 2006), 4% of UK citizens visit MySpace, 3% visit Bebo, and 1% visit both MySpace and Bebo. a. Draw a Venn diagram to illustrate the use of social networking sites in the United Kingdom. b. Find the probability that a UK citizen visits either the MySpace or Bebo social networking site. c. Use your answer to part b to find the probability that a UK citizen does not visit either social networking site.

Problem 41E Study of why EMS workers leave the job. An investigation into why emergency medical service (EMS) workers leave the profession was published in the Journal of Allied Health (Fall 2011). The researchers surveyed a sample of 244 former EMS workers, of which 127 were fully compensated while on the job, 45 were partially compensated, and 72 had noncompensated volunteer positions. The numbers of EMS workers who left because of retirement were 7 for fully compensated workers, 11 for partially compensated workers, and 10 for noncompensated volunteers. One of the 244 former EMS workers is selected at random. a. Find the probability that the former EMS worker was fully compensated while on the job. b. Find the probability that the former EMS worker was fully compensated while on the job and left due to retirement. c. Find the probability that the former EMS worker was not fully compensated while on the job. d. Find the probability that the former EMS worker was either fully compensated while on the job or left due to retirement.

Problem 43E 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, 2006). 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: 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.

Problem 44E Guilt in decision making. The effect of guilt emotion on how a decision maker focuses on a problem was investigated in the Jan. 2007 issue of the Journal of Behavioral Decision Making (see Exercise 1.30, p. 27). A total of 171 volunteer students participated in the experiment, where each was randomly assigned to one of three emotional states (guilt, anger, or neutral) through a reading/writing task. Immediately after the task, students were presented with a decision problem where the stated option had predominantly negative features (e.g., spending money on repairing a very old car). The results (number responding in each category) are summarized in the accompanying table. Suppose one of the 171 participants is selected at random. Emotional State Choose Stated Option Do Not Choose Stated Option Totals Guilt 45 12 57 Anger 8 50 58 Neutral 7 49 56 Totals 60 111 171 Source: Based on Gangemi, A., & Mancini, F. “Guilt and focusing in decisionmaking,” Journal of Behavioral Decision Making, Vol. 20, Jan. 2007 (Table 2). a. Find the probability that the respondent is assigned to the guilty state. b. Find the probability that the respondent chooses the stated option (repair the car). c. Find the probability that the respondent is assigned to the guilty state and chooses the stated option. d. Find the probability that the respondent is assigned to the guilty state or chooses the stated option.

Cell phone handoff behavior. A “handoff” is a term used in wireless communications to describe the process of a cell phone moving from a coverage area of one base station to another. Each base station has multiple channels (called color codes) that allow it to communicate with the cell phone. The Journal of Engineering, Computing and Architecture (Vol. 3., 2009) published a study of cell phone handoff behavior. During a sample driving trip which involved crossing from one base station to another, the different color codes accessed by the cell phone were monitored and recorded. The table below shows the number of times each color code was accessed for two identical driving trips, each using a different cell phone model. (Note: The table is similar to the one published in the article.) Suppose you randomly select one point during the combined driving trips. a. What is the probability that the cell phone is using color code 5? b. What is the probability that the cell phone is using color code 5 or color code 0? c. What is the probability that the cell phone used is Model 2 and the color code is 0?

Problem 46E Likelihood of a tax return audit. At the beginning of each year, the Internal Revenue Service (IRS) releases information on the likelihood of a tax return being audited. In 2010, the IRS audited 1,581,394 individual tax returns from the total of 142,823,105 filed returns; also, the IRS audited 29,803 returns from the total of 2,143,808 corporation returns filed (IRS 2010 Data Book). a. Suppose an individual tax return from 2010 is randomly selected. What is the probability that the return was audited by the IRS? b. Refer to part a. Determine the probability that an individual return was not audited by the IRS. c. Suppose a corporation tax return from 2010 is randomly selected. What is the probability that the return was audited by the IRS? d. Refer to part c. Determine the probability that a corporation return was not audited by the IRS.

Problem 42E 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 next table 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? 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 Grinblatt, M., Keloharju, M., & Linnainaa, J. “IQ and Stock Market Participation,” The Journal of Finance, Vol. 66, No. 6, December 2011 (adapted from Table 1 and Figure 1). 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?

Problem 48E Galileo’s Passedix game. Passedix is a game of chance played with three fair dice. Players bet whether the sum of the faces shown on the dice will be above or below 10. During the late sixteenth century, the astronomer and mathematician Galileo Galilei was asked by the Grand Duke of Tuscany to explain why “the chance of throwing a total of 9 with three fair dice was less than that of throwing a total of 10” (Interstat, Jan. 2004). The grand duke believed that the chance should be the same because “there are an equal number of partitions of the numbers 9 and 10.” Find the flaw in the grand duke’s reasoning and answer the question posed to Galileo.

Problem 47E Reliability of gas station air guages. Tire and automobile manufacturers and consumer safety experts all recommend that drivers maintain proper tire pressure in their cars. Consequently, many gas stations now provide air pumps and air gauges for their customers. In a Research Note (Nov. 2001), the National Highway Traffic Safety Administration studied the reliability of gas station air guages. The next table gives the percentage of gas stations that provide air gauges that overreport the pressure level in the tire. Station Gauge Pressure Overreport by 4 psi or More (%) Overreport by 6 psi or More (%) Overreport by 8 psi or More (%) 25psi 16 2 0 35psi 19 9 0 45psi 19 14 5 55psi 20 15 9 a. If the gas station air pressure gauge reads 35 psi, what is the probability that the pressure is overreported by 6 psi or more? b. If the gas station air pressure gauge reads 55 psi, what is the probability that the pressure is overreported by 8 psi or more? c. If the gas station air pressure gauge reads 25 psi, what is the probability that the pressure is not overreported by 4 psi or more? d. Are the events A = {overreport by 4 psi or more} and B = {overreport by 6 psi or more} mutually exclusive? Explain. e. Based on your answer to part d, why do the probabilities in the table not sum to 1?

For two events, A and B, P(A) = .4, P(B) = .2, and NW \(P(A \cap B)=.1\): a. Find P(A|B). b. Find P(B|A). c. Are A and B independent events? Text Transcription: P(A cap B) = .1

Problem 50E For two events, A and B, P(A) = .4, P(B) = .2, and P(A|B) = .6: a. Find P(A B). b. Find P(B|A).

For two independent events, A and B, P(A) = .4 and P(B) = .2: a. Find \(P(A \cup B)\) b. Find \(P(A \mid B)\). c. Find \(P(A \cup B)\)

Problem 54E Two fair coins are tossed, and the following events are defined: A: {Observe at least one head} B: {Observe exactly one head} a. Draw a Venn diagram for the experiment, showing events A and B. Assign probabilities to the sample points. b. Find P(A), P(B), and P(A B) c. Use the formula for conditional probability to find P(A|B) and P(B|A). Verify your answer by inspecting the Venn diagram and using the concept of reduced sample spaces.

Problem 53E Consider the experiment depicted by the Venn diagram, with the sample space S containing five sample points. The sample points are assigned the following probabilities: P(E1) = .20, P(E2) = .30, P(E3) = .30, P(E4) = .10, P(E5) = .10. a. Calculate P(A), P(B), and P(A ? B) b. Suppose we know that event A has occurred, so that the reduced sample space consists of the three sample points in A—namely, E1, E2, and E3. Use the formula for conditional probability to adjust the probabilities of these three sample points for the knowledge that A has occurred [i.e., P(Ei |A)]. Verify that the conditional probabilities are in the same proportion to one another as the original sample point probabilities. c. Calculate the conditional probability P(B|A) in two ways: (1) Add the adjusted (conditional) probabilities of the sample points in the intersection A ? B, as these represent the event that B occurs given that A has occurred; (2) use the formula for conditional probability: Verify that the two methods yield the same result. d. Are events A and B independent? Why or why not?

An experiment results in one of five sample points with the following probabilities: P(E1) = .22, P(E2) = .31, P(E3) = .15, P(E4) = .22, and P(E5) = .1. The following events have been defined: A: {E1, E3} B: {E2, E3, E4} C: {E1, E5} Find each of the following probabilities: a. P(A) b. P(B) c. P(A ? B) d. P(A|B) e. P(B ? C) f. P(C |B) g. Consider each pair of events: A and B, A and C, and B and C. Are any of the pairs of events independent? Why?

Two fair dice are tossed, and the following events are defined: A: {Sum of the numbers showing is odd} B: {Sum of the numbers showing is 9, 11, or 12} Are events A and B independent? Why?

Problem 52E An experiment results in one of three mutually exclusive events, A, B, or C. It is known that P(A) = .30, P(B) = .55, and P(C) = .15. Find each of the following probabilities: a. P(A ? B) b. P(A C) c. P(A|B) d. P(B ? C) e. Are B and C independent events? Explain.

Problem 57E A sample space contains six sample points and events A, B, and C as shown in the Venn diagram. The probabilities of the sample points are P(1) = .20, P(2) = .05, P(3) = .30, P(4) = .10, P(5) = .10, P(6) = .25. a. Which pairs of events, if any, are mutually exclusive? Why? b. Which pairs of events, if any, are independent? Why? c. Find P(A ? B) by adding the probabilities of the sample points and then by using the additive rule. Verify that the answers agree. Repeat for P(A ? C)

World’s largest public companies. Forbes (Apr. 20 , 2011) conducted a survey of the 20 largest public companies in the world. Of these 20 companies, 4 were banking or investment companies based in the United States. A total of 9 U.S. companies were on the top 20 list. Suppose we select one of these 20 companies at random. Given that the company is based in the United States, what is the probability that it is a banking or investment company?

Problem 58E On-the-job arrogance and task performance. Human Performance (Vol. 23, 2010) published the results of a study that found that arrogant workers are more likely to have poor performance ratings. Suppose that 15% of all full-time workers exhibit arrogant behaviors on the job and that 10% of all full-time workers will receive a poor performance rating. Also, assume that 5% of all full-time workers exhibit arrogant behaviors and receive a poor performance rating. Let A be the event that a full-time worker exhibits arrogant behavior on the job. Let B be the event that a fulltime worker will receive a poor performance rating. a. Are the events A and B mutually exclusive? Explain. b. Find P(B|A). c. Are the events A and B independent? Explain.

Guilt in decision making. Refer to the Journal of Behavioral Decision Making (Jan. 2007) study of the effect of guilt emotion on how a decision maker focuses on a problem, Exercise 3.44 (p. 153). The results (number responding in each category) for the 171 study participants are reproduced in the table below. Suppose one of the 171 participants is selected at random. a. Given that the respondent is assigned to the guilty state, what is the probability that the respondent chooses the stated option? b. If the respondent does not choose to repair the car, what is the probability that the respondent is in the anger state? c. Are the events {repair the car} and {guilty state} independent?

Problem 61E Twitter and wireless Internet. According to the Pew Internet&American Life Project (Oct. 2009), 54% of Internet users have a wireless connection to the Internet via a laptop, cell phone, game console, or other mobile device. Of these wireless users, 25% use Twitter to share updates about themselves. What is the probability that an Internet user has a wireless connection and uses Twitter?

Problem 62E Identity theft victims. According to The National Crime Victimization Survey (December 2010), published by the U.S. Department of Justice, about 11.7 million people were victims of identity theft over the past 2 years. This number represents 5% of all persons age 16 or older in the United States. Of these victims, 53% reported that the identity theft occurred from the unauthorized use of a credit card. Consider a randomly selected person of age 16 or older in the United States. a. What is the probability that this person was a victim of identity theft? b. What is the probability that this person was a victim of identity theft that occurred from the unauthorized use of a credit card?

Problem 63E Study of why EMS workers leave the job. Refer to the Journal of Allied Health (Fall 2011) study of why emergency medical service (EMS) workers leave the profession, Exercise 3.41 (p. 153). Recall that in a sample of 244 former EMS workers, 127 were fully compensated while on the job, 45 were partially compensated, and 72 had noncompensated volunteer positions. Also, the numbers of EMS workers who left because of retirement were 7 for fully compensated workers, 11 for partially compensated workers, and 10 for noncompensated volunteers. a. Given that the former EMS worker was fully compensated while on the job, estimate the probability that the worker left the EMS profession due to retirement. b. Given that the former EMS worker had a noncompensated volunteer position, estimate the probability that the worker left the EMS profession due to retirement. c. Are the events {a former EMS worker was fully compensated on the job} and {a former EMS worker left the job due to retirement} independent? Explain. 3.41 Study of why EMS workers leave the job. An investigation into why emergency medical service (EMS) workers leave the profession was published in the Journal of Allied Health (Fall 2011). The researchers surveyed a sample of 244 former EMS workers, of which 127 were fully compensated while on the job, 45 were partially compensated, and 72 had noncompensated volunteer positions. The numbers of EMS workers who left because of retirement were 7 for fully compensated workers, 11 for partially compensated workers, and 10 for noncompensated volunteers. One of the 244 former EMS workers is selected at random. a. Find the probability that the former EMS worker was fully compensated while on the job. b. Find the probability that the former EMS worker was fully compensated while on the job and left due to retirement. c. Find the probability that the former EMS worker was not fully compensated while on the job. d. Find the probability that the former EMS worker was either fully compensated while on the job or left due to retirement.

Stock market participation and IQ. Refer to The Journal of Finance (December 2011) study of whether the decision to invest in the stock market is dependent on IQ, Exercise 3.42 (p. 153). The summary table giving the number of the 158,044 Finnish citizens in each IQ score/investment category is reproduced below. Again, suppose one of the citizens is selected at random. Source: Based on Grinblatt, M., Keloharju, M., & Linnainaa, J. “IQ and Stock Market Participation,” The Journal of Finance, Vol. 66, No. 6, December 2011 (data from Table 1 and Figure 1). a. 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? b. 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? c. Based on the results, parts a and b, does it appear that investing in the stock market is dependent on IQ? Explain.

Problem 66E Degrees of best-paid CEOs. Refer to the results of the Forbes (April 13, 2011) survey of the top 40 best-paid CEOs shown in Table 2.1 (p. 39). The data on highest degree obtained are summarized in the SPSS printout below. a. What is the probability that the highest degree obtained by the first CEO you select is a bachelor’s degree? b. Suppose the highest degree obtained by each of the first four CEOs you select is a bachelor’s degree. What is the probability that the highest degree obtained by the fifth CEO you select is a bachelor’s degree?

Problem 67E Ambulance response time. Geographical Analysis (Jan. 2010) presented a study of Emergency Medical Services (EMS) ability to meet the demand for an ambulance. In one example, the researchers presented the following scenario. An ambulance station has one vehicle and two demand locations, A and B. The probability that the ambulance can travel to a location in under eight minutes is .58 for location A and .42 for location B. The probability that the ambulance is busy at any point in time is .3. a. Find the probability that EMS can meet demand for an ambulance at location A. ________________ b. Find the probability that EMS can meet demand for an ambulance at location B.

Problem 64E Working on summer vacation. Refer to the Adweek/Harris (July 2011) poll of whether U.S. adults work during their summer vacation, Exercise 3.14 (p. 141). Recall that the poll found that 46% of the respondents work during their summer vacation, 35% do not work at all while on vacation, and 19% were unemployed. Also, 35% of those who work while on vacation do so by monitoring their business e-mails. a. Given that a randomly selected poll respondent will work while on summer vacation, what is the probability that the respondent will monitor business e-mails? b. What is the probability that a randomly selected poll respondent will work while on summer vacation and will monitor business e-mails? c. What is the probability that a randomly selected poll respondent will not work while on summer vacation and will monitor business e-mails? 3.14 Working on summer vacation. Is summer vacation a break from work? Not according to an Adweek/Harris (July 2011) poll of 3,304 U.S. adults. The poll found that 46% of the respondents work during their summer vacation, 35% do not work at all while on vacation, and 19% were unemployed. Consider the work status during summer vacation of a randomly selected poll respondent. a. List the sample points for this experiment. b. Assign reasonable probabilities to the sample points. c. What is the probability that a randomly selected poll respondent will not work while on summer vacation or is unemployed?

Problem 68E Working mothers with children. The U.S. Census Bureau reports a decline in the percentage of mothers in the workforce who have infant children. The following table gives a breakdown of the marital status and working status of the 1.8 million mothers with infant children in the year 2010. (The numbers in the table are reported in thousands.) Consider the following events: A = {Mom with infant works}, B = {Mom with infant is married and living with husband}. Are A and B independent events? Working Not Working Married/living with husband 1,174 89 All other arrangements 416 121 Source: Data from U.S. Census Bureau, Bureau of Labor Statistics, 2010.

Problem 71E Are you really being served red snapper? Red snapper is a rare and expensive reef fish served at upscale restaurants. Federal law prohibits restaurants from serving a cheaper, look-alike variety of fish (e.g., vermillion snapper or lane snapper) to customers who order red snapper. Researchers at the University of North Carolina used DNA analysis to examine fish specimens labeled “red snapper” that were purchased from vendors across the country (Nature, July 15, 2004). The DNA tests revealed that 77% of the specimens were not red snapper but the cheaper, look-alike variety of fish. a. Assuming the results of the DNA analysis are valid, what is the probability that you are actually served red snapper the next time you order it at a restaurant? b. If there are five customers at a restaurant, all who have ordered red snapper, what is the probability that at least one customer is actually served red snapper?

Problem 69E Intrusion detection systems. A computer intrusion detection system (IDS) is designed to provide an alarm whenever an intrusion (e.g., unauthorized access) is being attempted into a computer system. A probabilistic evaluation of a system with two independently operating intrusion detection systems (a double IDS) was published in the Journal of Research of the National Institute of Standards and Technology (November/December 2003). Consider a double IDS with system A and system B. If there is an intruder, system A sounds an alarm with probability .9, and system B sounds an alarm with probability .95. If there is no intruder, the probability that system A sounds an alarm (i.e., a false alarm) is .2, and the probability that system B sounds an alarm is .1. Assume that under a given condition (intruder or not), systems A and B operate independently. a. Using symbols, express the four probabilities given in the example. b. If there is an intruder, what is the probability that both systems sound an alarm? c. If there is no intruder, what is the probability that both systems sound an alarm? d. Given an intruder, what is the probability that at least one of the systems sounds an alarm?

Lie detector test. The software for a lie detector based on voice stress levels—called the computerized voice stress analyzer (CVSA)—is available for about $10,000. The manufacturer claims that the CVSA is 98% accurate, and, unlike a polygraph machine, will not be thrown off by drugs and medical factors. However, laboratory studies by the U.S. Defense Department found that the CVSA had an accuracy rate of 49.8%—slightly less than pure chance. Suppose the CVSA is used to test the veracity of four suspects. Assume the suspects’ responses are independent. a. If the manufacturer’s claim is true, what is the probability that the CVSA will correctly determine the veracity of all four suspects? b. If the manufacturer’s claim is true, what is the probability that the CVSA will yield an incorrect result for at least one of the four suspects? c. Suppose that in a laboratory experiment conducted by the U.S. Defense Department on four suspects, the CVSA yielded incorrect results for two of the suspects. Use this result to make an inference about the true accuracy rate of the lie detector.

Problem 70E Wine quality and soil. The Journal of Wine Research (Vol. 21, 2010) published a study of the effects of soil and climate on the quality of wine produced in Spain. The soil at two vineyards— Llarga and Solar—was the focus of the analysis. Wine produced from grapes grown in each of the two vineyards was evaluated for each of three different years (growing seasons) by a wine-tasting panel. Based on the taste tests, the panel (as a group) selected the wine with the highest quality. a. How many different wines were evaluated by the panel, where one wine was produced for each vineyard/growing season combination? b. If the wines were all of equal quality, what is the probability that the panel selected a Llarga wine as the wine with the highest quality? c. If the wines were all of equal quality, what is the probability that the panel selected a wine produced in year 3 as the wine with the highest quality? d. The panel consisted of four different wine tasters who performed the evaluations independently of each other. If the wines were all of equal quality, what is the probability that all four tasters selected a Llarga wine as the wine with the highest quality?

Problem 73E Patient medical instruction sheets. Physicians and pharmacists sometimes fail to inform patients adequately about the proper application of prescription drugs and about the precautions to take in order to avoid potential side effects. One method of increasing patients’ awareness of the problem is for physicians to provide patient medication instruction (PMI) sheets. The American Medical Association, however, has found that only 20% of the doctors who prescribe drugs frequently distribute PMI sheets to their patients. Assume that 20% of all patients receive the PMI sheet with their prescriptions and that 12% receive the PMI sheet and are hospitalized because of a drug-related problem. What is the probability that a person will be hospitalized for a drug-related problem given that the person received the PMI sheet?

Problem 74E Risk of a natural gas pipeline accident. Process Safety Progress (Dec. 2004) published a risk analysis for a natural gas pipeline between Bolivia and Brazil. The most likely scenario for an accident would be natural gas leakage from a hole in the pipeline. The probability that the leak ignites immediately (causing a jet fire) is .01. If the leak does not immediately ignite, it may result in a delayed ignition of a gas cloud. Given no immediate ignition, the probability of delayed ignition (causing a flash fire) is .01. If there is no delayed ignition, the gas cloud will harmlessly disperse. Suppose a leak occurs in the natural gas pipeline. Find the probability that either a jet fire or a flash fire will occur. Illustrate with a tree diagram.

Problem 76E Software defects in NASA spacecraft instrument code. Portions of computer software code that may contain undetected defects are called blind spots. The issue of blind spots in software code evaluation was addressed at the 8th IEEE International Symposium on High Assurance Software Engineering (March 2004). The researchers developed guidelines for assessing methods of predicting software defects using data on 498 modules of software code written in “C” language for a NASA spacecraft instrument. One simple prediction algorithm is to count the lines of code in the module; any module with more than 50 lines of code is predicted to have a defect. The accompanying file contains the predicted and actual defect status of all 498 modules. A standard approach to evaluating a software defect prediction algorithm is to form a two-way summary table similar to the one shown here. In the table, a, b, c, and d represent the number of modules in each cell. Software engineers use these table entries to compute several probability measures, called accuracy, detection rate, false alarm rate, and precision. Module has defects False True Algorithm No a b Predicts Defects Yes c d a. Accuracy is defined as the probability that the prediction algorithm is correct. Write a formula for accuracy as a function of the table values a, b, c, and d. b. The detection rate is defined as the probability that the algorithm predicts a defect, given that the module actually is a defect. Write a formula for detection rate as a function of the table values a, b, c, and d. c. The false alarm rate is defined as the probability that the algorithm predicts a defect, given that the module actually has no defect. Write a formula for false alarm rate as a function of the table values a, b, c, and d. d. Precision is defined as the probability that the module has a defect, given that the algorithm predicts a defect. Write a formula for precision as a function of the table values a, b, c, and d. e. Access the accompanying file and compute the values of accuracy, detection rate, false alarm rate, and precision. Interpret the results.

Problem 75E Most likely coin-tossing sequence. In Parade Magazine’s (Nov. 26, 2000) column “Ask Marilyn,” the following question was posed: “I have just tossed a [balanced] coin 10 times, and I ask you to guess which of the following three sequences was the result. One (and only one) of the sequences is genuine.” (1) H H H H H H H H H H (2) H H T T H T T H H H (3) T T T T T T T T T T a. Demonstrate that prior to actually tossing the coins, the three sequences are equally likely to occur. b. Find the probability that the 10 coin tosses result in all heads or all tails. c. Find the probability that the 10 coin tosses result in a mix of heads and tails. d. Marilyn’s answer to the question posed was “Though the chances of the three specific sequences occurring randomly are equal . . . it’s reasonable for us to choose sequence (2) as the most likely genuine result.” If you know that only one of the three sequences actually occurred, explain why Marilyn’s answer is correct. [Hint: Compare the probabilities in parts b and c.]

Problem 77E Suppose the events B1 and B2 are mutually exclusive and complementary events, such that P(B1) = .75 and P(B2) = .25. Consider another event A such that P(A|B1) = .3 and P(A|B2) = .5. a. Find P(B1 ? A). b. Find P(B2 ? A). c. Find P(A) using the results in parts a and b. d. Find P(B1 |A). e. Find P(B2 |A).

Problem 78E Suppose the events B1, B2, and B3 are mutually exclusive and complementary events, such that P(B1) = .2, P(B2) = .15, and P(B3) = .65. Consider another event A such that P(A|B1) = .4, P(A|B2) = .25, and P(A|B3) = .6. Use Bayes’s Rule to find a. P(B1|A) b. P(B2 |A) c. P(B3 |A)

Problem 79E Suppose the events B1, B2, and B3 are mutually exclusive and complementary events, such that P(B1) = .2, P(B2) = .15, and P(B3) = .65. Consider another event A such that P(A) = .4. If A is independent of B1, B2, and B3, use Bayes’s Rule to show that P(B1 |A) = P(B1) = .2.

Problem 80E Tests for Down syndrome. Currently, there are three diagnostic tests available for chromosome abnormalities in a developing fetus: triple serum marker screening, ultrasound, and amniocentesis. The safest (to both the mother and fetus) and least expensive of the three is the ultrasound test. Two San Diego State University statisticians investigated the accuracy of using ultrasound to test for Down syndrome (Chance, Summer 2007). Let D denote that the fetus has a genetic marker for Down syndrome and N denote that the ultrasound test is normal (i.e., no indication of chromosome abnormalities). Then, the statisticians desired the probability P(D/N). Use Bayes’s Rule and the following probabilities (provided in the article) to find the desired probability: P(D) = 1/180, P(Dc) = 79/80, P(N|D) = 1/2 , P(Nc |D) = 1/2 , P(N|Dc) = 1, and P(Nc |Dc) = 0.

Problem 82E Errors in estimating job costs. A construction company employs three sales engineers. Engineers 1, 2, and 3 estimate the costs of 30%, 20%, and 50%, respectively, of all jobs bid by the company. For i = 1, 2, 3, define Ei to be the event that a job is estimated by engineer i. The following probabilities describe the rates at which the engineers make serious errors in estimating costs: P(error|E1) = .01, P(error|E2) = .03, and P(error|E3) = .02 a. If a particular bid results in a serious error in estimating job cost, what is the probability that the error was made by engineer 1? b. If a particular bid results in a serious error in estimating job cost, what is the probability that the error was made by engineer 2? c. If a particular bid results in a serious error in estimating job cost, what is the probability that the error was made by engineer 3? d. Based on the probabilities, parts a–c, which engineer is most likely responsible for making the serious error?

Problem 81E Fingerprint expertise. Contrary to what is presented on TV shows like CSI, fingerprint identification is not fully automated. Expert examiners are required to identify the person who left the fingerprint. A study published in Psychological Science (August 2011) tested the accuracy of experts and novices in identifying fingerprints. Participants were presented pairs of fingerprints and asked to judge whether the prints in each pair matched. The pairs were presented under three different conditions: prints from the same individual (match condition), nonmatching but similar prints (similar distracter condition), and nonmatching and very dissimilar prints (nonsimilar distracter condition). The percentages of correct decisions made by the two groups under each of the three conditions are listed in the table. Condition Fingerprint Experts Novices Match 92.12% 74.55% Similar Distracter 99.32% 44.82% Nonsimilar Distracter 100% 77.03% Source: Based on Tangen, J. M., Thompson, M. B., & McCarthy, D. J. “Identifying fingerprint expertise,” Psychological Science, Vol. 22, No. 8, August 2011 (Figure 1). a. Given a pair of matched prints, what is the probability that an expert failed to identify the match? b. Given a pair of matched prints, what is the probability that a novice failed to identify the match? c. Assume the study included 10 participants, 5 experts and 5 novices. Suppose that a pair of matched prints was presented to a randomly selected study participant and the participant failed to identify the match. Is the participant more likely to be an expert or a novice?

Problem 84E Drug testing in athletes. Due to inaccuracies in drugtesting procedures (e.g., false positives and false negatives) in the medical field, the results of a drug test represent only one factor in a physician’s diagnosis. Yet, 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 Chance (Spring 2004), University of Texas biostatisticians D. A. Berry and L. Chastain demonstrated the application of Bayes’s Rule for making inferences about testosterone abuse among Olympic athletes. They used the following example: 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.)

Problem 86E Nondestructive evaluation. Nondestructive evaluation (NDE) describes methods that quantitatively characterize materials, tissues, and structures by noninvasive means, such as X-ray computed tomography, ultrasonics, and acoustic emission. Recently, NDE was used to detect defects in steel castings (JOM, May 2005). Assume that the probability that NDE detects a “hit” (i.e., predicts a defect in a steel casting) when, in fact, a defect exists is .97. (This is often called the probability of detection.) Also assume that the probability that NDE detects a hit when, in fact, no defect exists is .005. (This is called the probability of a false call.) Past experience has shown a defect occurs once in every 100 steel castings. If NDE detects a hit for a particular steel casting, what is the probability that an actual defect exists?

Problem 83E Fish contaminated by a plant’s toxic discharge. Refer to the U.S. Army Corps of Engineers’ study on the DDT contamination of fish in the Tennessee River (Alabama), Example 1.5 (p. 13). Part of the investigation focused on how far upstream the contaminated fish have migrated. (A fish is considered to be contaminated if its measured DDT concentration is greater than 5.0 parts per million.) a. Considering only the contaminated fish captured from the Tennessee River, the data reveal that 52% of the fish are found between 275 and 300 miles upstream, 39% are found 305 to 325 miles upstream, and 9% are found 330 to 350 miles upstream. Use these percentages to determine the probabilities, P(275-300), P(305-325), and P(330-350). b. Given that a contaminated fish is found a certain distance upstream, the probability that it is a channel catfish (CC) is determined from the data as P(CC|275-300) = .775, P(CC|305-325) = .77, and P(CC|330-350) = .86. If a contaminated channel catfish is captured from the Tennessee River, what is the probability that it was captured 275–300 miles upstream?

Problem 85E Mining for dolomite. Dolomite is a valuable mineral that is found in sedimentary rock. During mining operations, dolomite is often confused with shale. The radioactivity features of rock can aid miners in distinguishing between dolomite and shale rock zones. For example, if the Gamma ray reading of a rock zone exceeds 60 API units, the area is considered to be mostly shale (and is not mined); if the Gamma ray reading of a rock zone is less than 60 API units, the area is considered to be abundant in dolomite (and is mined). Data on 771 core samples in a rock quarry collected by the Kansas Geological Survey revealed the following: 476 of the samples are dolomite and 295 of the samples are shale. Of the 476 dolomite core samples, 34 had a Gamma ray reading greater than 60. Of the 295 shale core samples, 280 had a Gamma ray reading greater than 60. Suppose you obtain a Gamma ray reading greater than 60 at a certain depth of the rock quarry. Should this area be mined?

Problem 87E Purchasing microchips. An important component of your desktop or laptop personal computer (PC) is a microchip. The table gives the proportions of microchips that a certain PC manufacturer purchases from seven suppliers. Supplier Proportion .15 .05 .10 .20 .12 .20 .18 a. It is known that the proportions of defective microchips produced by the seven suppliers are .001, .0003, .0007, .006, .0002, .0002, and .001, respectively. If a single PC microchip failure is observed, which supplier is most likely responsible? b. Suppose the seven suppliers produce defective microchips at the same rate, .0005. If a single PC microchip failure is observed, which supplier is most likely responsible?

Problem 88E Intrusion detection systems. Refer to the Journal of Research of the National Institute of Standards and Technology (Nov.–Dec. 2003) study of a double intrusion detection system with independent systems, Exercise 3.69 (p. 167). Recall that if there is an intruder, system A sounds an alarm with probability .9, and system B sounds an alarm with probability .95. If there is no intruder, system A sounds an alarm with probability .2, and system B sounds an alarm with probability .1. Now assume that the probability of an intruder is .4. Also assume that under a given condition (intruder or not), systems A and B operate independently. If both systems sound an alarm, what is the probability that an intruder is detected? 3.69 Intrusion detection systems. A computer intrusion detection system (IDS) is designed to provide an alarm whenever an intrusion (e.g., unauthorized access) is being attempted into a computer system. A probabilistic evaluation of a system with two independently operating intrusion detection systems (a double IDS) was published in the Journal of Research of the National Institute of Standards and Technology (November/December 2003). Consider a double IDS with system A and system B. If there is an intruder, system A sounds an alarm with probability .9, and system B sounds an alarm with probability .95. If there is no intruder, the probability that system A sounds an alarm (i.e., a false alarm) is .2, and the probability that system B sounds an alarm is .1. Assume that under a given condition (intruder or not), systems A and B operate independently. a. Using symbols, express the four probabilities given in the example. b. If there is an intruder, what is the probability that both systems sound an alarm? c. If there is no intruder, what is the probability that both systems sound an alarm? d. Given an intruder, what is the probability that at least one of the systems sounds an alarm?

Problem 89E Forensic analysis of JFK assassination bullets. Following the assassination of President John F. Kennedy (JFK) in 1963, the House Select Committee on Assassinations (HSCA) conducted an official government investigation. The HSCA concluded that although there was a probable conspiracy involving at least one shooter in addition to Lee Harvey Oswald, the additional shooter missed all limousine occupants. A recent analysis of assassination bullet fragments, reported in the Annals of Applied Statistics (Vol. 1, 2007), contradicted these findings, concluding that the evidence used by the HSCA to rule out a second assassin is fundamentally flawed. It is well documented that at least two different bullets were the source of bullet fragments found after the assassination. Let E = {bullet evidence used by the HSCA}, T = {two bullets used in the aassassination}, and Tc = {more than two bullets used in the assassination}. Given the evidence (E), which is more likely to have occurred—two bullets used (T) or more than two bullets used (Tc)? a. The researchers demonstrated that the ratio, P(T|E) >P(Tc|E), is less than 1. Explain why this result supports the theory of more than two bullets used in the assassination of JFK. b. To obtain the result, part a, the researchers first showed that P(T |E)>P(Tc|E) = [P(E|T) = P(T)]> [P(E|Tc) = P(Tc)] Demonstrate this equality using Bayes’s Rule.

Problem 90SE Which of the following pairs of events are mutually exclusive? Justify your response. a. {The Dow Jones Industrial Average increases on Monday.}, {A large New York bank decreases its prime interest rate on Monday.} b. {The next sale by a PC retailer is a notebook computer.}, {The next sale by a PC retailer is a desktop computer.} c. {You reinvest all your dividend income in a limited partnership.}, {You reinvest all your dividend income in a money market fund.}

A and B are mutually exclusive events, with P(A) = .2 and P(B) = .3. a. Find P(A|B). b. Are A and B independent events?

A sample space consists of four sample points, where \(P(S_1)=.2,\ P(S_2)=.1,\ P(S_3)=.3,\ \text{and} P(S_4)=.4\). a. Show that the sample points obey the two probability rules for a sample space. b. If an event \(A=\{S_1,\ S_4\}\), find P(A).

Two events, A and B, are independent, with P(A) = .3 and P(B) = .1. a. Are A and B mutually exclusive? Why? b. Find P(A| B) and P(B |A). c. Find \(P(A \cup B)\).

Given that \(P(A \cap B\) = .4 and \(P(A \mid B)\) = .8, find P(B).

For two events A and B, suppose P(A) = .7, P(B) = .5, and \(P(A \cap B)=.4\). Find \(P(A \cup B)\).

Problem 96SE A random sample of n = 5 is to be selected from N = 50. In how many different ways can the sample be drawn?

Find the numerical value of a. 6! b. \(\left(\begin{array}{c}10 \\ 9\end{array}\right)\) c. \(\left(\begin{array}{c}10 \\ 1\end{array}\right)\) d. \(\left(\begin{array}{l}6 \\ 3\end{array}\right)\) e. 0!

Problem 99SE Management system failures. Refer to the Process Safety Progress (Dec. 2004) study of 83 industrial accidents caused by management system failures, Exercise 2.146 (p. 115). A summary of the root causes of these 83 incidents is reproduced in the following table. One of the 83 incidents is randomly selected and the root cause is determined. Management System Cause Category Number of Incidents Engineering and Design 27 Procedures and Practices 24 Management and Oversight 22 Training and Communication 10 Total 83 Source: Based on Blair, A. S. “Management system failures identified in incidents investigated by the U.S. Chemical Safety and Hazard Investigation Board,” Process Safety Progress, Vol. 23, No. 4, Dec. 2004, pp. 232–236 (Table 1). a. List the sample points for this problem and assign reasonable probabilities to them. b. Find and interpret the probability that an industrial accident is caused by faulty engineering and design. c. Find and interpret the probability that an industrial accident is caused by something other than faulty procedures and practices. 2.146 Management system failures. The U.S. Chemical Safety and Hazard Investigation Board (CSB) is responsible for determining the root cause of industrial accidents (Process Safety Progress, Dec. 2004). The accompanying table gives a breakdown of the root causes of 83 incidents caused by management system failures. Management System Cause Category Number of Incidents Engineering& Design 27 Procedures& Practices 24 Management& Oversight 22 Training& Communication 10 Total 83 Source: Based on A. S. Blair, “Management system failures identified in incidents investigated by the U.S. Chemical Safety and Hazard Investigation Board,” PROCESS SAFETY PROGRESS, Vol. 23, No. 4, December 2004, pp. 232–236 (Table 1). a. Find the relative frequency of the number of incidents for each cause category. b. Construct a Pareto diagram for the data. c. From the Pareto diagram, identify the cause categories with the highest (and lowest) relative frequency of incidents.

Problem 97SE 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 P(1) = .3, P(2) = .2, P(3) = .1, P(4) = .1, P(5) = .1, and P(6) = .2. b. Are A and B independent? Mutually exclusive? Why? c. Are B and C independent? Mutually exclusive? Why?

Problem 100SE Annual compensation and benefits report. Each year, Hudson Institute conducts a survey of 10,000 U.S. workers about their total compensation packages. One question in Hudson’s Compensation and Benefits Report focused on employees who received raises in the past year. Of these employees, 35% reported that their raise was based on job performance, 50% reported that it was based on a standard cost of living, and the remainder (15%) were unsure how their raises were determined. Suppose we select (at random) one of the U.S. workers surveyed who received a raise last year and inquire about how that worker’s raise was determined. a. List the sample points for this experiment. b. Assign reasonable probabilities to the sample points. c. Find the probability that the raise was based either on job performance or a standard cost of living.

Condition of public school facilities. The National Center for Education Statistics (NCES) conducted a survey on the condition of America’s public school facilities. The survey revealed the following information. The probability that a public school building has inadequate plumbing is .25. Of the buildings with inadequate plumbing, the probability that the school has plans for repairing the building is .38. Find the probability that a public school building has inadequate plumbing and will be repaired.

Ownership of small businesses. According to the Journal of Business Venturing (Vol. 17, 2002), 27% of all small businesses owned by non-Hispanic whites nationwide are women-owned firms. If we select, at random, a small business owned by a non-Hispanic white, what is the probability that it is a male-owned firm?

Problem 105SE Survey on energy conservation. A state energy agency mailed questionnaires on energy conservation to 1,000 homeowners in the state capital. Five hundred questionnaires were returned. Suppose an experiment consists of randomly selecting and reviewing one of the returned questionnaires. Consider the events: A: {The home is constructed of brick.} B: {The home is more than 30 years old.} C: {The home is heated with oil.} Describe each of the following events in terms of unions, intersections, and complements (i.e., A ? B, A?B, Ac, etc.): a. The home is more than 30 years old and is heated with oil. b. The home is not constructed of brick. c. The home is heated with oil or is more than 30 years old. d. The home is constructed of brick and is not heated with oil.

Problem 104SE Speeding linked to fatal car crashes. According to 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, Aug.200). The probability that speeding is a cause of a fatal crash is .3. Furthermore, the probability that speeding and missing a curve are causes of a fatal crash is .12. Given that speeding is a cause of a fatal crash, what is the probability that the crash occurred on a curve?

Identifying urban counties. Urban and rural describe geographic areas for which land zoning regulations, school district policy, and public service policy are often set. However, the characteristics of urban/rural areas are not clearly defined. Researchers at the University of Nevada (Reno) asked a sample of county commissioners to give their perception of the single most important factor in identifying urban counties (Professional Geographer, Feb. 2000). In all, five factors were mentioned by the commissioners: total population, agricultural change, presence of industry, growth, and population concentration. The survey results are displayed in the pie chart below. Suppose one of the commissioners is selected at random and the most important factor specified by the commissioner is recorded. a. List the sample points for this experiment. b. Assign reasonable probabilities to the sample points. c. Find the probability that the most important factor specified by the commissioner is population related.

Problem 103SE New car crash tests. Refer to the National Highway Traffic Safety Administration (NHTSA) crash tests of new car models, Exercise 2.149 (p. 116). 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.

Problem 107SE Use of country club facilities. A local country club has a membership of 600 and operates facilities that include an 18-hole championship golf course and 12 tennis courts. Before deciding whether to accept new members, the club president would like to know how many members regularly use each facility. A survey of the membership indicates that 70% regularly use the golf course, 50% regularly use the tennis courts, and 5% use neither of these facilities regularly. a. Construct a Venn diagram to describe the results of the survey. b. If one club member is chosen at random, what is the probability that the member uses either the golf course or the tennis courts or both? c. If one member is chosen at random, what is the probability that the member uses both the golf and the tennis facilities? d. A member is chosen at random from among those known to use the tennis courts regularly. What is the probability that the member also uses the golf course regularly?

Problem 108SE Monitoring quality of power equipment. Mechanical Engineering (Feb. 2005) reported on the need for wireless networks to monitor the quality of industrial equipment. For example, consider Eaton Corp., a company that develops distribution products. Eaton estimates that 90% of the electrical switching devices it sells can monitor the quality of the power running through the device. Eaton further estimates that of the buyers of electrical switching devices capable of monitoring quality, 90% do not wire the equipment up for that purpose. Use this information to estimate the probability that an Eaton electrical switching device is capable of monitoring power quality and is wired up for that purpose.

Problem 109SE Federal civil trial appeals. The Journal of the American Law and Economics Association (Vol. 3, 2001) published the results of a study of appeals of federal civil trials. The accompanying table, extracted from the article, gives a breakdown of 2,143 civil cases that were appealed by either the plaintiff or the defendant. The outcome of the appeal, as well as the type of trial (judge or jury), was determined for each civil case. Suppose one of the 2,143 cases is selected at random and both the outcome of the appeal and the type of trial are observed. Jury Judge Totals Plaintiff trial win— reversed 194 71 265 Plaintiff trial win— affirmed/dismissed 429 240 669 Defendant trial win— reversed 111 68 179 Defendant trial win— affirmed/ dismissed 731 299 1,030 Totals 1,465 678 2,143 a. Find P(A), where A = {jurytrial} . ________________ b. Find P(B), where B = {plaintifftrialwinis reversed } . ________________ c. Are A and B mutually exclusive events? ________________ d. Find P(Ac). ________________ e. Find P(A U B) . ________________ f. Find P(A n B) .

Problem 110SE Assembling a panel of energy experts. The state legislature has appropriated $1 million to be distributed in the form of grants to individuals and organizations engaged in the research and development of alternative energy sources. You have been hired by the state’s energy agency to assemble a panel of five energy experts whose task it will be to determine which individuals and organizations should receive the grant money. You have identified 11 equally qualified individuals who are willing to serve on the panel. How many different panels of five experts could be formed from these 11 individuals?

Problem 111SE Testing a watch manufacturer’s claim. A manufacturer of electronic digital watches 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.

Problem 112SE Ranking razor blades. The corporations in the highly competitive razor blade industry do a tremendous amount of advertising each year. Corporation G gave a supply of three top-name brands, G, S, and W, to a consumer and asked her to use them and rank them in order of preference. The corporation was, of course, hoping the consumer would prefer its brand and rank it first, thereby giving them some material for a consumer interview advertising campaign. If the consumer did not prefer one blade over any other but was still required to rank the blades, what is the probability that a. The consumer ranked brand G first? b. The consumer ranked brand G last? c. The consumer ranked brand G last and brand W second? d. The consumer ranked brand W first, brand G second, and brand S third?

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. The results of consecutive tosses of a coin b. The opinions of randomly selected individuals in a preelection poll c. A Major League Baseball player’s results in two consecutive at-bats d. The amount of gain or loss associated with investments in different stocks if these stocks are bought on the same day and sold on the same day 1 month later e. The amount of gain or loss associated with investments in different stocks that are bought and sold in different time periods, 5 years apart f. The prices bid by two different development firms in response to a building construction proposal

Problem 115SE Home modifications for wheelchair users. The American Journal of Public Health (Jan. 2002) reported on a study of elderly wheelchair users who live at home. A sample of 306 wheelchair users, age 65 or older, were surveyed about whether they had an injurious fall during the year and whether their home features any one of five structural modifications: bathroom modifications, widened doorways/ hallways, kitchen modifications, installed railings, and easyopen doors. The responses are summarized in the accompanying table. Suppose we select, at random, one of the 306 surveyed wheelchair users. Home Features Injurious Fall(s) No Falls Totals All 5 2 7 9 At least 1 but not all 26 162 188 None 20 89 109 Totals 48 258 306 Source: Based on Berg, K., Hines, M., & Allen, S. “Wheelchair users at home: Few home modifications and many injurious falls,” American Journal of Public Health, Vol. 92, No. 1, Jan. 2002 (Table 1). a. Find the probability that the wheelchair user had an injurious fall. b. Find the probability that the wheelchair user had all five features installed in the home. c. Find the probability that the wheelchair user had no falls and none of the features installed in the home. d. Given the wheelchair user had all five features installed, what is the probability that the user had an injurious fall? e. Given the wheelchair user had none of the features installed, what is the probability that the user had an injurious fall?

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 2006 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 geographical 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 2006, 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. In 2006, 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 in 2006. 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?

Problem 113SE Link between cigar smoking and cancer. The Journal of the National Cancer Institute (Feb. 16, 2000) published the results of a study that investigated the association between cigar smoking and death from tobacco-related cancers. Data were obtained for a national sample of 137,243 American men. The results are summarized in the table. Each male in the study was classified according to his cigar-smoking status and whether or not he died from a tobacco-related cancer. Died from Cancer Did Not Die from Cancer Totals Never Smoked Cigars 782 120,747 121,529 Former Cigar Smoker 91 7,757 7,848 Current Cigar Smoker 141 7,725 7,866 Totals 1,014 136,229 137,243 Source: Based on Shapiro, J. A., Jacobs, E. J., & Thun, M. J. “Cigar smoking in men and risk of death from tobacco-related cancers,” Journal of the National Cancer Institute, Vol. 92, No. 4, Feb. 16, 2000. pp. 333–337 (Table 2). a. Find the probability that a man who never smoked cigars died from cancer. b. Find the probability that a former cigar smoker died from cancer. c. Find the probability that a current cigar smoker died from cancer.

Problem 117SE 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?

Repairing a computer network. The local area network (LAN) for the College of Business computing system at a large university is temporarily shut down for repairs. Previous shutdowns have been due to hardware failure, software failure, or power failure. Maintenance engineers have determined that the probabilities of hardware, software, and power problems are .01, .05, and .02, respectively. They have also determined that if the system experiences hardware problems, it shuts down 73% of the time. Similarly, if software problems occur, the system shuts down 12% of the time; and if a power failure occurs, the system shuts down 88% of the time. What is the probability that the current shutdown of the LAN is due to hardware failure? Software failure? Power failure?

Problem 119SE Profile of a sustainable farmer. Sustainable development or sustainable farming means finding ways to live and work the Earth without jeopardizing the future. Studies were conducted in five Midwestern states to develop a profile of a sustainable farmer. The results revealed that farmers can be classified along a sustainability scale, depending on whether they are likely (L) or unlikely (U) to engage in the following practices: (1) Raise a broad mix of crops; (2) raise livestock; (3) use chemicals sparingly; and (4) use techniques for regenerating the soil, such as crop rotation. a. List the different sets of classifications that are possible for the four practices (e.g., LUUL). b. Suppose you are planning to interview farmers across the country to determine the frequency with which they fall into the classification sets you listed for part a. Because no information is yet available, assume initially that there is an equal chance of a farmer falling into any single classification set. Using that assumption, what is the probability that a farmer will be classified as unlikely on all four criteria (i.e., classified as a nonsustainable farmer)? c. Using the same assumption as in part b, what is the probability that a farmer will be classified as likely on at least three of the criteria (i.e., classified as a nearsustainable farmer)?

Problem 120SE 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. 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?

Problem 121SE System components operating in series and parallel. Consider the two systems shown in the schematic at the top of the next page. System A operates properly only if all three components operate properly. (The three components are said to operate in series.) The probability of failure for system A components 1, 2, and 3 are .12, .09, and .11, respectively. Assume the components operate independently of each other. System B comprises two subsystems said to operate in parallel. Each subsystem has two components that operate in series. System B will operate properly as long as at least one of the subsystems functions properly. The probability of failure for each component in the system is .1. Assume the components operate independently of each other. a. Find the probability that system A operates properly. b. What is the probability that at least one of the components in system A will fail and therefore that the system will fail? c. Find the probability that system B operates properly. d. Find the probability that exactly one subsystem in system B fails. e. Find the probability that system B fails to operate properly. f. How many parallel subsystems like the two shown here would be required to guarantee that the system operates properly at least 99% of the time?

Problem 122SE Probability of winning a war. Before a country enters into a war, a prudent government will assess the cost, utility, and probability of a victory. University of Georgia professor P. L. Sullivan has developed a statistical model for determining the probability of winning a war based on a government’s capabilities and resources (Journal of Conflict Resolution, Vol. 51, 2007). Now consider the current U.S.-Iraq conflict. One researcher used the model to estimate that the probability of a successful regime change in Iraq was .70 prior to the start of the war. Of course, we now know that the successful regime change was achieved. However, the model also estimates that given the mission is extended to support a weak Iraq government, the probability of ultimate success is only .26. Assume these probabilities are accurate. a. Prior to the start of the U.S.-Iraq war, what is the probability that a successful regime change is not achieved? ________________ b. Given that the mission is extended to support a weak Iraq government, what is the probability that a successful regime change is ultimately achieved? ________________ c. Suppose the probability of the United States extending the mission to support a weak Iraq government was 55. Find the probability that the mission is extended and results in a successful regime change.

Problem 124SE Malfunctioning production lines. A manufacturing operation utilizes two production lines to assemble electronic fuses. Both lines produce fuses at the same rate and generally produce 2.5% defective fuses. However, production line 1 recently suffered mechanical difficulty and produced 6.0% defectives during a 3-week period. This situation was not known until several lots of electronic fuses produced in this period were shipped to customers. If one of the two fuses tested by a customer was found to be defective, what is the probability that the lot from which it came was produced on malfunctioning line 1? (Assume all the fuses in the lot were produced on the same line.)

Problem 123SE Detecting traces of TNT. University of Florida researchers in the Department of Materials Science and Engineering have invented a technique to rapidly detect traces of TNT (Today, Spring 2005). The method, which involves shining a laser light on a potentially contaminated object, provides instantaneous results and gives no false positives. In this application, a false positive would occur if the laser light detects traces of TNT when, in fact, no TNT is actually present on the object. Let A be the event that the laser light detects traces of TNT. Let B be the event that the object contains no traces of TNT. The probability of a false positive is 0. Write this probability in terms of A and B using symbols such as ?, ?, and |.

Problem 125SE Scrap rate of machine parts. A press produces parts used in the manufacture of large-screen plasma televisions. If the press is correctly adjusted, it produces parts with a scrap rate of 5%. If it is not adjusted correctly, it produces scrap at a 50% rate. From past company records, the machine is known to be correctly adjusted 90% of the time. A quality-control inspector randomly selects one part from those recently produced by the press and discovers it is defective. What is the probability that the machine is incorrectly adjusted?

Problem 126SE Chance of winning at “craps.” A version of the dice game “craps” is played in the following manner. A player starts by rolling two balanced dice. If the roll (the sum of the two numbers showing on the dice) results in a 7 or 11, the player wins. If the roll results in a 2 or a 3 (called craps), the player loses. For any other roll outcome, the player continues to throw the dice until the original roll outcome recurs (in which case the player wins) or until a 7 occurs (in which case the player loses). a. What is the probability that a player wins the game on the first roll of the dice? b. What is the probability that a player loses the game on the first roll of the dice? c. If the player throws a total of 4 on the first roll, what is the probability that the game ends (win or lose) on the next roll?

Problem 127SE Chance of winning blackjack. Blackjack, a favorite game of gamblers, is played by a dealer and at least one opponent (called a player). In one version of the game, 2 cards of a standard 52-card bridge deck are dealt to the player and 2 cards to the dealer. For this exercise, assume that drawing an ace and a face card is called blackjack. If the dealer does not draw a blackjack and the player does, the player wins. If both the dealer and player draw blackjack, a “push” (i.e., a tie) occurs. a. What is the probability that the dealer will draw a blackjack? b. What is the probability that the player wins with a blackjack?

Strategy in the game “Go.” “Go” is one of the oldest and most popular strategic board games in the world, especially in Japan and Korea. The University of Virginia requires MBA students to learn Go to understand how the Japanese conduct business. This two-player game is played on a flat surface marked with 19 vertical and 19 horizontal lines. The objective is to control territory by placing pieces, called stones, on vacant points on the board. Players alternate placing their stones. The player using black stones goes first, followed by the player using white stones. Chance (Summer 1995) published an article that investigated the advantage of playing first (i.e., using the black stones) in Go. The results of 577 games recently played by professional Go players were analyzed. a. In the 577 games, the player with the black stones won 319 times, and the player with the white stones won 258 times. Use this information to assess the probability of winning when you play first in Go. b. Professional Go players are classified by level. Group C includes the top-level players, followed by Group B (middle-level) and Group A (low-level) players. The previous table describes the number of games won by the player with the black stones, categorized by level of the black player and level of the opponent. Assess the probability of winning when you play first in Go for each combination of player and opponent level. Black Player Level Opponent Level Number of Wins Number of Games C A 34 34 C B 69 79 C C 66 118 B A 40 54 B B 52 95 B C 27 79 A A 15 28 A B 11 51 A C 5 39 Totals 319 577 Source: Kim, J., & Kim, H. J. “The advantage of playing first in Go,” Chance, Vol. 8, No. 3, Summer 1995, p. 26 (Table 3). Copyright © 1995 by the American Statistical Association. Reprinted with permission. c. If the player with the black stones is ranked higher than the player with the white stones, what is the probability that black wins? d. Given the players are of the same level, what is the probability that the player with the black stones wins?

“Let’s Make a Deal.” Marilyn vos Savant, who is listed in Guinness Book of World Records Hall of Fame for “Highest IQ,” writes a weekly column in the Sunday newspaper supplement Parade Magazine. Her column, “Ask Marilyn,” is devoted to games of skill, puzzles, and mind bending riddles. In one issue (Parade Magazine, Feb. 24, 1991), vos Savant posed the following question: Suppose you’re on a game show, and you’re given a choice of three doors. Behind one door is a car; behind the others, goats. You pick a door—say, #1—and the host, who knows what’s behind the doors, opens another door—say #3—which has a goat. He then says to you, “Do you want to pick door #2?” Is it to your advantage to switch your choice? Marilyn’s answer: “Yes, you should switch. The first door has a 13 chance of winning [the car], but the second has a 23 chance [of winning the car].” Predictably, vos Savant’s surprising answer elicited thousands of critical letters, many of them from PhD mathematicians, who disagreed with her. Who is correct, the PhDs or Marilyn?