In a Little League baseball game, team A’s pitcher throws

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 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?.? ist 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. ? ist 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??

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?? ist 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 ?A?1, ?A?2, and ?A?3, draw a Venn diagram, and shade the region corresponding to each one. a?.? t least one plant is completed by the contract date. b?. ?All plants are completed by the contract date. c?.? nly the plant at site 1 is completed by the contract date. d?.? xactly 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.]

Problem 10E a?? n 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?

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 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(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?sub>1?A?2?A3 f.( A?1?A?;2) ?A3

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 16E 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 tha? ?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?

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?? hat 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 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?

Problem 23E 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?), thenP(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?B)and ?

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.

Problem 26E 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 P(A1)= .12 P(A2) = .07 P(A3) =.05 P(A1 ?A2)= .13 P(A1 ?A3)= .14 P(A2 ?A3) = .10 P(A1 ?A2 ?A3) = .01

Problem 28E 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.? ll three family members are assigned to the same station? b.?? t most two family members are assigned to the same station? c.? very family member is assigned to a different station?

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 ? ?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?

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 31E 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 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?

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 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?

Problem 35E 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?

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 entire vote count (e.g., this event occurs if the ordering is AABAB, but not for ABBAA)?

Problem 37E 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?

Problem 38E 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 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—?A?1, ?A?2, ?A?3—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 ? BBAAADDDCCC) ? ?

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.

Problem 42E 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?

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??? ?), 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 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?

Problem 47E

Problem 48E 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? Reference exercise- 26 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 P(A1)= .12 P(A2) = .07 P(A3) =.05 P(A1 ?A2)= .13 P(A1 ?A3)= .14 P(A2 ?A3) = .10 P(A1 ?A2 ?A3) = .01

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?

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?

Problem 54E 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 53E 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 iscontained in ?? . Suppose that P(A) = .6and P(B) = .5. What isP(B|A) ?

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? nd ?B ?with P(B)>0, show that P(A|B)+ P(A ’|B)= 1

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 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).

At a certain gas station, 40% of the customers use regular gas(\(A_1\)), 35% use plus gas (\(A_2\)) and 25 % usepremium (\(A_3\)). 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 premium, 50% fill their tanks. a) What is the probability that the next customer will request plus gas and fill their tank ? 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 the regular gas is requested? Plus ?Premium?

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 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 ??? ?). e. ?Compute P(A|B?C), the probability of a U.S. purchase given that an icemaker and extended warranty are also purchased.

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 50% of them are short, 30% are medium, and the other 20% are long. Reviews are submitted in either Word or LaTeX. For short reviews, 70% 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? (Round your answers to three decimal places.) 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 67E 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 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 Return to the credit card scenario of Exercise 12 (Section 2.2), where A = {Visa}, B = { MasterCard}, P(A) = .5, P(B) = .4, and P(A ?B) = .25. Calculate and interpret each of the following probabilities (a Venn diagram might help). a. P(B|A) b. P(B ?|A). c. P(A|B) d. P(A ?|B) e. ?Given that the selected individual has at least one card, what is the probability that he or she has a Visa card?

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 ?A?i ?independent of any other ?A?j? 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 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 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 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 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 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?

Problem 78E 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 79E 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 le? ?= ? . 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

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)?

Problem 84E 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 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 ?n?th 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.

Problem 86E a. A lumber company has just taken delivery on a lot of 10, 002 × 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) ?

Problem 87E Consider randomly selecting a single individual and having that person test drive 3 different vehicles. Define events A1, A2, and A3 by A? 1 =likes vehicle #1 A? 2 =likes vehicle #2 A? 3 =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 interpretP(A2|A3) . c.? re 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 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 ?C?1 and ?C?2 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 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 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?

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), \(P(A\ \cup\ B) = .626\), and \(P(A\ \cap\ B) =.144\), 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 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.? hat 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 96E 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 ? ? n an aircraft structure: 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.?? hat 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 2?c?*. 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?(? as ? ?? ?

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?

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?

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 \(A_1\) = Jay wins each of the next two points, \(A_2\) = Maurice wins each of the next two points, and \(A_3\) = 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 \(A_3\))?] b. If Jay wins the game, what is the probability that he needed only two points to do so?

Problem 101E 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?

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 \(E_1\)?

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). 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 \(P(F\ \cap\ C)\). 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?

Problem 104E A company uses three different assembly lines—?A?1,?A?2, and ?A?3—to manufacture a particular component. Of those manufactured by line ?A?1, 5% need rework to remedy a defect, whereas 8% of ?A?2’s components need rework and 10% of ?A?3’s need rework. Suppose that 50% of all components are produced by line ?A?1, 30% are produced by line ?A?2, and 20% come from line ?A?3. If a randomly selected component needs rework, what is the probability that it came from line ?A?1? From line ?A?2? From lin? ?3?

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? [? ote: T he 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 ?R?1, ?R?2, and ?R?3 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 ?R?is 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?

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 ?n?th glimpse. [?Note: ?This model is discussed in “Predicting Aircraft Detectability,” ?Human Factors, ?1979: 277–291.]

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 109E 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 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?

Problem 111E 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 ?A?1, ?A?2, ?A?3, and ?A?4, 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 ?p?is, and do the same for the probability that at least two of the events occur.

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 ?A?1 and ?A?2 are independent, that ?A?1 and ?A?3 are independent and that ?A?2 and ?A?3 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? ot? utually independent.

Problem 114E Show that if ?A?1, ?A?2, and ?A?3 are independent events, then ?P?(?A?1 | ?A?2 ??A?3) 5 P?(A1?).

List the elements of each of the following sample spaces: (a) the set of integers between 1 and 50 divisible by 8; (b) the set S = {x | x2 +4x5=0 }; (c) the set of outcomes when a coin is tossed until a tail or three heads appear; (d) the set S = {x | x is a continent}; (e) the set S = {x | 2x4 0 and x<1}.

Use the rule method to describe the sample space S consisting of all points in the rst quadrant inside a circle of radius 3 with center at the origin. 2.3 Which of the following events are equal? (a) A = {1,3}; (b) B = {x | x is a number on a die }; (c) C = {x | x2 4x +3=0 }; (d) D = {x | x is the number of heads when six coins are tossed}.

Which of the following events are equal? (a) A = {1,3}; (b) B = {x | x is a number on a die }; (c) C = {x | x2 4x +3=0 }; (d) D = {x | x is the number of heads when six coins are tossed}.

An experiment involves tossing a pair of dice, one green and one red, and recording the numbers that come up. If x equals the outcome on the green die and y the outcome on the red die, describe the sample space S (a) by listing the elements (x,y); (b) by using the rule method.

An experiment consists of tossing a die and then flipping a coin once if the number on the die is even. If the number on the die is odd, the coin is flipped twice. Using the notation 4H, for example, to denote the outcome that the die comes up 4 and then the coin comes up heads, and 3HT to denote the outcome that the die comes up 3 followed by a head and then a tail on the coin, construct a tree diagram to show the 18 elements of the sample space S.

Two jurors are selected from 4 alternates to serve at a murder trial. Using the notation A1A3, for example, to denote the simple event that alternates 1 and 3 are selected, list the 6 elements of the sample space S.

Four students are selected at random from a chemistry class and classified as male or female. List the elements of the sample space \(S_1\), using the letter M for male and F for female. Define a second sample space \(S_2\) where the elements represent the number of females selected.

For the sample space of Exercise 2.4, (a) list the elements corresponding to the event A that the sum is greater than 8; (b) list the elements corresponding to the event B that a 2 occurs on either die; (c) list the elements corresponding to the event C that a number greater than 4 comes up on the green die; (d) list the elements corresponding to the event AC; (e) list the elements corresponding to the event AB; (f) list the elements corresponding to the event BC; (g) construct a Venn diagram to illustrate the intersections and unions of the events A, B, and C.

For the sample space of Exercise 2.5, (a) list the elements corresponding to the event A that a number less than 3 occurs on the die; (b) list the elements corresponding to the event B that two tails occur; (c) list the elements corresponding to the event A; (d) list the elements corresponding to the event \(A\ \cap\ B\); (e) list the elements corresponding to the event \(A\ \cup\ B\).

An engineering rm is hired to determine if certain waterways in Virginia are safe for shing. Samples are taken from three rivers. (a) List the elements of a sample space S, using the letters F for safe to sh andN for not safe to sh. (b) List the elements of S corresponding to event E that at least two of the rivers are safe for shing. (c) Dene an event that has as its elements the points {FFF,NFF,FFN,NFN}.

The resumes of two male applicants for a college teaching position in chemistry are placed in the same le as the resumes of two female applicants. Two positions become available, and the rst, at the rank of assistant professor, is lled by selecting one of the four applicants at random. The second position, at the rank of instructor, is then lled by selecting at random one of the remaining three applicants. Using the notation M2F1, for example, to denote the simple event that the rst position is lled by the second male applicant and the second position is then lled by the rst female applicant, (a) list the elements of a sample space S; (b) list the elements of S corresponding to event A that the position of assistant professor is lled by a male applicant; (c) list the elements of S corresponding to event B that exactly one of the two positions is lled by a male applicant; (d) list the elements of S corresponding to event C that neither position is lled by a male applicant; (e) list the elements of S corresponding to the event AB; (f) list the elements of S corresponding to the event AC; (g) construct a Venn diagram to illustrate the intersections and unions of the events A, B, and C.

Exercise and diet are being studied as possible substitutes for medication to lower blood pressure. Three groups of subjects will be used to study the effect of exercise. Group 1 is sedentary, while group 2 walks and group 3 swims for 1 hour a day. Half of each of the three exercise groups will be on a salt-free diet. An additional group of subjects will not exercise or restrict their salt, but will take the standard medication. Use Z for sedentary, W for walker, S for swimmer, Y for salt, N for no salt, M for medication, and F for medication free. (a) Show all of the elements of the sample space S. (b) Given that A is the set of nonmedicated subjects and B is the set of walkers, list the elements of AB. (c) List the elements of AB.

Construct a Venn diagram to illustrate the possible intersections and unions for the following events relative to the sample space consisting of all automobiles made in the United States. F : Four door, S : Sun roof, P : Power steering.

If S = {0,1,2,3,4,5,6,7,8,9} and A = {0,2,4,6,8}, B = {1,3,5,7,9}, C = {2,3,4,5}, and D = {1,6,7}, list the elements of the sets corresponding to the following events: (a) AC; (b) AB; (c) C; (d) (CD)B; (e) (S C); (f) AC D.

Consider the sample space S ={copper, sodium, nitrogen, potassium, uranium, oxygen, zinc} and the events A = {copper, sodium, zinc}, B = {sodium, nitrogen, potassium}, C = {oxygen}. List the elements of the sets corresponding to the following events: (a) A; (b) AC; (c) (AB)C; (d) BC; (e) AB C; (f) (AB)(AC).

If S = {x | 0 < x < 12}, M = {x | 1 < x < 9}, and N = {x | 0 < x < 5}, find (a) \(M\ \cup\ N\); (b) \(M\ \cap\ N\); (c) \(M’\ \cap\ N’\).

Let A, B, and C be events relative to the sample space S. Using Venn diagrams, shade the areas representing the following events: (a) (\(A\ \cap\ B\))’; (b) (\(A\ \cup\ B\))’; (c) \((A\ \cap\ C)\ \cup\ B\).

Which of the following pairs of events are mutually exclusive? (a) A golfer scoring the lowest 18-hole round in a 72hole tournament and losing the tournament. (b) A poker player getting a ush (all cards in the same suit) and 3 of a kind on the same 5-card hand. (c) A mother giving birth to a baby girl and a set of twin daughters on the same day. (d) A chess player losing the last game and winning the match.

Suppose that a family is leaving on a summer vacation in their camper and that M is the event that they will experience mechanical problems, T is the event that they will receive a ticket for committing a trac violation, and V is the event that they will arrive at a campsite with no vacancies. Referring to the Venn diagram of Figure 2.5, state in words the events represented by the following regions: (a) region 5;(b) region 3; (c) regions 1 and 2 together; (d) regions 4 and 7 together; (e) regions 3, 6, 7, and 8 together.

Referring to Exercise 2.19 and the Venn diagram of Figure 2.5, list the numbers of the regions that represent the following events: (a) The family will experience no mechanical problems and will not receive a ticket for a trac violation but will arrive at a campsite with no vacancies. (b) The family will experience both mechanical problems and trouble in locating a campsite with a vacancy but will not receive a ticket for a trac violation. (c) The family will either have mechanical trouble or arrive at a campsite with no vacancies but will not receive a ticket for a trac violation. (d) The family will not arrive at a campsite with no vacancies.

Registrants at a large convention are oered 6 sightseeing tours on each of 3 days. In how many ways can a person arrange to go on a sightseeing tour planned by this convention?

In a medical study, patients are classied in 8 ways according to whether they have blood type AB+, AB, A+, A, B+, B, O+, orO, and also according to whether their blood pressure is low, normal, or high. Find the number of ways in which a patient can be classied.

If an experiment consists of throwing a die and then drawing a letter at random from the English alphabet, how many points are there in the sample space?

Students at a private liberal arts college are classied as being freshmen, sophomores, juniors, or seniors, and also according to whether they are male or female. Find the total number of possible classications for the students of that college.

A certain brand of shoes comes in 5 dierent styles, with each style available in 4 distinct colors. If the store wishes to display pairs of these shoes showing all of its various styles and colors, how many dierent pairs will the store have on display?

A California study concluded that following 7 simple health rules can extend a mans life by 11 years on the average and a womans life by 7 years. These 7 rules are as follows: no smoking, get regular exercise, use alcohol only in moderation, get 7 to 8 hours of sleep, maintain proper weight, eat breakfast, and do not eat between meals. In how many ways can a person adopt 5 of these rules to follow (a) if the person presently violates all 7 rules? (b) if the person never drinks and always eats breakfast?

A developer of a new subdivision offers a prospective home buyer a choice of 4 designs, 3 different heating systems, a garage or carport, and a patio or screened porch. How many different plans are available to this buyer?

A drug for the relief of asthma can be purchased from 5 dierent manufacturers in liquid, tablet, or capsule form, all of which come in regular and extra strength. How many dierent ways can a doctor prescribe the drug for a patient suering from asthma?

In a fuel economy study, each of 3 race cars is tested using 5 different brands of gasoline at 7 test sites located in different regions of the country. If 2 drivers are used in the study, and test runs are made once under each distinct set of conditions, how many test runs are needed?

In how many dierent ways can a true-false test consisting of 9 questions be answered?

A witness to a hit-and-run accident told the police that the license number contained the letters RLH followed by 3 digits, the rst of which was a 5. If the witness cannot recall the last 2 digits, but is certain that all 3 digits are dierent, nd the maximum number of automobile registrations that the police may have to check.

(a) In how many ways can 6 people be lined up to get on a bus? (b) If 3 specific persons, among 6, insist on following each other, how many ways are possible? (c) If 2 specific persons, among 6, refuse to follow each other, how many ways are possible?

If a multiple-choice test consists of 5 questions, each with 4 possible answers of which only 1 is correct, (a) in how many dierent ways can a student check o one answer to each question? (b) in how many ways can a student check o one answer to each question and get all the answers wrong?

(a) How many distinct permutations can be made from the letters of the word COLUMNS? (b) How many of these permutations start with the letter M?

A contractor wishes to build 9 houses, each different in design. In how many ways can he place these houses on a street if 6 lots are on one side of the street and 3 lots are on the opposite side?

(a) How many three-digit numbers can be formed from the digits 0, 1, 2, 3, 4, 5, and 6 if each digit can be used only once? (b) How many of these are odd numbers? (c) How many are greater than 330?

In how many ways can 4 boys and 5 girls sit in a row if the boys and girls must alternate?

Four married couples have bought 8 seats in the same row for a concert. In how many dierent ways can they be seated (a) with no restrictions? (b) if each couple is to sit together? (c) if all the men sit together to the right of all the women?

In a regional spelling bee, the 8 nalists consist of 3 boys and 5 girls. Find the number of sample points in the sample space S for the number of possible orders at the conclusion of the contest for (a) all 8 nalists; (b) the rst 3 positions.

In how many ways can 5 starting positions on a basketball team be lled with 8 men who can play any of the positions?

Find the number of ways that 6 teachers can be assigned to 4 sections of an introductory psychology course if no teacher is assigned to more than one section.

Three lottery tickets for rst, second, and third prizes are drawn from a group of 40 tickets. Find the number of sample points in S for awarding the 3 prizes if each contestant holds only 1 ticket.

In how many ways can 5 different trees be planted in a circle?

In how many ways can a caravan of 8 covered wagons from Arizona be arranged in a circle?

How many distinct permutations can be made from the letters of the word INFINITY?

In how many ways can 3 oaks, 4 pines, and 2 maples be arranged along a property line if one does not distinguish among trees of the same kind?

How many ways are there to select 3 candidates from 8 equally qualied recent graduates for openings in an accounting rm?

How many ways are there that no two students will have the same birth date in a class of size 60?

Find the errors in each of the following statements: (a) The probabilities that an automobile salesperson will sell 0, 1, 2, or 3 cars on any given day in February are, respectively, 0.19, 0.38, 0.29, and 0.15. (b) The probability that it will rain tomorrow is 0.40, and the probability that it will not rain tomorrow is 0.52. (c) The probabilities that a printer will make 0, 1, 2, 3, or 4 or more mistakes in setting a document are, respectively, 0.19,0.34,0.25,0.43, and 0.29. (d) On a single draw from a deck of playing cards, the probability of selecting a heart is 1/4, the probability of selecting a black card is 1/2, and the probability of selecting both a heart and a black card is 1/8.

Assuming that all elements of S in Exercise 2.8 on page 42 are equally likely to occur, nd (a) the probability of event A; (b) the probability of event C; (c) the probability of event AC.

A box contains 500 envelopes, of which 75 contain $100 in cash, 150 contain $25, and 275 contain $10. An envelope may be purchased for $25. What is the sample space for the dierent amounts of money? Assign probabilities to the sample points and then nd the probability that the rst envelope purchased contains less than $100.

Suppose that in a senior college class of 500 students it is found that 210 smoke, 258 drink alcoholic beverages, 216 eat between meals, 122 smoke and drink alcoholic beverages, 83 eat between meals and drink alcoholic beverages, 97 smoke and eat between meals, and 52 engage in all three of these bad health practices. If a member of this senior class is selected at random, nd the probability that the student (a) smokes but does not drink alcoholic beverages; (b) eats between meals and drinks alcoholic beverages but does not smoke; (c) neither smokes nor eats between meals.

The probability that an American industry will locate in Shanghai, China, is 0.7, the probability that it will locate in Beijing, China, is 0.4, and the probability that it will locate in either Shanghai or Beijing or both is 0.8. What is the probability that the industry will locate (a) in both cities? (b) in neither city?

From past experience, a stockbroker believes that under present economic conditions a customer will invest in tax-free bonds with a probability of 0.6, will invest in mutual funds with a probability of 0.3, and will invest in both tax-free bonds and mutual funds with a probability of 0.15. At this time, find the probability that a customer will invest (a) in either tax-free bonds or mutual funds; (b) in neither tax-free bonds nor mutual funds.

If each coded item in a catalog begins with 3 distinct letters followed by 4 distinct nonzero digits, nd the probability of randomly selecting one of these coded items with the rst letter a vowel and the last digit even.

An automobile manufacturer is concerned about a possible recall of its best-selling four-door sedan. If there were a recall, there is a probability of 0.25 of a defect in the brake system, 0.18 of a defect in the transmission, 0.17 of a defect in the fuel system, and 0.40 of a defect in some other area. (a) What is the probability that the defect is the brakes or the fueling system if the probability of defects in both systems simultaneously is 0.15? (b) What is the probability that there are no defects in either the brakes or the fueling system?

If a letter is chosen at random from the English alphabet, nd the probability that the letter (a) is a vowel exclusive of y; (b) is listed somewhere ahead of the letter j; (c) is listed somewhere after the letter g.

A pair of fair dice is tossed. Find the probability of getting (a) a total of 8; (b) at most a total of 5.

In a poker hand consisting of 5 cards, find the probability of holding (a) 3 aces; (b) 4 hearts and 1 club.

If 3 books are picked at random from a shelf containing 5 novels, 3 books of poems, and a dictionary, what is the probability that (a) the dictionary is selected? (b) 2 novels and 1 book of poems are selected?

In a high school graduating class of 100 students, 54 studied mathematics, 69 studied history, and 35 studied both mathematics and history. If one of these students is selected at random, nd the probability that (a) the student took mathematics or history; (b) the student did not take either of these subjects; (c) the student took history but not mathematics.

Doms Pizza Company uses taste testing and statistical analysis of the data prior to marketing any new product. Consider a study involving three types of crusts (thin, thin with garlic and oregano, and thin with bits of cheese). Doms is also studying three sauces (standard, a new sauce with more garlic, and a new sauce with fresh basil). (a) How many combinations of crust and sauce are involved? (b) What is the probability that a judge will get a plain thin crust with a standard sauce for his rst taste test?

According to Consumer Digest (July/August 1996), the probable location of personal computers (PC) in the home is as follows: Adult bedroom: 0.03 Child bedroom: 0.15 Other bedroom: 0.14 Oce or den: 0.40 Other rooms: 0.28 (a) What is the probability that a PC is in a bedroom? (b) What is the probability that it is not in a bedroom? (c) Suppose a household is selected at random from households with a PC; in what room would you expect to nd a PC?

Interest centers around the life of an electronic component. Suppose it is known that the probability that the component survives for more than 6000 hours is 0.42. Suppose also that the probability that the component survives no longer than 4000 hours is 0.04. (a) What is the probability that the life of the component is less than or equal to 6000 hours? (b) What is the probability that the life is greater than 4000 hours?

Consider the situation of Exercise 2.64. Let A be the event that the component fails a particular test and B be the event that the component displays strain but does not actually fail. Event A occurs with probability 0.20, and event B occurs with probability 0.35. (a) What is the probability that the component does not fail the test? (b) What is the probability that the component works perfectly well (i.e., neither displays strain nor fails the test)? (c) What is the probability that the component either fails or shows strain in the test?

Factory workers are constantly encouraged to practice zero tolerance when it comes to accidents in factories. Accidents can occur because the working environment or conditions themselves are unsafe. On the other hand, accidents can occur due to carelessness or so-called human error. In addition, the worker’s shift, 7:00 A.M.–3:00 P.M. (day shift), 3:00 P.M.–11:00 P.M. (evening shift), or 11:00 P.M.–7:00 A.M. (graveyard shift), may be a factor. During the last year, 300 accidents have occurred. The percentages of the accidents for the condition combinations are as follows: If an accident report is selected randomly from the 300 reports, (a) what is the probability that the accident occurred on the graveyard shift? (b) what is the probability that the accident occurred due to human error? (c) what is the probability that the accident occurred due to unsafe conditions? (d) what is the probability that the accident occurred on either the evening or the graveyard shift?

Interest centers around the nature of an oven purchased at a particular department store. It can be either a gas or an electric oven. Consider the decisions made by six distinct customers. (a) Suppose that the probability is 0.40 that at most two of these individuals purchase an electric oven. What is the probability that at least three purchase the electric oven? (b) Suppose it is known that the probability that all six purchase the electric oven is 0.007 while 0.104 is the probability that all six purchase the gas oven. What is the probability that at least one of each type is purchased?

It is common in many industrial areas to use a lling machine to ll boxes full of product. This occurs in the food industry as well as other areas in which the product is used in the home, for example, detergent. These machines are not perfect, and indeed they may A, ll to specication, B, underll, and C, overll. Generally, the practice of underlling is that which one hopes to avoid. Let P(B)=0 .001 while P(A)=0 .990. (a) Give P(C). (b) What is the probability that the machine does not underll? (c) What is the probability that the machine either overlls or underlls?

It is common in many industrial areas to use a lling machine to ll boxes full of product. This occurs in the food industry as well as other areas in which the product is used in the home, for example, detergent. These machines are not perfect, and indeed they may A, ll to specication, B, underll, and C, overll. Generally, the practice of underlling is that which one hopes to avoid. Let P(B)=0 .001 while P(A)=0 .990. (a) Give P(C). (b) What is the probability that the machine does not underll? (c) What is the probability that the machine either overlls or underlls?

Consider the situation of Exercise 2.69. Suppose 50,000 boxes of detergent are produced per week and suppose also that those underlled are sent back, with customers requesting reimbursement of the purchase price. Suppose also that the cost of production is known to be $4.00 per box while the purchase price is $4.50 per box. (a) What is the weekly prot under the condition of no defective boxes? (b) What is the loss in prot expected due to underlling?

As the situation of Exercise 2.69 might suggest, statistical procedures are often used for control of quality (i.e., industrial quality control). At times, the weight of a product is an important variable to control. Specications are given for the weight of a certain packaged product, and a package is rejected if it is either too light or too heavy. Historical data suggest that 0.95 is the probability that the product meets weight specications whereas 0.002 is the probability that the product is too light. For each single packaged product, the manufacturer invests $20.00 in production and the purchase price for the consumer is $25.00. (a) What is the probability that a package chosen randomly from the production line is too heavy? (b) For each 10,000 packages sold, what prot is received by the manufacturer if all packages meet weight specication? (c) Assuming that all defective packages are rejected and rendered worthless, how much is the prot reduced on 10,000 packages due to failure to meet weight specication?

Prove that P(AB)=1+P(AB)P(A)P(B).

If R is the event that a convict committed armed robbery and D is the event that the convict pushed dope, state in words what probabilities are expressed by (a) P(R|D); (b) P(D|R); (c) P(R|D).

A class in advanced physics is composed of 10 juniors, 30 seniors, and 10 graduate students. The nal grades show that 3 of the juniors, 10 of the seniors, and 5 of the graduate students received an A for the course. If a student is chosen at random from this class and is found to have earned an A, what is the probability that he or she is a senior?

A random sample of 200 adults are classied below by sex and their level of education attained. Education Male Female Elementary 38 45 Secondary 28 50 College 22 17 If a person is picked at random from this group, nd the probability that (a) the person is a male, given that the person has a secondary education; (b) the person does not have a college degree, given that the person is a female.

In an experiment to study the relationship of hypertension and smoking habits, the following data are collected for 180 individuals: Moderate Heavy Nonsmokers Smokers Smokers H 21 36 30 NH 48 26 19 where H and NH in the table stand for Hypertension and Nonhypertension, respectively. If one of these individuals is selected at random, nd the probability that the person is (a) experiencing hypertension, given that the person is a heavy smoker; (b) a nonsmoker, given that the person is experiencing no hypertension.

In the senior year of a high school graduating class of 100 students, 42 studied mathematics, 68 studied psychology, 54 studied history, 22 studied both mathematics and history, 25 studied both mathematics and psychology, 7 studied history but neither mathematics nor psychology, 10 studied all three subjects, and 8 did not take any of the three. Randomly select a student from the class and nd the probabilities of the following events. (a) A person enrolled in psychology takes all three subjects. (b) A person not taking psychology is taking both history and mathematics.

A manufacturer of a u vaccine is concerned about the quality of its u serum. Batches of serum are processed by three dierent departments having rejection rates of 0.10, 0.08, and 0.12, respectively. The inspections by the three departments are sequential and independent. (a) What is the probability that a batch of serum survives the rst departmental inspection but is rejected by the second department? (b) What is the probability that a batch of serum is rejected by the third department?

In USA Today (Sept. 5, 1996), the results of a survey involving the use of sleepwear while traveling were listed as follows: Male Female Total Underwear 0.220 0.024 0.244 Nightgown 0.002 0.180 0.182 Nothing 0.160 0.018 0.178 Pajamas 0.102 0.073 0.175 T-shirt 0.046 0.088 0.134 Other 0.084 0.003 0.087 (a) What is the probability that a traveler is a female who sleeps in the nude? (b) What is the probability that a traveler is male? (c) Assuming the traveler is male, what is the probability that he sleeps in pajamas? (d) What is the probability that a traveler is male if the traveler sleeps in pajamas or a T-shirt?

The probability that an automobile being lled with gasoline also needs an oil change is 0.25; the probability that it needs a new oil lter is 0.40; and the probability that both the oil and the lter need changing is 0.14. (a) If the oil has to be changed, what is the probability that a new oil lter is needed? (b) If a new oil lter is needed, what is the probability that the oil has to be changed?

The probability that a married man watches a certain television show is 0.4, and the probability that a married woman watches the show is 0.5. The probability that a man watches the show, given that his wife does, is 0.7. Find the probability that (a) a married couple watches the show; (b) a wife watches the show, given that her husband does; (c) at least one member of a married couple will watch the show.

For married couples living in a certain suburb, the probability that the husband will vote on a bond referendum is 0.21, the probability that the wife will vote on the referendum is 0.28, and the probability that both the husband and the wife will vote is 0.15. What is the probability that (a) at least one member of a married couple will vote? (b) a wife will vote, given that her husband will vote? (c) a husband will vote, given that his wife will not vote?

The probability that a vehicle entering the Luray Caverns has Canadian license plates is 0.12; the probability that it is a camper is 0.28; and the probability that it is a camper with Canadian license plates is 0.09. What is the probability that (a) a camper entering the Luray Caverns has Canadian license plates? (b) a vehicle with Canadian license plates entering the Luray Caverns is a camper? (c) a vehicle entering the Luray Caverns does not have Canadian plates or is not a camper?

The probability that the head of a household is home when a telemarketing representative calls is 0.4. Given that the head of the house is home, the probability that goods will be bought from the company is 0.3. Find the probability that the head of the house is home and goods are bought from the company.

The probability that a doctor correctly diagnoses a particular illness is 0.7. Given that the doctor makes an incorrect diagnosis, the probability that the patient files a lawsuit is 0.9. What is the probability that the doctor makes an incorrect diagnosis and the patient sues?

In 1970, 11% of Americans completed four years of college; 43% of them were women. In 1990, 22% of Americans completed four years of college; 53% of them were women (Time, Jan. 19, 1996). (a) Given that a person completed four years of college in 1970, what is the probability that the person was a woman? (b) What is the probability that a woman nished four years of college in 1990? (c) What is the probability that a man had not nished college in 1990?

A real estate agent has 8 master keys to open several new homes. Only 1 master key will open any given house. If 40% of these homes are usually left unlocked, what is the probability that the real estate agent can get into a specific home if the agent selects 3 master keys at random before leaving the office?

Before the distribution of certain statistical software, every fourth compact disk (CD) is tested for accuracy. The testing process consists of running four independent programs and checking the results. The failure rates for the four testing programs are, respectively, 0.01, 0.03, 0.02, and 0.01. (a) What is the probability that a CD was tested and failed any test? (b) Given that a CD was tested, what is the probability that it failed program 2 or 3? (c) In a sample of 100, how many CDs would you expect to be rejected? (d) Given that a CD was defective, what is the probability that it was tested?

A town has two fire engines operating independently. The probability that a specific engine is available when needed is 0.96. (a) What is the probability that neither is available when needed? (b) What is the probability that a fire engine is available when needed?

Pollution of the rivers in the United States has been a problem for many years. Consider the following events: A: the river is polluted, B: a sample of water tested detects pollution, C: shing is permitted. Assume P(A)=0 .3, P(B|A)=0 .75, P(B|A)=0 .20, P(C|AB)=0 .20, P(C|AB)=0 .15, P(C|AB)=0 .80, and P(C|AB)=0 .90. (a) Find P(AB C). (b) Find P(BC). (c) Find P(C). (d) Find the probability that the river is polluted, given that shing is permitted and the sample tested did not detect pollution.

Find the probability of randomly selecting 4 good quarts of milk in succession from a cooler containing 20 quarts of which 5 have spoiled, by using (a) the first formula of Theorem 2.12 on page 68; (b) the formulas of Theorem 2.6 and Rule 2.3 on pages 50 and 54, respectively.

Suppose the diagram of an electrical system is as given in Figure 2.10. What is the probability that the system works? Assume the components fail independently.

A circuit system is given in Figure 2.11. Assume the components fail independently. (a) What is the probability that the entire system works? (b) Given that the system works, what is the probability that the component A is not working?

In the situation of Exercise 2.93, it is known that the system does not work. What is the probability that the component A also does not work?

In a certain region of the country it is known from past experience that the probability of selecting an adult over 40 years of age with cancer is 0.05. If the probability of a doctor correctly diagnosing a person with cancer as having the disease is 0.78 and the probability of incorrectly diagnosing a person without cancer as having the disease is 0.06, what is the prob ability that an adult over 40 years of age is diagnosed as having cancer?

Police plan to enforce speed limits by using radar traps at four different locations within the city limits. The radar traps at each of the locations \(L_1\), \(L_2\), \(L_3\), and \(L_4\) will be operated 40%, 30%, 20%, and 30% of the time. If a person who is speeding on her way to work has probabilities of 0.2, 0.1, 0.5, and 0.2, respectively, of passing through these locations, what is the probability that she will receive a speeding ticket?

Referring to Exercise 2.95, what is the probability that a person diagnosed as having cancer actually has the disease?

If the person in Exercise 2.96 received a speeding ticket on her way to work, what is the probability that she passed through the radar trap located at L2?

Suppose that the four inspectors at a lm factory are supposed to stamp the expiration date on each package of lm at the end of the assembly line. John, who stamps 20% of the packages, fails to stamp the expiration date once in every 200 packages; Tom, who stamps 60% of the packages, fails to stamp the expiration date once in every 100 packages; Je, who stamps 15% of the packages, fails to stamp the expiration date once in every 90 packages; and Pat, who stamps 5% of the packages, fails to stamp the expiration date once in every 200 packages. If a customer complains that her package of lm does not show the expiration date, what is the probability that it was inspected by John

A regional telephone company operates three identical relay stations at dierent locations. During a one-year period, the number of malfunctions reported by each station and the causes are shown below. Station ABC Problems with electricity supplied 2 1 1 Computer malfunction 4 3 2 Malfunctioning electrical equipment 5 4 2 Caused by other human errors 7 7 5 Suppose that a malfunction was reported and it was found to be caused by other human errors. What is the probability that it came from station C?

A paint-store chain produces and sells latex and semigloss paint. Based on long-range sales, the probability that a customer will purchase latex paint is 0.75. Of those that purchase latex paint, 60% also purchase rollers. But only 30% of semigloss paint buyers purchase rollers. A randomly selected buyer purchases a roller and a can of paint. What is the probability that the paint is latex?

Denote by A, B, and C the events that a grand prize is behind doors A, B, and C, respectively. Suppose you randomly picked a door, say A. The game host opened a door, say B, and showed there was no prize behind it. Now the host oers you the option of either staying at the door that you picked (A) or switching to the remaining unopened door (C). Use probability to explain whether you should switch or not.

A truth serum has the property that 90% of the guilty suspects are properly judged while, of course, 10% of the guilty suspects are improperly found innocent. On the other hand, innocent suspects are misjudged 1% of the time. If the suspect was selected from a group of suspects of which only 5% have ever committed a crime, and the serum indicates that he is guilty, what is the probability that he is innocent?

An allergist claims that 50% of the patients she tests are allergic to some type of weed. What is the probability that (a) exactly 3 of her next 4 patients are allergic to weeds? (b) none of her next 4 patients is allergic to weeds?

By comparing appropriate regions of Venn diagrams, verify that (a) \((A \cap B) \cup\left(A \cap B^{\prime}\right)=A\); (b) \(A^{\prime} \cap\left(B^{\prime} \cup C\right)=\left(A^{\prime} \cap B^{\prime}\right) \cup\left(A^{\prime} \cap C\right)\).

The probabilities that a service station will pump gas into 0, 1, 2, 3, 4, or 5 or more cars during a certain 30-minute period are 0.03, 0.18, 0.24, 0.28, 0.10, and 0.17, respectively. Find the probability that in this 30-minute period (a) more than 2 cars receive gas; (b) at most 4 cars receive gas; (c) 4 or more cars receive gas.

How many bridge hands are possible containing 4 spades, 6 diamonds, 1 club, and 2 hearts?

If the probability is 0.1 that a person will make a mistake on his or her state income tax return, nd the probability that (a) four totally unrelated persons each make a mistake; (b) Mr. Jones and Ms. Clark both make mistakes, and Mr. Roberts and Ms. Williams do not make a mistake.

A large industrial rm uses three local motels to provide overnight accommodations for its clients. From past experience it is known that 20% of the clients are assigned rooms at the Ramada Inn, 50% at the Sheraton, and 30% at the Lakeview Motor Lodge. If the plumbing is faulty in 5% of the rooms at the Ramada Inn, in 4% of the rooms at the Sheraton, and in 8% of the rooms at the Lakeview Motor Lodge, what is the probability that (a) a client will be assigned a room with faulty plumbing? (b) a person with a room having faulty plumbing was assigned accommodations at the Lakeview Motor Lodge?

The probability that a patient recovers from a delicate heart operation is 0.8. What is the probability that (a) exactly 2 of the next 3 patients who have this operation survive? (b) all of the next 3 patients who have this operation survive?

In a certain federal prison, it is known that 2/3 of the inmates are under 25 years of age. It is also known that 3/5 of the inmates are male and that 5/8 of the inmates are female or 25 years of age or older. What is the probability that a prisoner selected at random from this prison is female and at least 25 years old?

From 4 red, 5 green, and 6 yellow apples, how many selections of 9 apples are possible if 3 of each color are to be selected?

From a box containing 6 black balls and 4 green balls, 3 balls are drawn in succession, each ball being replaced in the box before the next draw is made. What is the probability that (a) all 3 are the same color? (b) each color is represented?

A shipment of 12 television sets contains 3 defective sets. In how many ways can a hotel purchase 5 of these sets and receive at least 2 of the defective sets?

A certain federal agency employs three consulting rms (A, B, and C) with probabilities 0.40, 0.35, and 0.25, respectively. From past experience it is known that the probability of cost overruns for the rms are 0.05, 0.03, and 0.15, respectively. Suppose a cost overrun is experienced by the agency. (a) What is the probability that the consulting rm involved is company C? (b) What is the probability that it is company A?

A manufacturer is studying the eects of cooking temperature, cooking time, and type of cooking oil for making potato chips. Three dierent temperatures, 4 dierent cooking times, and 3 dierent oils are to be used. (a) What is the total number of combinations to be studied? (b) How many combinations will be used for each type of oil? (c) Discuss why permutations are not an issue in this exercise.

Consider the situation in Exercise 2.116, and suppose that the manufacturer can try only two combinations in a day. (a) What is the probability that any given set of two runs is chosen? (b) What is the probability that the highest temperature is used in either of these two combinations?

A certain form of cancer is known to be found in women over 60 with probability 0.07. A blood test exists for the detection of the disease, but the test is not infallible. In fact, it is known that 10% of the time the test gives a false negative (i.e., the test incorrectly gives a negative result) and 5% of the time the test gives a false positive (i.e., incorrectly gives a positive result). If a woman over 60 is known to have taken the test and received a favorable (i.e., negative) result, what is the probability that she has the disease?

A producer of a certain type of electronic component ships to suppliers in lots of twenty. Suppose that 60% of all such lots contain no defective components, 30% contain one defective component, and 10% contain two defective components. A lot is picked, two components from the lot are randomly selected and tested, and neither is defective. (a) What is the probability that zero defective components exist in the lot? (b) What is the probability that one defective exists in the lot? (c) What is the probability that two defectives exist in the lot?

A rare disease exists with which only 1 in 500 is aected. A test for the disease exists, but of course it is not infallible. A correct positive result (patient actually has the disease) occurs 95% of the time, while a false positive result (patient does not have the disease) occurs 1% of the time. If a randomly selected individual is tested and the result is positive, what is the probability that the individual has the disease?

A construction company employs two sales engineers. Engineer 1 does the work of estimating cost for 70% of jobs bid by the company. Engineer 2 does the work for 30% of jobs bid by the company. It is known that the error rate for engineer 1 is such that 0.02 is the probability of an error when he does the work, whereas the probability of an error in the work of engineer 2 is 0.04. Suppose a bid arrives and a serious error occurs in estimating cost. Which engineer would you guess did the work? Explain and show all work.

In the field of quality control, the science of statistics is often used to determine if a process is “out of control.” Suppose the process is, indeed, out of control and 20% of items produced are defective. (a) If three items arrive off the process line in succession, what is the probability that all three are defective? (b) If four items arrive in succession, what is the probability that three are defective?

An industrial plant is conducting a study to determine how quickly injured workers are back on the job following injury. Records show that 10% of all injured workers are admitted to the hospital for treatment and 15% are back on the job the next day. In addition, studies show that 2% are both admitted for hospital treatment and back on the job the next day. If a worker is injured, what is the probability that the worker will either be admitted to a hospital or be back on the job the next day or both?

A rm is accustomed to training operators who do certain tasks on a production line. Those operators who attend the training course are known to be able to meet their production quotas 90% of the time. New operators who do not take the training course only meet their quotas 65% of the time. Fifty percent of new operators attend the course. Given that a new operator meets her production quota, what is the probability that she attended the program?

A survey of those using a particular statistical software system indicated that 10% were dissatised. Half of those dissatised purchased the system from vendor A. It is also known that 20% of those surveyed purchased from vendor A. Given that the software was purchased from vendor A, what is the probability that that particular user is dissatised?

During bad economic times, industrial workers are dismissed and are often replaced by machines. The history of 100 workers whose loss of employment is attributable to technological advances is reviewed. For each of these individuals, it is determined if he or she was given an alternative job within the same company, found a job with another company in the same eld, found a job in a new eld, or has been unemployed for 1 year. In addition, the union status of each worker is recorded. The following table summarizes the results. Union Nonunion Same Company New Company (same eld) New Field Unemployed 40 13 4 2 15 10 11 5 (a) If the selected worker found a job with a new company in the same eld, what is the probability that the worker is a union member? (b) If the worker is a union member, what is the probability that the worker has been unemployed for a year?

There is a 50-50 chance that the queen carries the gene of hemophilia. If she is a carrier, then each prince has a 50-50 chance of having hemophilia independently. If the queen is not a carrier, the prince will not have the disease. Suppose the queen has had three princes without the disease. What is the probability the queen is a carrier?

Group Project: Give each student a bag of chocolate M&Ms. Divide the students into groups of 5 or 6. Calculate the relative frequency distribution for color of M&Ms for each group. (a) What is your estimated probability of randomly picking a yellow? a red? (b) Redo the calculations for the whole classroom. Did the estimates change? (c) Do you believe there is an equal number of each color in a process batch? Discuss.