Answer: Show that if one event A is contained in another

Problem 1E Four universities—1, 2, 3, and 4—are participating in a holiday basketball tournament. In the first round, 1 will play 2 and 3 will play 4. Then the two winners will play for the championship, and the two losers will also play. One possible outcome can be denoted by 1324 (1 beats 2 and 3 beats 4 in first-round games, and then 1 beats 3 and 2 beats 4). a. List all outcomes in . b. Let A denote the event that 1 wins the tournament. List outcomes in A. c. Let B denote the event that 2 gets into the championship game. List outcomes in B. d. What are the outcomes inA ?B and in A?B? What are the outcomes in A?

Problem 2E Suppose that vehicles taking a particular freeway exit can turn right (R), turn left (L), or go straight (S). Consider observing the direction for each of three successive vehicles. a. List all outcomes in the event A that all three vehicles go in the same direction. b. List all outcomes in the event B that all three vehicles take different directions. c. List all outcomes in the event C that exactly two of the three vehicles turn right. d. List all outcomes in the event D that exactly two vehicles go in the same direction. e. List outcomes in D,C ?D , and C?D.

Problem 3E Three components are connected to form a system as shown in the accompanying diagram. Because the components in the 2–3 subsystem are connected in parallel, that subsystem will function if at least one of the two individual components functions. For the entire system to function, component 1 must function and so must the 2–3 subsystem. The experiment consists of determining the condition of each component [S (success) for a functioning component and F (failure) for a nonfunctioning component]. a. Which outcomes are contained in the event A that exactly two out of the three components function? b. Which outcomes are contained in the event B that at least two of the components function? c. Which outcomes are contained in the event C that the system functions? d. List outcomes in

Problem 22E The route used by a certain motorist in commuting to work contains two intersections with traffic signals. The probability that he must stop at the first signal is .4, the analogous probability for the second signal is .5, and the probability that he must stop at at least one of the two signals is .6. What is the probability that he must stop a. At both signals? b. At the first signal but not at the second one? c. At exactly one signal?

The computers of six faculty members in a certain department are to be replaced. Two of the faculty members have selected laptop machines and the other four have chosen desktop machines. Suppose that only two of the setups can be done on a particular day, and the two computers to be set up are randomly selected from the six (implying 15 equally likely outcomes; if the computers are numbered 1, 2,..., 6, then one outcome consists of computers 1 and 2, another consists of computers 1 and 3, and so on). a. What is the probability that both selected setups are for laptop computers? b. What is the probability that both selected setups are desktop machines? c. What is the probability that at least one selected setup is for a desktop computer? d. What is the probability that at least one computer of each type is chosen for setup?

Problem 24E Show that if one event A is contained in another event B (i.e., A is a subset of B), then P(A) ? P(B) . [Hint: For such A and B, A and B?A ’, are disjoint and B = A ?(B?A ’),as can be seen from a Venn diagram.] For general A and B, what does this imply about the relationship among P(A?B), P(A) and P(A?B) ?

Problem 43E In five-card poker, a straight consists of five cards with adjacent denominations (e.g., 9 of clubs, 10 of hearts, jack of hearts, queen of spades, and king of clubs). Assuming that aces can be high or low, if you are dealt a five-card hand, what is the probability that it will be a straight with high card 10? What is the probability that it will be a straight? What is the probability that it will be a straight flush (all cards in the same suit)?

Problem 44E Show that . Give an interpretation involving subsets.

Problem 45E The population of a particular country consists of three ethnic groups. Each individual belongs to one of the four major blood groups. The accompanying joint probability table gives the proportions of individuals in the various ethnic group–blood group combinations. Suppose that an individual is randomly selected from the population, and define events by A = { type A selected}, B = { type B selected}, and C = { ethnic group 3 selected}. a. Calculate P(A), P(C), andP(A ?C) . b. Calculate both P(A|C) and P(C|A) , and explain in context what each of these probabilities represents. c.If the selected individual does not have type B blood, what is the probability that he or she is from ethnic group 1?

Problem 64E The Reviews editor for a certain scientific journal decides whether the review for any particular book should be short (1–2 pages), medium (3–4 pages), or long (5–6 pages). Data on recent reviews indicates that 60% of them are short, 30% are medium, and the other 10% are long. Reviews are submitted in either Word or LaTeX. For short reviews, 80% are in Word, whereas 50% of medium reviews are in Word and 30% of long reviews are in Word. Suppose a recent review is randomly selected. a. What is the probability that the selected review was submitted in Word format? b. If the selected review was submitted in Word format, what are the posterior probabilities of it being short, medium, or long?

Problem 65E A large operator of timeshare complexes requires anyone interested in making a purchase to first visit the site of interest. Historical data indicates that 20% of all potential purchasers select a day visit, 50% choose a one-night visit, and 30% opt for a two-night visit. In addition, 10% of day visitors ultimately make a purchase, 30% of one night visitors buy a unit, and 20% of those visiting for two nights decide to buy. Suppose a visitor is randomly selected and is found to have made a purchase. How likely is it that this person made a day visit? A one-night visit? A two-night visit?

Problem 66E Consider the following information about travelers on vacation (based partly on a recent Travelocity poll): 40% check work email, 30% use a cell phone to stay connected to work, 25% bring a laptop with them, 23% both check work email and use a cell phone to stay connected, and 51% neither check work email nor use a cell phone to stay connected nor bring a laptop. In addition, 88 out of every 100 who bring a laptop also check work email, and 70 out of every 100 who use a cell phone to stay connected also bring a laptop. a. What is the probability that a randomly selected traveler who checks work email also uses a cell phone to stay connected? b. What is the probability that someone who brings a laptop on vacation also uses a cell phone to stay connected? c. If the randomly selected traveler checked work email and brought a laptop, what is the probability that he/she uses a cell phone to stay connected?

Problem 85E A quality control inspector is inspecting newly produced items for faults. The inspector searches an item for faults in a series of independent fixations, each of a fixed duration. Given that a flaw is actually present, let p denote the probability that the flaw is detected during any one fixation (this model is discussed in “Human Performance in Sampling Inspection,” Human Factors, 1979: 99–105). a. Assuming that an item has a flaw, what is the probability that it is detected by the end of the second fixation (once a flaw has been detected, the sequence of fixations terminates)? b. Give an expression for the probability that a flaw will be detected by the end of the nth fixation. c. If when a flaw has not been detected in three fixations, the item is passed, what is the probability that a flawed item will pass inspection? d. Suppose 10% of all items contain a flaw [P (randomly chosen item is flawed) = .1]. With the assumption of part (c), what is the probability that a randomly chosen item will pass inspection (it will automatically pass if it is not flawed, but could also pass if it s flawed)? e. Given that an item has passed inspection (no flaws in three fixations), what is the probability that it is actually flawed? Calculate for p = .5.

a.) A lumber company has just taken delivery on a lot of 10, 002 x 4 boards. Suppose that 20% of these boards (2,000) are actually too green to be used in first-quality construction. Two boards are selected at random, one after the other. Let A = { the first board is green} and B = { the second board is green}. Compute P(A), P(B), and P (A?B) (a tree diagram might help). Are A and B independent? b.) With A and B independent and P(A) = P(B) = .2, what is P(A?B)? How much difference is there between this answer and P(A?B)? in part (a)? For purposes of calculating P(A? B)?, can we assume that A and B of part (a) are independent to obtain essentially the correct probability? c.) Suppose the lot consists of ten boards, of which two are green. Does the assumption of independence now yield approximately the correct answer for P (A?B)? What is the critical difference between the situation here and that of part (a)? When do you think an independence assumption would be valid in obtaining an approximately correct answer to P (A?B)?

Consider randomly selecting a single individual and having that person test drive 3 different vehicles. Define events A1, A2, and A3 by A1 =likes vehicle #1 A2 = likes vehicle #2 A3 =likes vehicle #3 Suppose that P(A1)= .55,P(A2) = .65,P(A3 )= .70,P(A1?A2) = .80, P(A2?A3) = .40, and P(A1?A2?A3)= .88. a. What is the probability that the individual likes both vehicle #1 and vehicle #2? b. Determine and interpret P(A2|A3) . c. Are A2 and A3 independent events? Answer in two different ways. d. If you learn that the individual did not like vehicle #1, what now is the probability that he/she liked at least one of the other two vehicles?

Problem 108E In a Little League baseball game, team A’s pitcher throws a strike 50% of the time and a ball 50% of the time, successive pitches are independent of one another, and the pitcher never hits a batter. Knowing this, team B’s manager has instructed the first batter not to swing at anything. Calculate the probability that a. The batter walks on the fourth pitch b. The batter walks on the sixth pitch (so two of the first five must be strikes), using a counting argument or constructing a tree diagram c. The batter walks d. The first batter up scores while no one is out (assuming that each batter pursues a no-swing strategy)

Problem 107E A subject is allowed a sequence of glimpses to detect a target . Let Gi = {the target is detected on the ith glimpse},with pi = P(Gi). Suppose the Gi ' s are independent events, and write an expression for the probability that the target has been detected by the end of the nth glimpse. [Note: This model is discussed in “Predicting Aircraft Detectability,” Human Factors, 1979: 277–291.]

Four engineers, A, B, C, and D, have been scheduled for job interviews at 10 A.M. on Friday, January 13, at Random Sampling, Inc. The personnel manager has scheduled the four for interview rooms 1, 2, 3, and 4, respectively. However, the manager’s secretary does not know this, so assigns them to the four rooms in a completely random fashion (what else!). What is the probability that a. All four end up in the correct rooms? b. None of the four ends up in the correct room?

Problem 4E Each of a sample of four home mortgages is classified as fixed rate (F) or variable rate (V). a. What are the 16 outcomes in ? b. Which outcomes are in the event that exactly three of the selected mortgages are fixed rate? c. Which outcomes are in the event that all four mortgages are of the same type? d. Which outcomes are in the event that at most one of the four is a variable-rate mortgage? e. What is the union of the events in parts (c) and (d), and what is the intersection of these two events? f. What are the union and intersection of the two events in parts (b) and (c)?

Problem 5E A family consisting of three persons—A, B, and C—goes to a medical clinic that always has a doctor at each of stations 1, 2, and 3. During a certain week, each member of the family visits the clinic once and is assigned at random to a station. The experiment consists of recording the station number for each member. One outcome is (1, 2, 1) for A to station 1, B to station 2, and C to station 1. a. List the 27 outcomes in the sample space. b. List all outcomes in the event that all three members go to the same station. c. List all outcomes in the event that all members go to different stations. d. List all outcomes in the event that no one goes to station 2.

Problem 6E A college library has five copies of a certain text on reserve. Two copies (1 and 2) are first printings, and the other three (3, 4, and 5) are second printings. A student examines these books in random order, stopping only when a second printing has been selected. One possible outcome is 5, and another is 213. a. List the outcomes in . b. Let A denote the event that exactly one book must be examined. What outcomes are in A? c. Let B be the event that book 5 is the one selected. What outcomes are in B? d. Let C be the event that book 1 is not examined. What outcomes are in C?

The three most popular options on a certain type of new car are a built-in GPS (A), a sunroof (B), and an automatic transmission (C). If 40% of all purchasers request A, 55% request B, 70% request C, 63% request A or B, 77% request A or C, 80% request B or C, and 85% request A or B or C, determine the probabilities of the following events. [Hint: “A or B” is the event that at least one of the two options is requested; try drawing a Venn diagram and labeling all regions.] a. The next purchaser will request at least one of the three options. b. The next purchaser will select none of the three options. c. The next purchaser will request only an automatic transmission and not either of the other two options. d. The next purchaser will select exactly one of these three options.

A certain system can experience three different types of defects. Let Ai (i= 1,2,3) denote the event that the system has a defect of type i. Suppose that a. What is the probability that the system does not have a type 1 defect? b. What is the probability that the system has both type 1 and type 2 defects? c. What is the probability that the system has both type 1 and type 2 defects but not a type 3 defect? d. What is the probability that the system has at most two of these defects?

Problem 27E An academic department with five faculty members—Anderson, Box, Cox, Cramer, and Fisher—must select two of its members to serve on a personnel review committee. Because the work will be time-consuming, no one is anxious to serve, so it is decided that the representative will be selected by putting the names on identical pieces of paper and then randomly selecting two. a. What is the probability that both Anderson and Box will be selected? [Hint: List the equally likely outcomes.] b. What is the probability that at least one of the two members whose name begins with C is selected? c. If the five faculty members have taught for 3, 6, 7, 10, and 14 years, respectively, at the university, what is the probability that the two chosen representatives have a total of at least 15 years’ teaching experience there?

Problem 46E Suppose an individual is randomly selected from the population of all adult males living in the United States. Let A be the event that the selected individual is over 6 ft in height, and let B be the event that the selected individual is a professional basketball player. Which do you think is larger, P(A|B) or P(B|A) ? Why?

Reconsider the system defect situation described in Exercise 26 (Section 2.2). a. Given that the system has a type 1 defect, what is the probability that it has a type 2 defect? b. Given that the system has a type 1 defect, what is the probability that it has all three types of defects? c. Given that the system has at least one type of defect, what is the probability that it has exactly one type of defect? d. Given that the system has both of the first two types of defects, what is the probability that it does not have the third type of defect?

There has been a great deal of controversy over the last several years regarding what types of surveillance are appropriate to prevent terrorism. Suppose a particular surveillance system has a 99% chance of correctly identifying a future terrorist and a 99.9% chance of correctly identifying someone who is not a future terrorist. If there are 1000 future terrorists in a population of 300 million, and one of these 300 million is randomly selected, scrutinized by the system, and identified as a future terrorist, what is the probability that he/she actually is a future terrorist? Does the value of this probability make you uneasy about using the surveillance system? Explain.

Problem 68E A friend who lives in Los Angeles makes frequent consulting trips to Washington, D.C.; 50% of the time she travels on airline #1, 30% of the time on airline #2, and the remaining 20% of the time on airline #3. For airline #1, flights are late into D.C. 30% of the time and late into L.A. 10% of the time. For airline #2, these percentages are 25% and 20%, whereas for airline #3 the percentages are 40% and 25%. If we learn that on a particular trip she arrived late at exactly one of the two destinations, what are the posterior probabilities of having flown on airlines #1, #2, and #3?Assume that the chance of a late arrival in L.A. is unaffected by what happens on the flight to D.C. [Hint: From the tip of each first-generation branch on a tree diagram, draw three second-generation branches labeled, respectively, 0 late, 1 late, and 2 late.]

Problem 69E In Exercise 59, consider the following additional information on credit card usage: 70% of all regular fill-up customers use a credit card. 50% of all regular non-fill-up customers use a credit card. 60% of all plus fill-up customers use a credit card. 50% of all plus non-fill-up customers use a credit card. 50% of all premium fill-up customers use a credit card. 40% of all premium non-fill-up customers use a credit card. Compute the probability of each of the following events for the next customer to arrive (a tree diagram might help). a. {plus and fill-up and credit card} b. {premium and non-fill-up and credit card} c. {premium and credit card} d. {fill-up and credit card} e. {credit card} f. If the next customer uses a credit card, what is the probability that premium was requested?

Problem 89E Suppose identical tags are placed on both the left ear and the right ear of a fox. The fox is then let loose for a period of time. Consider the two events C1={left ear tag is lost}and C2={left ear tag is lost}. Let ? = P(C1)= P(C2), and assume C1 and C2 are independent events. Derive an expression (involving _) for the probability that exactly one tag is lost, given that at most one is lost (“Ear Tag Loss in Red Foxes,” J. Wildlife Mgmt., 1976: 164–167). [Hint: Draw a tree diagram in which the two initial branches refer to whether the left ear tag was lost.]

Problem 90E A certain legislative committee consists of 10 senators. A subcommittee of 3 senators is to be randomly selected. a. How many different such subcommittees are there? b. If the senators are ranked 1, 2,..., 10 in order of seniority, how many different subcommittees would include the most senior senator? c. What is the probability that the selected subcommittee has at least 1 of the 5 most senior senators? d. What is the probability that the subcommittee includes neither of the two most senior senators?

Problem 47E

Problem 91E A factory uses three production lines to manufacture cans of a certain type. The accompanying table gives percentages of nonconforming cans, categorized by type of nonconformance, for each of the three lines during a particular time period. During this period, line 1 produced 500 nonconforming cans, line 2 produced 400 such cans, and line 3 was responsible for 600 nonconforming cans. Suppose that one of these 1500 cans is randomly selected. a. What is the probability that the can was produced by line 1? That the reason for nonconformance is a crack? b. If the selected can came from line 1, what is the probability that it had a blemish? c. Given that the selected can had a surface defect, what is the probability that it came from line 1?

Problem 110E A particular airline has 10 A.M. flights from Chicago to New York, Atlanta, and Los Angeles. Let A denote the event that the New York flight is full and define events B and C analogously for the other two flights. Suppose P(A) = .6, P(B) = .5, P(C) = .4 and the three events are independent. What is the probability that a. All three flights are full? That at least one flight is not full? b. Only the New York flight is full? That exactly one of the three flights is full?

A personnel manager is to interview four candidates for a job. These are ranked 1, 2, 3, and 4 in order of preference and will be interviewed in random order. However, at the conclusion of each interview, the manager will know only how the current candidate compares to those previously interviewed. For example, the interview order 3, 4, 1, 2 generates no information after the first interview, shows that the second candidate is worse than the first, and that the third is better than the first two. However, the order 3, 4, 2, 1 would generate the same information after each of the first three interviews. The manager wants to hire the best candidate but must make an irrevocable hire/no hire decision after each interview. Consider the following strategy: Automatically reject the first s candidates and then hire the first subsequent candidate who is best among those already interviewed (if no such candidate appears, the last one interviewed is hired). For example, with s =2 , the order 3, 4, 1, 2 would result in the best being hired, whereas the order 3, 1, 2, 4 would not. Of the four possible s values (0, 1, 2, and 3), which one maximizes P(best is hired)? [Hint: Write out the 24 equally likely interview orderings: s=0 means that the first candidate is automatically hired.]

Problem 112E Consider four independent events A1, A2, A3, and A4, and let pi = P(A) for I = 1,2,3,4. Express the probability that at least one of these four events occurs in terms of the pis, and do the same for the probability that at least two of the events occur.

Problem 7E An academic department has just completed voting by secret ballot for a department head. The ballot box contains four slips with votes for candidate A and three slips with votes for candidate B. Suppose these slips are removed from the box one by one. a. List all possible outcomes. b. Suppose a running tally is kept as slips are removed. For what outcomes does A remain ahead of B throughout the tally?

Problem 8E An engineering construction firm is currently working on power plants at three different sites. Let Ai denote the event that the plant at site i is completed by the contract date. Use the operations of union, intersection, and complementation to describe each of the following events in terms of A1, A2, and A3, draw a Venn diagram, and shade the region corresponding to each one. a. At least one plant is completed by the contract date. b. All plants are completed by the contract date. c. Only the plant at site 1 is completed by the contract date. d. Exactly one plant is completed by the contract date. e. Either the plant at site 1 or both of the other two plants are completed by the contract date.

Problem 9E Use Venn diagrams to verify the following two relationships for any events A and B (these are called De Morgan’s laws): a.(A?B)?=A ??B ? b.(A?B) ?= A ? ?B ? [Hint: In each part, draw a diagram corresponding to the left side and another corresponding to the right side.]

In Exercise 5, suppose that any incoming individual is equally likely to be assigned to any of the three stations irrespective of where other individuals have been assigned. What is the probability that a. All three family members are assigned to the same station? b. At most two family members are assigned to the same station? c. Every family member is assigned to a different station?

Problem 29E As of April 2006, roughly 50 million .com web domain names were registered (e.g., yahoo.com). a. How many domain names consisting of just two letters in sequence can be formed? How many domain names of length two are there if digits as well as letters are permitted as characters? [Note: A character length of three or more is now mandated.] b. How many domain names are there consisting of three letters in sequence? How many of this length are there if either letters or digits are permitted? [Note: All are currently taken.] c. Answer the questions posed in (b) for four-character sequences. d. As of April 2006, 97,786 of the four-character sequences using either letters or digits had not yet been claimed. If a four-character name is randomly selected, what is the probability that it is already owned?

Problem 30E A friend of mine is giving a dinner party. His current wine supply includes 8 bottles of zinfandel, 10 of merlot, and 12 of cabernet (he only drinks red wine), all from different wineries. a. If he wants to serve 3 bottles of zinfandel and serving order is important, how many ways are there to do this? b. If 6 bottles of wine are to be randomly selected from the 30 for serving, how many ways are there to do this? c. If 6 bottles are randomly selected, how many ways are there to obtain two bottles of each variety? d. If 6 bottles are randomly selected, what is the probability that this results in two bottles of each variety being chosen? e. If 6 bottles are randomly selected, what is the probability that all of them are the same variety?

Problem 49E The accompanying table gives information on the type of coffee selected by someone purchasing a single cup at a particular airport kiosk. Consider randomly selecting such a coffee purchaser. a. What is the probability that the individual purchased a small cup? A cup of decaf coffee? b. If we learn that the selected individual purchased a small cup, what now is the probability that he/she chose decaf coffee, and how would you interpret this probability? c. If we learn that the selected individual purchased decaf, what now is the probability that a small size was selected, and how does this compare to the corresponding unconditional probability of (a)?

Problem 50E A department store sells sport shirts in three sizes (small, medium, and large), three patterns (plaid, print, and stripe), and two sleeve lengths (long and short). The accompanying tables give the proportions of shirts sold in the various category combinations. a. What is the probability that the next shirt sold is a medium, long-sleeved, print shirt? b. What is the probability that the next shirt sold is a medium print shirt? c. What is the probability that the next shirt sold is a shortsleeved shirt? A long-sleeved shirt? d. What is the probability that the size of the next shirt sold is medium? That the pattern of the next shirt sold is a print? e. Given that the shirt just sold was a short-sleeved plaid, what is the probability that its size was medium? f. Given that the shirt just sold was a medium plaid, what is the probability that it was short-sleeved? Longsleeved?

Problem 51E According to a July 31, 2013, posting on cnn.com sub-sequent to the death of a child who bit into a peanut, a 2010 study in the journal Pediatrics found that 8% of children younger than 18 in the United States have at least one food allergy. Among those with food allergies, about 39% had a history of severe reaction. a. If a child younger than 18 is randomly selected, what is the probability that he or she has at least one food allergy and a history of severe reaction? b. It was also reported that 30% of those with an allergy in fact are allergic to multiple foods. If a child younger than 18 is randomly selected, what is the probability that he or she is allergic to multiple foods?

Problem 70E Reconsider the credit card scenario of Exercise 47 (Section 2.4), and show that A and B are dependent first by using the definition of independence and then by verifying that the multiplication property does not hold. Reference Exercise 47 Reference Exercise 12

Problem 71E An oil exploration company currently has two active projects, one in Asia and the other in Europe. Let A be the event that the Asian project is successful and B be the event that the European project is successful. Suppose that A and B are independent events with P(A) = .4and P(B) = .7. a. If the Asian project is not successful, what is the probability that the European project is also not successful? Explain your reasoning. b. What is the probability that at least one of the two projects will be successful? c. Given that at least one of the two projects is successful, what is the probability that only the Asian project is successful?

Problem 72E In Exercise 13, is any Ai independent of any other Aj? Answer using the multiplication property for independent events. Reference exercise- 13 A computer consulting firm presently has bids out on three projects. Let Ai = { awarded project i}, for i = 1,2,3, and suppose that P(A1) =.22, P(A2) = .25, P(A3) = .28, P(A1?A2) = .11,P?(A1?A3)= .05, P(A2?A3) = .07, P(A1?A2?A3)= .01. Express in words each of the following events, and compute the probability of each event: a. A1?A2 b.A?1?A’2[Hint:( A1?A2) ?= ( A?1?A?2] c.A1?A2?A3 d. A?1?A?2?A’3 e. A?1?A?2?A3 f.( A?1?A?2) ?A3

Problem 92E An employee of the records office at a certain university currently has ten forms on his desk awaiting processing. Six of these are withdrawal petitions and the other four are course substitution requests. a. If he randomly selects six of these forms to give to a subordinate, what is the probability that only one of the two types of forms remains on his desk? b. Suppose he has time to process only four of these forms before leaving for the day. If these four are randomly selected one by one, what is the probability that each succeeding form is of a different type from its predecessor?

Problem 93E One satellite is scheduled to be launched from Cape Canaveral in Florida, and another launching is scheduled for Vandenberg Air Force Base in California. Let A denote the event that the Vandenberg launch goes off on schedule, and let B represent the event that the Cape Canaveral launch goes off on schedule. If A and B are independent events withP(A)>P(B), P(A ?B) = .626, and P(A ?B) =.144, determine the values of P(A) and P(B).

Problem 94E A transmitter is sending a message by using a binary code, namely, a sequence of 0’s and 1’s. Each transmitted bit (0 or 1) must pass through three relays to reach the receiver. At each relay, the probability is .20 that the bit sent will be different from the bit received (a reversal). Assume that the relays operate independently of one another. Transmitter ? Relay 1 ? Relay 2 ? Relay 3 ? Receiver a. If a 1 is sent from the transmitter, what is the probability that a 1 is sent by all three relays? b. If a 1 is sent from the transmitter, what is the probability that a 1 is received by the receiver? [Hint: The eight experimental outcomes can be displayed on a tree diagram with three generations of branches, one generation for each relay.] c. Suppose 70% of all bits sent from the transmitter are 1s. If a 1 is received by the receiver, what is the probability that a 1 was sent?

Problem 113E A box contains the following four slips of paper, each having exactly the same dimensions: (1) win prize 1; (2) win prize 2; (3) win prize 3; (4) win prizes 1, 2, and 3. One slip will be randomly selected. Let A1 ={win prize 1},A2={win prize2}, and A3 ={win prize 3},. Show that A1 and A2 are independent, that A1 and A3 are independent and that A2 and A3 are also independent (this is pairwise independence). However, show that P(A1 ? A2 ? A3) ?P(A1).P(A2).P(A3), so the three events are not mutually independent.

Problem 114E Show that if A1, A2, and A3 are independent events, then P(A1 | A2 ?A3) 5 P(A1).

Problem 10E a. In Example 2.10, identify three events that are mutually exclusive. b. Suppose there is no outcome common to all three of the events A, B, and C. Are these three events necessarily mutually exclusive? If your answer is yes, explain why; if your answer is no, give a counterexample using the experiment of Example 2.10. Reference example 2.10 A small city has three automobile dealerships: a GM dealer selling Chevrolets and Buicks; a Ford dealer selling Fords and Lincolns; and a Toyota dealer. If an experiment consists of observing the brand of the next car sold, then the events A ={Chevrolet, Buick}and B={Ford, Lincoln} are mutually exclusive because the next car sold cannot be both a GM product and a Ford product (at least until the two companies merge!). The operations of union and intersection can be extended to more than two events. For any three events A, B, and C, the event A ?B & ?C is the set of outcomes contained in at least one of the three events, whereas A ´ ?B ´ ?C is the set of outcomes contained in all three events. Given events A1,A2,A3,…….., these events are said to be mutually exclusive (or pairwise disjoint) if no two events have any outcomes in common. A pictorial representation of events and manipulations with events is obtained by using Venn diagrams. To construct a Venn diagram, draw a rectangle whose interior will represent the sample space . Figure 2.1 shows examples of Venn diagrams.

Problem 11E A mutual fund company offers its customers a variety of funds: a money-market fund, three different bond funds (short, intermediate, and long-term), two stock funds (moderate and high-risk), and a balanced fund. Among customers who own shares in just one fund, the percentages of customers in the different funds are as follows: A customer who owns shares in just one fund is randomly selected. a. What is the probability that the selected individual owns shares in the balanced fund? b. What is the probability that the individual owns shares in a bond fund? c. What is the probability that the selected individual does not own shares in a stock fund?

The composer Beethoven wrote 9 symphonies, 5 piano concertos (music for piano and orchestra), and 32 piano sonatas (music for solo piano). a. How many ways are there to play first a Beethoven symphony and then a Beethoven piano concerto? b. The manager of a radio station decides that on each successive evening ( 7 days per week), a Beethoven symphony will be played followed by a Beethoven piano concerto followed by a Beethoven piano sonata. For how many years could this policy be continued before exactly the same program would have to be repeated?

Problem 12E Consider randomly selecting a student at a certain university, and let A denote the event that the selected individual has a Visa credit card and B be the analogous event for a MasterCard. Suppose that P(A)=.5. P(B)= .4. and P(A ?B) =25. a. Compute the probability that the selected individual has at least one of the two types of cards (i.e., the probability of the event (A ?B). b. What is the probability that the selected individual has neither type of card? c. Describe, in terms of A and B, the event that the selected student has a Visa card but not a MasterCard, and then calculate the probability of this event.

Problem 32E An electronics store is offering a special price on a complete set of components (receiver, compact disc player, speakers, turntable). A purchaser is offered a choice of manufacturer for each component: Receiver: Kellwood, Onkyo, Pioneer, Sony, Sherwood Compact disc player: Onkyo, Pioneer, Sony, Technics Speakers: Boston, Infinity, Polk Turntable: Onkyo, Sony, Teac, Technics A switchboard display in the store allows a customer to hook together any selection of components (consisting of one of each type). Use the product rules to answer the following questions: a. In how many ways can one component of each type be selected? b. In how many ways can components be selected if both the receiver and the compact disc player are to be Sony? c. In how many ways can components be selected if none is to be Sony? d. In how many ways can a selection be made if at least one Sony component is to be included? e. If someone flips switches on the selection in a completely random fashion, what is the probability that the system selected contains at least one Sony component? Exactly one Sony component?

Problem 33E Again consider a Little League team that has 15 players on its roster. a. How many ways are there to select 9 players for the starting lineup? b. How many ways are there to select 9 players for the starting lineup and a batting order for the 9 starters? c. Suppose 5 of the 15 players are left-handed. How many ways are there to select 3 left-handed outfielders and have all 6 other positions occupied by right-handed players?

Problem 52E A system consists of two identical pumps, #1 and #2. If one pump fails, the system will still operate. However, because of the added strain, the remaining pump is now more likely to fail than was originally the case. That is, r = P(#2 fails ??#1 fails) > P(#2 fails) = q. If at least one pump fails by the end of the pump design life in 7% of all systems and both pumps fail during that period in only 1%, what is the probability that pump #1 will fail during the pump design life?

A certain shop repairs both audio and video components. Let A denote the event that the next component brought in for repair is an audio component, and let B be the event that the next component is a compact disc player (so the event B is contained in A). Suppose that P(A) = .6 and P(B) = .05. What is P(B|A)?

In Exercise 13, Ai={ awarded project i} , for i =1,2,3. Use the probabilities given there to compute the following probabilities, and explain in words the meaning of each one. A. P(A2|A1) B. P(A2 ?A3|A1) C. P(A2 ?A3|A1) D. P(A1 ?A2 ?A3|A1 ?A2 ?A3)

Problem 73E If A and B are independent events, show that A_ and B are also independent. [Hint: First establish a relationship between P(A ’?B), P(B), and P(A?B).]

Problem 75E One of the assumptions underlying the theory of control charting (see Chapter 16) is that successive plotted points are independent of one another. Each plotted point can signal either that a manufacturing process is operating correctly or that there is some sort of malfunction. Even when a process is running correctly, there is a small probability that a particular point will signal a problem with the process. Suppose that this probability is .05. What is the probability that at least one of 10 successive points indicates a problem when in fact the process is operating correctly? Answer this question for 25 successive points.

Problem 74E The proportions of blood phenotypes in the U.S. population are as follows: Assuming that the phenotypes of two randomly selected individuals are independent of one another, what is the probability that both phenotypes are O? What is the probability that the phenotypes of two randomly selected individuals match?

Problem 95E Individual A has a circle of five close friends (B, C, D, E, and F). A has heard a certain rumor from outside the circle and has invited the five friends to a party to circulate the rumor. To begin, A selects one of the five at random and tells the rumor to the chosen individual. That individual then selects at random one of the four remaining individuals and repeats the rumor. Continuing, a new individual is selected from those not already having heard the rumor by the individual who has just heard it, until everyone has been told. a. What is the probability that the rumor is repeated in the order B, C, D, E, and F? b. What is the probability that F is the third person at the party to be told the rumor? c. What is the probability that F is the last person to hear the rumor? d. If at each stage the person who currently “has” the rumor does not know who has already heard it and selects the next recipient at random from all five possible individuals, what is the probability that F has still not heard the rumor after it has been told ten times at the party?

Problem 97E A chemical engineer is interested in determining whether a certain trace impurity is present in a product. An experiment has a probability of .80 of detecting the impurity if it is present. The probability of not detecting the impurity if it is absent is .90. The prior probabilities of the impurity being present and being absent are .40 and .60, respectively. Three separate experiments result in only two detections. What is the posterior probability that the impurity is present?

According to the article “Optimization of Distribution Parameters for Estimating Probability of Crack Detection” (J. of Aircraft, 2009: 2090–2097), the following “Palmberg” equation is commonly used to determine the probability Pd(c) of detecting a crack of size c in an aircraft structure: \(P_{d}(c)=\frac{\left(c / c^{*}\right)^{\beta}}{1+\left(c / c^{*}\right)^{\beta}}\) where \(c^{*}\) is the crack size that corresponds to a .5 detection probability (and thus is an assessment of the quality of the inspection process). a. Verify that Pd(c*)= .5 b. What is Pd(2c*)= when ?= 4 ? c. Suppose an inspector inspects two different panels, one with a crack size of c* and the other with a crack size of 2c*. Again assuming ?= 4 and also that the results of the two inspections are independent of one another, what is the probability that exactly one of the two cracks will be detected? d. What happens to Pd(c) as ? ?? ?

Problem 13E A computer consulting firm presently has bids out on three projects. Let Ai = { awarded project i}, for i = 1,2,3, and suppose that P(A1) =.22, P(A2) = .25, P(A3) = .28, P(A1A2) = .11, P(A1A3)= .05, P(A2?A3) = .07, P(A1?A2?A3)= .01. Express in words each of the following events, and compute the probability of each event:

Problem 14E Suppose that 55% of all adults regularly consume coffee, 45% regularly consume carbonated soda, and 70% regularly consume at least one of these two products. a. What is the probability that a randomly selected adult regularly consumes both coffee and soda? b. What is the probability that a randomly selected adult doesn’t regularly consume at least one of these two products?

Problem 15E Consider the type of clothes dryer (gas or electric) purchased by each of five different customers at a certain store. a. If the probability that at most one of these purchases an electric dryer is .428, what is the probability that at least two purchase an electric dryer? b.If and P(all five purchase gas) = .116 and P(all five purchase electric) =.005, what is the probability that at least one of each type is purchased?

Problem 34E Computer keyboard failures can be attributed to electrical defects or mechanical defects. A repair facility currently has 25 failed keyboards, 6 of which have electrical defects and 19 of which have mechanical defects. a. How many ways are there to randomly select 5 of these keyboards for a thorough inspection (without regard to order)? b. In how many ways can a sample of 5 keyboards be selected so that exactly two have an electrical defect? c. If a sample of 5 keyboards is randomly selected, what is the probability that at least 4 of these will have a mechanical defect?

A production facility employs 20 workers on the day shift, 15 workers on the swing shift, and 10 workers on the graveyard shift. A quality control consultant is to select 6 of these workers for in-depth interviews. Suppose the selection is made in such a way that any particular group of 6 workers has the same chance of being selected as does any other group (drawing 6 slips without replacement from among 45). a. How many selections result in all 6 workers coming from the day shift? What is the probability that all 6 selected workers will be from the day shift? b. What is the probability that all 6 selected workers will be from the same shift? c. What is the probability that at least two different shifts will be represented among the selected workers? d. What is the probability that at least one of the shifts will be unrepresented in the sample of workers?

Problem 36E An academic department with five faculty members narrowed its choice for department head to either candidate A or candidate B. Each member then voted on a slip of paper for one of the candidates. Suppose there are actually three votes for A and two for B. If the slips are selected for tallying in random order, what is the probability that A remains ahead of B throughout the vote count (e.g., this event occurs if the selected ordering is AABAB, but not for ABBAA)?

Problem 55E Deer ticks can be carriers of either Lyme disease or human granulocytic ehrlichiosis (HGE). Based on a recent study, suppose that 16% of all ticks in a certain location carry Lyme disease, 10% carry HGE, and 10% of the ticks that carry at least one of these diseases in fact carry both of them. If a randomly selected tick is found to have carried HGE, what is the probability that the selected tick is also a carrier of Lyme disease?

Problem 56E For any events A and B with P(B)>0, show that P(A|B)+ P(A ’|B)= 1

Problem 76E In October, 1994, a flaw in a certain Pentium chip installed in computers was discovered that could result in a wrong answer when performing a division. The manufacturer initially claimed that the chance of any particular division being incorrect was only 1 in 9 billion, so that it would take thousands of years before a typical user encountered a mistake. However, statisticians are not typical users; some modern statistical techniques are so computationally intensive that a billion divisions over a short time period is not outside the realm of possibility. Assuming that the 1 in 9 billion figure is correct and that results of different divisions are independent of one another, what is the probability that at least one error occurs in one billion divisions with this chip?

Problem 57E If P(B|A)> P(B), show that P( B ’A) < P(B ’). [Hint: Add P( B’|A) to both sides of the given inequality and then use the result of Exercise 56.] Reference exercise -56 For any events A and B with P(B) > 0, show that P(A|B)+ P(A’|B)= 1

Problem 77E An aircraft seam requires 25 rivets. The seam will have to be reworked if any of these rivets is defective. Suppose rivets are defective independently of one another, each with the same probability. a. If 15% of all seams need reworking, what is the probability that a rivet is defective? b. How small should the probability of a defective rivet be to ensure that only 10% of all seams need reworking?

A boiler has five identical relief valves. The probability that any particular valve will open on demand is .95. Assuming independent operation of the valves, calculate P(at least one valve opens) and P(at least one valve fails to open).

Problem 98E Five friends—Allison, Beth, Carol, Diane, and Evelyn—have identical calculators and are studying for a statistics exam. They set their calculators down in a pile before taking a study break and then pick them up in random order when they return from the break. What is the probability that at least one of the five gets her own calculator? [Hint: Let A be the event that Alice gets her own calculator, and define events B, C, D, and E analogously for the other four students.] How can the event (at least one gets her own calculator} be expressed in terms of the five events A, B, C, D, and E? Now use a general law of probability. [Note: This is called the matching problem. Its solution is easily extended to n individuals. Can you recognize the result when n is large (the approximation to the resulting series)?]

Problem 99E Fasteners used in aircraft manufacturing are slightly crimped so that they lock enough to avoid loosening during vibration. Suppose that 95% of all fasteners pass an initial inspection. Of the 5% that fail, 20% are so seriously defective that they must be scrapped. The remaining fasteners are sent to a recrimping operation, where 40% cannot be salvaged and are discarded. The other 60% of these fasteners are corrected by the recrimping process and subsequently pass inspection. a. What is the probability that a randomly selected incoming fastener will pass inspection either initially or after recrimping? b. Given that a fastener passed inspection, what is the probability that it passed the initial inspection and did not need recrimping?

Problem 100E Jay and Maurice are playing a tennis match. In one particular game, they have reached deuce, which means each player has won three points. To finish the game, one of the two players must get two points ahead of the other. For example, Jay will win if he wins the next two points (JJ), or if Maurice wins the next point and Jay the three points after that (MJJJ), or if the result of the next six points is JMMJJJ, and so on. a. Suppose that the probability of Jay winning a point is .6 and outcomes of successive points are independent of one another. What is the probability that Jay wins the game? [Hint: In the law of total probability, let AI = Jay wins each of the next two points, A2 = Maurice wins each of the next two points, and A3 = each player wins one of the next two points. Also let p = P(Jay wins the game). How does p compare to P(Jay wins the game I A3)?] b. If Jay wins the game, what is the probability that he needed only two points to do so?

An individual is presented with three different glasses of cola, labeled C, D, and P. He is asked to taste all three and then list them in order of preference. Suppose the same cola has actually been put into all three glasses. a. What are the simple events in this ranking experiment, and what probability would you assign to each one? b. What is the probability that C is ranked first? c. What is the probability that C is ranked first and D is ranked last?

Problem 17E Let A denote the event that the next request for assistance from a statistical software consultant relates to the SPSS package, and let B be the event that the next request is for help with SAS. Suppose that P(A) =. 30 and P(B) = .50. a. Why is it not the case that P(A) + (B) = 1? b. Calculate P(A ?) . c. Calculate P(A ?B). d.Calculate P(A? ?B?).

A wallet contains five $10 bills, four $5 bills, and six $1 bills (nothing larger). If the bills are selected one by one in random order, what is the probability that at least two bills must be selected to obtain a first $10 bill?

An experimenter is studying the effects of temperature, pressure, and type of catalyst on yield from a certain chemical reaction. Three different temperatures, four different pressures, and five different catalysts are under consideration. a. If any particular experimental run involves the use of a single temperature, pressure, and catalyst, how many experimental runs are possible? b. How many experimental runs are there that involve use of the lowest temperature and two lowest pressures? c. Suppose that five different experimental runs are to be made on the first day of experimentation. If the five are randomly selected from among all the possibilities, so that any group of five has the same probability of selection, what is the probability that a different catalyst is used on each run?

A sonnet is a 14-line poem in which certain rhyming patterns are followed. The writer Raymond Queneau published a book containing just 10 sonnets, each on a different page. However, these were structured such that other sonnets could be created as follows: the first line of a sonnet could come from the first line on any of the 10 pages, the second line could come from the second line on any of the 10 pages, and so on (successive lines were perforated for this purpose). a. How many sonnets can be created from the 10 in the book? b. If one of the sonnets counted in part (a) is selected at random, what is the probability that none of its lines came from either the first or the last sonnet in the book?

Problem 39E A box in a supply room contains 15 compact fluorescent lightbulbs, of which 5 are rated 13-watt, 6 are rated 18-watt, and 4 are rated 23-watt. Suppose that three of these bulbs are randomly selected. a. What is the probability that exactly two of the selected bulbs are rated 23-watt? b. What is the probability that all three of the bulbs have the same rating? c. What is the probability that one bulb of each type is selected? d. If bulbs are selected one by one until a 23-watt bulb is obtained, what is the probability that it is necessary to examine at least 6 bulbs?

Problem 58E Show that for any three events A, B, and C with P(C) > 0, P(A ?B|C) = P(A|C) + P(B|C) – P(A?B|C).

Problem 60E Seventy percent of the light aircraft that disappear while in flight in a certain country are subsequently discovered. Of the aircraft that are discovered, 60% have an emergency locator, whereas 90% of the aircraft not discovered do not have such a locator. Suppose a light aircraft has disappeared. a. If it has an emergency locator, what is the probability that it will not be discovered? b. If it does not have an emergency locator, what is the probability that it will be discovered?

Problem 59E At a certain gas station, 40% of the customers use regular gas (A1), 35% use plus gas (A2), and 25% use premium (A3). Of those customers using regular gas, only 30% fill their tanks (event B). Of those customers using plus, 60% fill their tanks, whereas of those using premium, 50% fill their tanks. a. What is the probability that the next customer will request plus gas and fill the tank (A2?B)? b. What is the probability that the next customer fills the tank? c. If the next customer fills the tank, what is the probability that regular gas is requested? Plus? Premium?

Two pumps connected in parallel fail independently of one another on any given day. The probability that only the older pump will fail is .10, and the probability that only the newer pump will fail is .05. What is the probability that the pumping system will fail on any given day (which happens if both pumps fail)?

Problem 80E Consider the system of components connected as in the accompanying picture. Components 1 and 2 are connected in parallel, so that subsystem works iff either 1 or 2 works; since 3 and 4 are connected in series, that subsystem works iff both 3 and 4 work. If components work independently of one another and P(component works) = .9, calculate P(system works).

Problem 81E Refer back to the series-parallel system configuration introduced in Example 2.35, and suppose that there are only two cells rather than three in each parallel subsystem [in Figure 2.14(a), eliminate cells 3 and 6, and renumber cells 4 and 5 as 3 and 4]. Using P(Ai) = .9, the probability that system lifetime exceeds t0 is easily seen to be .9639. To what value would .9 have to be changed in order to increase the system lifetime reliability from .9639 to .99? [Hint: LetP(Ai) = .9 , express system reliability in terms of p, and then letx = p2 . Reference example 2.35 Each day, Monday through Friday, a batch of components sent by a first supplier arrives at a certain inspection facility. Two days a week, a batch also arrives from a second supplier. Eighty percent of all supplier 1’s batches pass inspection, and 90% of supplier 2’s do likewise. What is the probability that, on a randomly selected day, two batches pass inspection? We will answer this assuming that on days when two batches are tested, whether the first batch passes is independent of whether the second batch does so. Figure 2.13 displays the relevant information. Reference 2.13 Reference 2.14

A system consists of two components. The probability that the second component functions in a satisfactory manner during its design life is .9, the probability that at least one of the two components does so is .96, and the probability that both components do so is .75. Given that the first component functions in a satisfactory manner throughout its design life, what is the probability that the second one does also?

Problem 102E

a. A certain company sends 40% of its overnight mail parcels via express mail service \(E_{1}\). Of these parcels, 2% arrive after the guaranteed delivery time (denote the event "late delivery" by L) . If a record of an overnight mailing is randomly selected from the company's file, what is the probability that the parcel went via \(E_{1}\) and was late? b. Suppose that 50% of the overnight parcels are sent via express mail service \(E_{2}\) and the remaining 10% are sent via \(E_{3}\). Of those sent via \(E_{2}\), only 1% arrive late, whereas 5% of the parcels handled by \(F_{3}\) arrive late. What is the probability that a randomly selected parcel arrived late? c. If a randomly selected parcel has arrived on time, what is the probability that it was not sent via \(E_{1}\)?

Problem 19E Human visual inspection of solder joints on printed circuit boards can be very subjective. Part of the problem stems from the numerous types of solder defects (e.g., pad nonwetting, knee visibility, voids) and even the degree to which a joint possesses one or more of these defects. Consequently, even highly trained inspectors can disagree on the disposition of a particular joint. In one batch of 10,000 joints, inspector A found 724 that were judged defective, inspector B found 751 such joints, and 1159 of the joints were judged defective by at least one of the inspectors. Suppose that one of the 10,000 joints is randomly selected. a. What is the probability that the selected joint was judged to be defective by neither of the two inspectors? b. What is the probability that the selected joint was judged to be defective by inspector B but not by inspector A?

Problem 20E A certain factory operates three different shifts. Over the last year, 200 accidents have occurred at the factory. Some of these can be attributed at least in part to unsafe working conditions, whereas the others are unrelated to working conditions. The accompanying table gives the percentage of accidents falling in each type of accident– shift category. Suppose one of the 200 accident reports is randomly selected from a file of reports, and the shift and type of accident are determined. a. What are the simple events? b. What is the probability that the selected accident was attributed to unsafe conditions? c. What is the probability that the selected accident did not occur on the day shift?

Problem 21E An insurance company offers four different deductible levels—none, low, medium, and high—for its homeowner’s policyholders and three different levels—low, medium, and high—for its automobile policyholders. The accompanying table gives proportions for the various categories of policyholders who have both types of insurance. For example, the proportion of individuals with both low homeowner’s deductible and low auto deductible is .06 (6% of all such individuals). Suppose an individual having both types of policies is randomly selected. a. What is the probability that the individual has a medium auto deductible and a high homeowner’s deductible? b. What is the probability that the individual has a low auto deductible? A low homeowner’s deductible? c. What is the probability that the individual is in the same category for both auto and homeowner’s deductibles? d. Based on your answer in part (c), what is the probability that the two categories are different? e. What is the probability that the individual has at least one low deductible level? f. Using the answer in part (e), what is the probability that neither deductible level is low?

Problem 40E Three molecules of type A, three of type B, three of type C, and three of type D are to be linked together to form a chain molecule. One such chain molecule is ABCDABCDABCD, and another is BCDDAAABDBCC. a. How many such chain molecules are there? [Hint: If the three A’s were distinguishable from one another—A1, A2, A3—and the B’s, C’s, and D’s were also, how many molecules would there be? How is this number reduced when the subscripts are removed from the A’s?] b. Suppose a chain molecule of the type described is randomly selected. What is the probability that all three molecules of each type end up next to one another (such as in BBBAAADDDCCC)?

Problem 41E An ATM personal identification number (PIN) consists of four digits, each a 0, 1, 2, . . . 8, or 9, in succession. a. How many different possible PINs are there if there are no restrictions on the choice of digits? b. According to a representative at the author’s local branch of Chase Bank, there are in fact restrictions on the choice of digits. The following choices are prohibited: (i) all four digits identical (ii) sequences of consecutive ascending or descending digits, such as 6543 (iii) any sequence starting with 19 (birth years are too easy to guess). So if one of the PINs in (a) is randomly selected, what is the probability that it will be a legitimate PIN (that is, not be one of the prohibited sequences)? c. Someone has stolen an ATM card and knows that the first and last digits of the PIN are 8 and 1, respectively. He has three tries before the card is retained by the ATM (but does not realize that). So he randomly selects the 2nd and 3rd digits for the first try, then randomly selects a different pair of digits for the second try, and yet another randomly selected pair of digits for the third try (the individual knows about the restrictions described in (b) so selects only from the legitimate possibilities). What is the probability that the individual gains access to the account? d. Recalculate the probability in (c) if the first and last digits are 1 and 1, respectively.

A starting lineup in basketball consists of two guards, two forwards, and a center. a. A certain college team has on its roster three centers, four guards, four forwards, and one individual (X) who can play either guard or forward. How many different starting lineups can be created? [Hint: Consider lineups without X, then lineups with X as guard, then lineups with X as forward.] b. Now suppose the roster has 5 guards, 5 forwards, 3 centers, and 2 “swing players” (X and Y) who can play either guard or forward. If 5 of the 15 players are randomly selected, what is the probability that they constitute a legitimate starting lineup?

Problem 61E Components of a certain type are shipped to a supplier in batches of ten. Suppose that 50% of all such batches contain no defective components, 30% contain one defective component, and 20% contain two defective components. Two components from a batch are randomly selected and tested. What are the probabilities associated with 0, 1, and 2 defective components being in the batch under each of the following conditions? a. Neither tested component is defective. b. One of the two tested components is defective. [Hint: Draw a tree diagram with three first-generation branches for the three different types of batches.]

Problem 62E Blue Cab operates 15% of the taxis in a certain city, and Green Cab operates the other 85%. After a nighttime hit-and-run accident involving a taxi, an eyewitness said the vehicle was blue. Suppose, though, that under night vision conditions, only 80% of individuals can correctly distinguish between a blue and a green vehicle. What is the (posterior) probability that the taxi at fault was blue? In answering, be sure to indicate which probability rules you are using. [Hint: A tree diagram might help. Note: This is based on an actual incident.]

Problem 63E For customers purchasing a refrigerator at a certain appliance store, let A be the event that the refrigerator was manufactured in the U.S., B be the event that the refrigerator had an icemaker, and C be the event that the customer purchased an extended warranty. Relevant probabilities are a. Construct a tree diagram consisting of first-, second-, and third-generation branches, and place an event label and appropriate probability next to each branch. b. Compute P(A?B?C). c. Compute P(B?C). d. Compute P(C). e. Compute P(A|B?C), the probability of a U.S. purchase given that an icemaker and extended warranty are also purchased.

Problem 82E Consider independently rolling two fair dice, one red and the other green. Let A be the event that the red die shows 3 dots, B be the event that the green die shows 4 dots, and C be the event that the total number of dots showing on the two dice is 7. Are these events pairwise independent (i.e., are A and B independent events, are A and C independent, and are B and C independent)? Are the three events mutually independent?

Problem 83E Components arriving at a distributor are checked for defects by two different inspectors (each component is checked by both inspectors). The first inspector detects 90% of all defectives that are present, and the second inspector does likewise. At least one inspector does not detect a defect on 20% of all defective components. What is the probability that the following occur? a. A defective component will be detected only by the first inspector? By exactly one of the two inspectors? b. All three defective components in a batch escape detection by both inspectors (assuming inspections of different components are independent of one another)?

Consider purchasing a system of audio components consisting of a receiver, a pair of speakers, and a CD player. Let A1 be the event that the receiver functions properly throughout the warranty period, A2 be the event that the speakers function properly throughout the warranty period, and A3 be the event that the CD player functions properly throughout the warranty period. Suppose that these events are (mutually) independent with P(A1) = .95, P(A2) = .98, and P(A3) = .80. a. What is the probability that all three components function properly throughout the warranty period? b. What is the probability that at least one component needs service during the warranty period? c. What is the probability that all three components need service during the warranty period? d. What is the probability that only the receiver needs service during the warranty period? e. What is the probability that exactly one of the three components needs service during the warranty period? f. What is the probability that all three components function properly throughout the warranty period but that at least one fails within a month after the warranty expires?

Problem 104E A company uses three different assembly lines—A1,A2, and A3—to manufacture a particular component. Of those manufactured by line A1, 5% need rework to remedy a defect, whereas 8% of A2’s components need rework and 10% of A3’s need rework. Suppose that 50% of all components are produced by line A1, 30% are produced by line A2, and 20% come from line A3. If a randomly selected component needs rework, what is the probability that it came from line A1? From line A2? From line A3?

Problem 105E Disregarding the possibility of a February 29 birthday, suppose a randomly selected individual is equally likely to have been born on any one of the other 365 days. a. If ten people are randomly selected, what is the probability that all have different birthdays? That at least two have the same birthday? b. With k replacing ten in part (a), what is the smallest k for which there is at least a 50-50 chance that two or more people will have the same birthday? c. If ten people are randomly selected, what is the probability that either at least two have the same birthday or at least two have the same last three digits of their Social Security numbers? [Note: The article “Methods for Studying Coincidences” (F. Mosteller and P. Diaconis, J. Amer. Stat. Assoc., 1989: 853–861) discusses problems of this type.]

Problem 106E One method used to distinguish between granitic (G) and basaltic (B) rocks is to examine a portion of the infrared spectrum of the sun’s energy reflected from the rock surface. Let R1, R2, and R3 denote measured spectrum intensities at three different wavelengths; typically, for graniteR123 , whereas for basalt R123. When measurements are made remotely (using aircraft), various orderings of the Ris may arise whether the rock is basalt or granite. Flights over regions of known composition have yielded the following information: Suppose that for a randomly selected rock in a certain region P(granite) =.25 , and P(basalt) = .75.. a. Show thatP(granite| R123) >P(basalt| R123). If measurements yielded R123 , would you classify the rock as granite or basalt? b. If measurements yielded R123, how would you classify the rock? Answer the same question for R123. c. Using the classification rules indicated in parts (a) and (b), when selecting a rock from this region, what is the probability of an erroneous classification? [Hint: Either G could be classified as B or B as G, and P(B) and P(G) are known.] d. If P(granite) = p rather than .25, are there values of p (other than 1) for which one would always classify a rock as granite?

A certain legislative committee consists of 10 senators. A subcommittee of 3 senators is to be randomly selected. a. How many different such subcommittees are there? b. If the senators are ranked 1, 2, . . . , 10 in order of seniority, how many different subcommittees would include the most senior senator? c. What is the probability that the selected subcommittee has at least 1 of the 5 most senior senators? d. What is the probability that the subcommittee includes neither of the two most senior senators?

A factory uses three production lines to manufacture cans of a certain type. The accompanying table gives percentages of nonconforming cans, categorized by type of nonconformance, for each of the three lines during a particular time period. During this period, line 1 produced 500 nonconforming cans, line 2 produced 400 such cans, and line 3 was responsible for 600 nonconforming cans. Suppose that one of these 1500 cans is randomly selected. a. What is the probability that the can was produced by line 1? That the reason for nonconformance is a crack? b. If the selected can came from line 1, what is the probability that it had a blemish? c. Given that the selected can had a surface defect, what is the probability that it came from line 1?

An employee of the records office at a certain university currently has ten forms on his desk awaiting processing. Six of these are withdrawal petitions and the other four are course substitution requests. a. If he randomly selects six of these forms to give to a subordinate, what is the probability that only one of the two types of forms remains on his desk? b. Suppose he has time to process only four of these forms before leaving for the day. If these four are randomly selected one by one, what is the probability that each succeeding form is of a different type from its predecessor?

One satellite is scheduled to be launched from Cape Canaveral in Florida, and another launching is scheduled for Vandenberg Air Force Base in California. Let A denote the event that the Vandenberg launch goes off on schedule, and let B represent the event that the Cape Canaveral launch goes off on schedule. If A and B are independent events with P(A) > P(B), P(A B) = .626, and P(A B) = .144, determine the values of P(A) and P(B).

A transmitter is sending a message by using a binary code, namely, a sequence of 0’s and 1’s. Each transmitted bit (0 or 1) must pass through three relays to reach the receiver. At each relay, the probability is .20 that the bit sent will be different from the bit received (a reversal). Assume that the relays operate independently of one another. Transmitter \(\rightarrow\) Relay 1 \(\rightarrow\) Relay 2 \(\rightarrow\) Relay 3 \(\rightarrow\) Receiver a. If a 1 is sent from the transmitter, what is the probability that a 1 is sent by all three relays? b. If a 1 is sent from the transmitter, what is the probability that a 1 is received by the receiver? [Hint: The eight experimental outcomes can be displayed on a tree diagram with three generations of branches, one generation for each relay.] c. Suppose 70% of all bits sent from the transmitter are 1s. If a 1 is received by the receiver, what is the probability that a 1 was sent?

Individual A has a circle of five close friends (B, C, D, E, and F). A has heard a certain rumor from outside the circle and has invited the five friends to a party to circulate the rumor. To begin, A selects one of the five at random and tells the rumor to the chosen individual. That individual then selects at random one of the four remaining individuals and repeats the rumor. Continuing, a new individual is selected from those not already having heard the rumor by the individual who has just heard it, until everyone has been told. a. What is the probability that the rumor is repeated in the order B, C, D, E, and F? b. What is the probability that F is the third person at the party to be told the rumor? c. What is the probability that F is the last person to hear the rumor? d. If at each stage the person who currently has the rumor does not know who has already heard it and selects the next recipient at random from all five possible individuals, what is the probability that F has still not heard the rumor after it has been told ten times at the party?

According to the article Optimization of Distribution Parameters for Estimating Probability of CrackDetection (J. of Aircraft, 2009: 20902097), the followingPalmberg equation is commonly used to determine the probability Pd(c) of detecting a crack of size c in an aircraft structure: \(P_{d}(c)=\frac{\left(c / c^{*}\right)^{\beta}}{1+\left(c / c^{*}\right)^{\beta}}\) where c* is the crack size that corresponds to a .5 detection probability (and thus is an assessment of the quality of the inspection process). a. Verify that Pd(c*) = .5 b. What is Pd(2c*) when \(\beta\) = 4? c. Suppose an inspector inspects two different panels,one with a crack size of c* and the other with a crack size of 2c*. Again assuming \(\beta\) = 4 and also that the results of the two inspections are independent of one another, what is the probability that exactly one of the two cracks will be detected? d. What happens to Pd(c) as \(\beta \rightarrow \infty \text { ? }\)

A chemical engineer is interested in determining whether a certain trace impurity is present in a product. An experiment has a probability of .80 of detecting the impurity if it is present. The probability of not detecting the impurity if it is absent is .90. The prior probabilities of the impurity being present and being absent are .40 and .60, respectively. Three separate experiments result in only two detections. What is the posterior probability that the impurity is present?

Five friendsAllison, Beth, Carol, Diane, and Evelyn have identical calculators and are studying for a statistics exam. They set their calculators down in a pile before taking a study break and then pick them up in random order when they return from the break. What is the probability that at least one of the five gets her own calculator? [Hint: Let A be the event that Alice gets her own calculator, and define events B, C, D, and E analogously for the other four students.] How can the event {at least one gets her own calculator} be expressed in terms of the five events A, B, C, D, and E? Now use a general law of probability. [Note: This is called the matching problem. Its solution is easily extended to n individuals. Can you recognize the result when n is large (the approximation to the resulting series)?]

Fasteners used in aircraft manufacturing are slightly crimped so that they lock enough to avoid loosening during vibration. Suppose that 95% of all fasteners pass an initial inspection. Of the 5% that fail, 20% are so seriously defective that they must be scrapped. The remaining fasteners are sent to a recrimping operation, where 40% cannot be salvaged and are discarded. The other 60% of these fasteners are corrected by the recrimping process and subsequently pass inspection. a. What is the probability that a randomly selected incoming fastener will pass inspection either initially or after recrimping? b. Given that a fastener passed inspection, what is the probability that it passed the initial inspection and did not need recrimping?

Jay and Maurice are playing a tennis match. In one particular game, they have reached deuce, which means each player has won three points. To finish the game, one of the two players must get two points ahead of the other.For example, Jay will win if he wins the next two points(JJ), or if Maurice wins the next point and Jay the three points after that (MJJJ), or if the result of the next six points is JMMJJJ, and so on. a. Suppose that the probability of Jay winning a point is .6 and outcomes of successive points are independent of one another. What is the probability thatJay wins the game? [Hint: In the law of total probability,let A1 = Jay wins each of the next two points, A2 = Maurice wins each of the next two points, and A3 = each player wins one of the next two points. Also let p = P(Jay wins the game).How does p compare to P(Jay wins the game | A3)?] b. If Jay wins the game, what is the probability that he needed only two points to do so?

A system consists of two components. The probability that the second component functions in a satisfactory manner during its design life is .9, the probability that at least one of the two components does so is .96, and the probability that both components do so is .75. Given that the first component functions in a satisfactory manner throughout its design life, what is the probability that the second one does also?

The accompanying table categorizing each student in a sample according to gender and eye color appeared in the article Does Eye Color Depend on Gender? It Might Depend on Who or How You Ask (J. of Statistics Educ., 2013, Vol. 21, Num. 2). Eye Color Gender Blue Brown Green Hazel Total Male 370 352 198 187 1107 Female 359 290 110 160 919 Total 729 642 308 347 2026 Suppose that one of these 2026 students is randomly selected. Let F denote the event that the selected individual is a female, and A, B, C, and D represent the events that he or she has blue, brown, green, and hazel eyes, respectively. a. Calculate both P(F) and P(C). b. Calculate P(F > C). Are the events F and C independent? Why or why not? c. If the selected individual has green eyes, what is the probability that he or she is a female? d. If the selected individual is female, what is the probability that she has green eyes? e. What is the conditional distribution of eye color for females (i.e., P(AuF), P(BuF), P(CuF), and P(DuF)), and what is it for males? Compare the two distributions.

a. A certain company sends 40% of its overnight mail parcels via express mail service E1. Of these parcels, 2% arrive after the guaranteed delivery time (denote the event late delivery by L). If a record of an overnight mailing is randomly selected from the companys file, what is the probability that the parcel went via E1 and was late? b. Suppose that 50% of the overnight parcels are sent via express mail service E2 and the remaining 10% are sent via E3. Of those sent via E2, only 1% arrive late, whereas 5% of the parcels handled by E3 arrive late. What is the probability that a randomly selected parcel arrived late? c. If a randomly selected parcel has arrived on time, what is the probability that it was not sent via E1?

A company uses three different assembly linesA1, A2, and A3to manufacture a particular component. Of those manufactured by line A1, 5% need rework to remedy a defect, whereas 8% of A2s components need rework and 10% of A3s need rework. Suppose that 50% of all components are produced by line A1, 30% are produced by line A2, and 20% come from line A3. If a randomly selected component needs rework, what is the probability that it came from line A1? From line A2? From line A3?

Disregarding the possibility of a February 29 birthday, suppose a randomly selected individual is equally likely to have been born on any one of the other 365 days. a. If ten people are randomly selected, what is the probability that all have different birthdays? That at leasttwo have the same birthday? b. With k replacing ten in part (a), what is the smallest k for which there is at least a 50-50 chance that two or more people will have the same birthday? c. If ten people are randomly selected, what is the probability that either at least two have the same birthday or at least two have the same last three digits of their Social Security numbers? [Note: The article Methods for Studying Coincidences (F. Mosteller and P. Diaconis, J. Amer. Stat. Assoc., 1989: 853861) discusses problems of this type.]

One method used to distinguish between granitic (G) and basaltic (B) rocks is to examine a portion of the infrared spectrum of the suns energy reflected from the rock surface. Let R1, R2, and R3 denote measured spectrum intensities at three different wavelengths; typically, for granite R1 , R2 , R3, whereas for basalt R3 , R1 , R2. When measurements are made remotely (using aircraft), various orderings of the Ri s may arise whether the rock is basalt or granite. Flights over regions of known composition have yielded the following information: Granite Basalt R1 , R2 , R3 60% 10% R1 , R3 , R2 25% 20% R3 , R1 , R2 15% 70% Suppose that for a randomly selected rock in a certain region, P(granite) = .25 and P(basalt) = .75. a. Show that P(granite u R1 , R2 , R3) . P(basalt u R1 ,R2 , R3). If measurements yielded R1 , R2 , R3, would you classify the rock as granite or basalt? b. If measurements yielded R1 , R3 , R2, how would you classify the rock? Answer the same question for R3 , R1 , R2. c. Using the classification rules indicated in parts (a) and (b), when selecting a rock from this region, what is the probability of an erroneous classification? [Hint: Either G could be classified as B or B as G, and P(B) and P(G) are known.]

A subject is allowed a sequence of glimpses to detect a target. Let Gi 5 {the target is detected on the ith glimpse}, with pi 5 P(Gi ). Suppose the Gi 9s are independent events, and write an expression for the probability that the target has been detected by the end of the nth glimpse. [Note: This model is discussed in Predicting Aircraft Detectability, Human Factors, 1979: 277291.]

In a Little League baseball game, team As pitcher throws a strike 50% of the time and a ball 50% of the time, successive pitches are independent of one another, and the pitcher never hits a batter. Knowing this, team Bs manager has instructed the first batter not to swing at anything. Calculate the probability that a. The batter walks on the fourth pitch b. The batter walks on the sixth pitch (so two of the first five must be strikes), using a counting argument or constructing a tree diagram c. The batter walks d. The first batter up scores while no one is out (assuming that each batter pursues a no-swing strategy)

Four engineers, A, B, C, and D, have been scheduled for job interviews at 10 A.M. on Friday, January 13, at Random Sampling, Inc. The personnel manager has scheduled the four for interview rooms 1, 2, 3, and 4, respectively. However, the manager’s secretary does not know this, so assigns them to the four rooms in a completely random fashion (what else!). What is the probability that a. All four end up in the correct rooms? b. None of the four ends up in the correct room?

A particular airline has 10 a.m. flights from Chicago to New York, Atlanta, and Los Angeles. Let A denote the event that the New York flight is full and define events B and C analogously for the other two flights. Suppose P(A) = .9, P(B) = .7, P(C) = .8 and the three events are independent. What is the probability that a. All three flights are full? That at least one flight is not full? b. Only the New York flight is full? That exactly one of the three flights is full?

A personnel manager is to interview four candidates for a job. These are ranked 1, 2, 3, and 4 in order of preference and will be interviewed in random order. However, at the conclusion of each interview, the manager will know only how the current candidate compares to those previously interviewed. For example, the interview order 3, 4, 1, 2 generates no information after the first interview, shows that the second candidate is worse than the first, and that the third is better than the first two. However, the order 3, 4, 2, 1 would generate the same information after each of the first three interviews. The manager wants to hire the best candidate but must make an irrevocable hire/no hire decision after each interview. Consider the following strategy: Automatically reject the first s candidates and then hire the first subsequent candidate who is best among those already interviewed (if no such candidate appears, the last one interviewed is hired). For example, with s = 2, the order 3, 4, 1, 2 would result in the best being hired, whereas the order 3, 1, 2, 4 would not. Of the four possible s values (0, 1, 2, and 3), which one maximizes P(best is hired)? [Hint: Write out the 24 equally likely interview orderings: s = 0 means that the first candidate is automatically hired.]

Consider four independent events A1, A2, A3, and A4, and let pi = P(Ai ) for i = 1,2,3,4. Express the probability that at least one of these four events occurs in terms of the pi s, and do the same for the probability that at least two of the events occur

A box contains the following four slips of paper, each having exactly the same dimensions: (1) win prize 1; (2) win prize 2; (3) win prize 3; (4) win prizes 1, 2, and 3. One slip will be randomly selected. Let A1 = {win prize 1}, A2 = {win prize 2}, and A3 = {win prize 3}. Show that A1 and A2 are independent, that A1 and A3 are independent, and that A2 and A3 are also independent (this is pairwise independence). However, show that P(A1 A2 A3) P(A1) ? P(A2) ? P(A3), so the three events are not mutually independent.

Show that if A1, A2, and A3 are independent events, then \(P\left(A_{1} \mid A_{2} \cap \overline{A_{3}}\right)=P\left(A_{1}\right)\)