Suppose two dice are tossed and the numbers on the upper

Suppose that n indistinguishable balls are to be arranged in N distinguishable boxes so that each distinguishable arrangement is equally likely. If \(n \geq N\), show that the probability no box will be empty is given by \(\frac{\left(\begin{array}{c} n-1 \\ N-1 \end{array}\right)}{\left(\begin{array}{c} N+n-1 \\ N-1 \end{array}\right)} \) Equation Transcription: Text Transcription: n >/= N (n-1 N-1) over (N+n-1 N-1)

Problem 2E Suppose that A and B are two events. Write expressions involving unions, intersections, and complements that describe the following: a Both events occur. b At least one occurs. c Neither occurs. d Exactly one occurs.

Draw Venn diagrams to verify DeMorgan's laws. That is, for any two sets A and B, \(\overline{(A \cup B})=\bar{A} \cap \bar{B}\) and \((\overline{A \cap B})=\bar{A} \cup \bar{B}\). Equation Transcription: Text Transcription: (A union B hbar)=A bar intersection B bar (A intersection B hbar)=A bar union B bar

Suppose a family contains two children of different ages, and we are interested in the gender of these children. Let F denote that a child is female and M that the child is male and let a pair such as FM denote that the older child is female and the younger is male. There are four points in the set S of possible observations: \(S=\{F F\), \(F M\), \(M F\), \(M M\}\). Let A denote the subset of possibilities containing no males; B, the subset containing two males; and C, the subset containing at least one male. List the elements of A, B,C, \(A \cap B\), \(A \cup B\), \(A \cap C\), \(A \cup C\), \(B \cap C\), \(B \cup C\), and \(C \cap \bar{B}\). Equation Transcription: Text Transcription: S={FF,FM,MF,MM} A cap B A U B A cap C A UC B cap C B U C C cap B bar

2.4 If A and B are two sets, draw Venn diagrams to verify the following: a \(A=(A \cap B) \cup(A \cap \bar{B})\). b If \(B \subset A\) then \(A=B \cup(A \cap \bar{B})\). Equation Transcription: Text Transcription: A=(A cap B)U(A cap B) B subset A A=B U (A cap B bar)

Refer to Exercise 2.4. Use the identities \(A=A \cap S\) and \(S=B \cup \bar{B}\) and a distributive law to prove that a \(A=(A \cap B) \cup(A \cap \bar{B})\). b If \(B \subset A\) then \(A=(A \cap B) \cup(A \cap \bar{B})\). c Further, show that \((A \cap B)\) and \((A \cap \bar{B})\) are mutually exclusive and therefore that A is the union of two mutually exclusive sets, \((A \cap B)\) and \((A \cap \bar{B})\). d Also show that B and \((A \cap \bar{B})\) are mutually exclusive and if \(B \subset A\), A is the union of two mutually exclusive sets, B and \((A \cap \bar{B})\). Equation Transcription: Text Transcription: A=A cap S S=B U B bar A=(A cap B)U(A cap B bar) B subset A A=B U(A cap B bar) (A cap B) (A cap B bar) (A cap B) (A cap B bar) (A cap B bar) B subset A (A cap B bar)

2.8 Suppose two dice are tossed and the numbers on the upper faces are observed. Let S denote the set of all possible pairs that can be observed. [These pairs can be listed, for example, by letting (2, 3) denote that a 2 was observed on the first die and a 3 on the second.] a Define the following subsets of S: A: The number on the second die is even. B: The sum of the two numbers is even. C: At least one number in the pair is odd. b List the points in A, \(\bar{C}\), \(A \cap B\), \(A \cap \bar{B}\), \(\bar{A} \cup B\), and \(\bar{A} \cap C\). Equation Transcription: Text Transcription: C bar A cap B A cap B bar A bar U B A bar cap C

Problem 9E Every person’s blood type is A, B, AB, or O. In addition, each individual either has the Rhesus (Rh) factor (+) or does not (?). A medical technician records a person’s blood type and Rh factor. List the sample space for this experiment.

A group of five applicants for a pair of identical jobs consists of three men and two women. The employer is to select two of the five applicants for the jobs. Let S denote the set of all possible outcomes for the employer’s selection. Let A denote the subset of outcomes corresponding to the selection of two men and B the subset corresponding to the selection of at least one woman. List the outcomes in A, \(\bar{B}\), \(A \cup B\), \(A \cap B\), and \(A \cap \bar{B}\). (Denote the different men and women by \(M_{1}\), \(M_{2}\), \(M_{3}\) and \(W_{1}\), \(W_{2}\), respectively.) Equation Transcription: Text Transcription: B bar A U B A cap B A cap B bar M_1 M_2 M_3 W_1 W_2

Problem 12E A vehicle arriving at an intersection can turn right, turn left, or continue straight ahead. The experiment consists of observing the movement of a single vehicle through the intersection. a List the sample space for this experiment. b Assuming that all sample points are equally likely, find the probability that the vehicle turns.

Problem 8E From a survey of 60 students attending a university, it was found that 9 were living off campus, 36 were undergraduates, and 3 were undergraduates living off campus. Find the number of these students who were a undergraduates, were living off campus, or both. b undergraduates living on campus. c graduate students living on campus.

Problem 11E A sample space consists of five simple events, E1, E2, E3, E4, and E5. a If P(E1) = P(E2) = 0.15, P(E3) = 0.4, and P(E4) = 2P(E5), find the probabilities of E4 and E5. b If P(E1) = 3P(E2) = 0.3, find the probabilities of the remaining simple events if you know that the remaining simple events are equally probable.

Problem 10E The proportions of blood phenotypes, A, B, AB, and O, in the population of all Caucasians in the United States are approximately .41, .10, .04, and .45, respectively. A single Caucasian is chosen at random from the population. a List the sample space for this experiment. b Make use of the information given above to assign probabilities to each of the simple events. c What is the probability that the person chosen at random has either type A or type AB blood?

Americans can be quite suspicious, especially when it comes to government conspiracies. On the question of whether the U.S. Air Force has withheld proof of the existence of intelligent life on other planets, the proportions of Americans with varying opinions are given in the table. Opinion Proportion Very likely .24 Somewhat likely .24 Unlikely .40 Other .12 Suppose that one American is selected and his or her opinion is recorded. a What are the simple events for this experiment? b Are the simple events that you gave in part (a) all equally likely? If not, what are the probabilities that should be assigned to each? c What is the probability that the person selected finds it at least somewhat likely that the Air Force is withholding information about intelligent life on other planets?

A survey classified a large number of adults according to whether they were diagnosed as needing eyeglasses to correct their reading vision and whether they use eyeglasses when reading. The proportions falling into the four resulting categories are given in the following table: Uses Eyeglasses for Reading Needs Glasses Yes No Yes .44 .14 No .02 .40 If a single adult is selected from the large group, find the probabilities of the events defined below. The adult a needs glasses. b needs glasses but does not use them. c uses glasses whether the glasses are needed or not.

An oil prospecting firm hits oil or gas on 10% of its drillings. If the firm drills two wells, the four possible simple events and three of their associated probabilities are as given in the accompanying table. Find the probability that the company will hit oil or gas a on the first drilling and miss on the second. b on at least one of the two drillings. Simple Event Outcome of First Drilling Outcome of Second Drilling Probability \(E_{1}\) Hit (oil or gas) Hit (oil or gas) .01 \(E_{2}\) Hit Miss ? \(E_{3}\) Miss Hit .09 \(E_{4}\) Miss Miss .81 Equation Transcription: Text Transcription: E_1 E_2 E_3 E_4

Problem 16E Of the volunteers coming into a blood center, 1 in 3 have O+ blood, 1 in 15 have O?, 1 in 3 have A+, and 1 in 16 have A?. The name of one person who previously has donated blood is selected from the records of the center. What is the probability that the person selected has a type O+ blood? b type O blood? c type A blood? d neither type A nor type O blood?

Problem 20E The following game was played on a popular television show. The host showed a contestant three large curtains. Behind one of the curtains was a nice prize (maybe a new car) and behind the other two curtains were worthless prizes (duds). The contestant was asked to choose one curtain. If the curtains are identified by their prizes, they could be labeled G, D1, and D2 (Good Prize, Dud1, and Dud2). Thus, the sample space for the contestants choice is S = {G, D1, D2}.1 a If the contestant has no idea which curtains hide the various prizes and selects a curtain at random, assign reasonable probabilities to the simple events and calculate the probability that the contestant selects the curtain hiding the nice prize. b Before showing the contestant what was behind the curtain initially chosen, the game show host would open one of the curtains and show the contestant one of the duds (he could always do this because he knew the curtain hiding the good prize). He then offered the contestant the option of changing from the curtain initially selected to the other remaining unopened curtain. Which strategy maximizes the contestant’s probability of winning the good prize: stay with the initial choice or switch to the other curtain? In answering the following sequence of questions, you will discover that, perhaps surprisingly, this question can be answered by considering only the sample space above and using the probabilities that you assigned to answer part (a). i If the contestant choses to stay with her initial choice, she wins the good prize if and only if she initially chose curtain G. If she stays with her initial choice, what is the probability that she wins the good prize? ii If the host shows her one of the duds and she switches to the other unopened curtain, what will be the result if she had initially selected G? iii Answer the question in part (ii) if she had initially selected one of the duds. iv If the contestant switches from her initial choice (as the result of being shown one of the duds), what is the probability that the contestant wins the good prize? v Which strategy maximizes the contestant’s probability of winning the good prize: stay with the initial choice or switch to the other curtain?

Problem 17E Hydraulic landing assemblies coming from an aircraft rework facility are each inspected for defects. Historical records indicate that 8% have defects in shafts only, 6% have defects in bushings only, and2%have defects in both shafts and bushings. One of the hydraulic assemblies is selected randomly. What is the probability that the assembly has a a bushing defect? b a shaft or bushing defect? c exactly one of the two types of defects? d neither type of defect?

Suppose two balanced coins are tossed and the upper faces are observed. a List the sample points for this experiment. b Assign a reasonable probability to each sample point. (Are the sample points equally likely?) c Let A denote the event that exactly one head is observed and B the event that at least one head is observed. List the sample points in both A and B. d From your answer to part (c), find \(P(A)\), \(P(B)\), \(P(A \cap B)\), \(P(A \cup B)\), and \(P(\bar{A} \cup B)\). Equation Transcription: Text Transcription: P(A) P(B) P(A cap B) P(A U B) P(A bar U B)

If A and B are events, use the result derived in Exercise 2.5(a) and the Axioms in Definition 2.6 to prove that \(P(A)=P(A \cap B)+P(A \cap \bar{B})\). Equation Transcription: Text Transcription: P(A)=P(A cap B)+P(A cap B bar)

Problem 19E A business office orders paper supplies from one of three vendors, V1, V2, or V3. Orders are to be placed on two successive days, one order per day. Thus, (V2, V3) might denote that vendor V2 gets the order on the first day and vendor V3 gets the order on the second day. a List the sample points in this experiment of ordering paper on two successive days. b Assume the vendors are selected at random each day and assign a probability to each sample point. c Let A denote the event that the same vendor gets both orders and B the event that V2 gets at least one order. Find P( A), P( B), P( A ? B), and P( A ? B) by summing the probabilities of the sample points in these events.

Problem 23E If A and B are events and B ? A, why is it “obvious” that P( B) ? P( A)?

Problem 25E A single car is randomly selected from among all of those registered at a local tag agency. What do you think of the following claim? “All cars are either Volkswagens or they are not. Therefore, the probability is 1/2 that the car selected is a Volkswagen.”

Problem 27E In Exercise 2.12 we considered a situation where cars entering an intersection each could turn right, turn left, or go straight. An experiment consists of observing two vehicles moving through the intersection. a How many sample points are there in the sample space? List them. b Assuming that all sample points are equally likely, what is the probability that at least one car turns left? c Again assuming equally likely sample points, what is the probability that at most one vehicle turns? Reference A vehicle arriving at an intersection can turn right, turn left, or continue straight ahead. The experiment consists of observing the movement of a single vehicle through the intersection. a List the sample space for this experiment. b Assuming that all sample points are equally likely, find the probability that the vehicle turns.

Problem 26E According to Webster’s New Collegiate Dictionary, a divining rod is “a forked rod believed to indicate [divine] the presence of water or minerals by dipping downward when held over a vein.” To test the claims of a divining rod expert, skeptics bury four cans in the ground, two empty and two filled with water. The expert is led to the four cans and told that two contain water. He uses the divining rod to test each of the four cans and decide which two contain water. a List the sample space for this experiment. b If the divining rod is completely useless for locating water, what is the probability that the expert will correctly identify (by guessing) both of the cans containing water?

Problem 28E Four equally qualified people apply for two identical positions in a company. One and only one applicant is a member of a minority group. The positions are filled by choosing two of the applicants at random. a List the possible outcomes for this experiment. b Assign reasonable probabilities to the sample points. c Find the probability that the applicant from the minority group is selected for a position.

2.24 Use the result in Exercise 2.22 and the Axioms in Definition 2.6 to prove the “obvious” result in Exercise 2.23. Reference 2.23 If A and B are events and \(B \subset A\), why is it “obvious” that \(P(B) \leq(A)\)? 2.21 If A and B are events, use the result derived in Exercise 2.5(a) and the Axioms in Definition 2.6 to prove that \(P(A)=P(A \cap B)+P(A \cap \bar{B})\). Equation Transcription: Text Transcription: B subset A P(B)</=(A) P(A)=P(A cap B)+P(A cap B bar)

Problem 29E Two additional jurors are needed to complete a jury for a criminal trial. There are six prospective jurors, two women and four men. Two jurors are randomly selected from the six available. a Define the experiment and describe one sample point. Assume that you need describe only the two jurors chosen and not the order in which they were selected. b List the sample space associated with this experiment. c What is the probability that both of the jurors selected are women?

If A and B are events and \(B \subset A\), use the result derived in Exercise 2.5(b) and the Axioms in Definition 2.6 to prove that \(P(A)=P(B)+P(A \cap \bar{B})\). Equation Transcription: Text Transcription: B subset A P(A)=P(B)+P(A cap B bar)

Problem 30E Three imported wines are to be ranked from lowest to highest by a purported wine expert. That is, one wine will be identified as best, another as second best, and the remaining wine as worst. a Describe one sample point for this experiment. b List the sample space. c Assume that the “expert” really knows nothing about wine and randomly assigns ranks to the three wines. One of the wines is of much better quality than the others. What is the probability that the expert ranks the best wine no worse than second best?

Problem 31E A boxcar contains six complex electronic systems. Two of the six are to be randomly selected for thorough testing and then classified as defective or not defective. a If two of the six systems are actually defective, find the probability that at least one of the two systems tested will be defective. Find the probability that both are defective. b If four of the six systems are actually defective, find the probabilities indicated in part (a).

Problem 32E A retailer sells only two styles of stereo consoles, and experience shows that these are in equal demand. Four customers in succession come into the store to order stereos. The retailer is interested in their preferences. a List the possibilities for preference arrangements among the four customers (that is, list the sample space). b Assign probabilities to the sample points. c Let A denote the event that all four customers prefer the same style. Find P( A).

Problem 33E The Bureau of the Census reports that the median family income for all families in the United States during the year 2003 was $43,318. That is, half of all American families had incomes exceeding this amount, and half had incomes equal to or below this amount. Suppose that four families are surveyed and that each one reveals whether its income exceeded $43,318 in 2003. a List the points in the sample space. b Identify the simple events in each of the following events: A: At least two had incomes exceeding $43,318. B: Exactly two had incomes exceeding $43,318. C: Exactly one had income less than or equal to $43,318. c Make use of the given interpretation for the median to assign probabilities to the simple events and find P( A), P( B), and P(C).

Problem 34E Patients arriving at a hospital outpatient clinic can select one of three stations for service. Suppose that physicians are assigned randomly to the stations and that the patients therefore have no station preference. Three patients arrive at the clinic and their selection of stations is observed. a List the sample points for the experiment. b Let A be the event that each station receives a patient. List the sample points in A. c Make a reasonable assignment of probabilities to the sample points and find P( A).

Problem 36E An assembly operation in a manufacturing plant requires three steps that can be performed in any sequence. How many different ways can the assembly be performed?

Problem 35E An airline has six flights from New York to California and seven flights from California to Hawaii per day. If the flights are to be made on separate days, how many different flight arrangements can the airline offer from New York to Hawaii?

qQqProblem 37E A businesswoman in Philadelphia is preparing an itinerary for a visit to six major cities. The distance traveled, and hence the cost of the trip, will depend on the order in which she plans her route. a How many different itineraries (and trip costs) are possible? b If the businesswoman randomly selects one of the possible itineraries and Denver and San Francisco are two of the cities that she plans to visit, what is the probability that she will visit Denver before San Francisco?

Problem 38E An upscale restaurant offers a special fixe prix menu in which, for a fixed dinner cost, a diner can select from four appetizers, three salads, four entrees, and five desserts. How many different dinners are available if a dinner consists of one appetizer, one salad, one entree, and one dessert?

Problem 39E An experiment consists of tossing a pair of dice. a Use the combinatorial theorems to determine the number of sample points in the sample space S. b Find the probability that the sum of the numbers appearing on the dice is equal to 7.

Problem 42E A personnel director for a corporation has hired ten new engineers. If three (distinctly different) positions are open at a Cleveland plant, in how many ways can she fill the positions?

Problem 43E A fleet of nine taxis is to be dispatched to three airports in such a way that three go to airport A, five go to airport B, and one goes to airport C. In how many distinct ways can this be accomplished?

Problem 44E Refer to Exercise 2.43. Assume that taxis are allocated to airports at random. a If exactly one of the taxis is in need of repair, what is the probability that it is dispatched to airport C? b If exactly three of the taxis are in need of repair, what is the probability that every airport receives one of the taxis requiring repairs? Reference A fleet of nine taxis is to be dispatched to three airports in such a way that three go to airport A, five go to airport B, and one goes to airport C. In how many distinct ways can this be accomplished?

Problem 41E How many different seven-digit telephone numbers can be formed if the first digit cannot be zero?

Problem 40E A brand of automobile comes in five different styles, with four types of engines, with two types of transmissions, and in eight colors. a How many autos would a dealer have to stock if he included one for each style–engine– transmission combination? b How many would a distribution center have to carry if all colors of cars were stocked for each combination in part (a)?

Problem 45E Suppose that we wish to expand (x + y + z)17. What is the coefficient of x2 y5 z10?

Problem 46E Ten teams are playing in a basketball tournament. In the first round, the teams are randomly assigned to games 1, 2, 3, 4 and 5. In how many ways can the teams be assigned to the games?

Refer to Exercise 2.46. If \(2n\) teams are to be assigned to games 1, 2, . . . ,\(n\), in how many ways can the teams be assigned to the games?

Problem 49E Students attending the University of Florida can select from 130 major areas of study. A student’s major is identified in the registrar’s records with a two-or three-letter code (for example, statistics majors are identified by STA, math majors by MS). Some students opt for a double major and complete the requirements for both of the major areas before graduation. The registrar was asked to consider assigning these double majors a distinct two- or three-letter code so that they could be identified through the student records’ system. a What is the maximum number of possible double majors available to University of Florida students? b If any two- or three-letter code is available to identify majors or double majors, how many major codes are available? c How many major codes are required to identify students who have either a single major or a double major? d Are there enough major codes available to identify all single and double majors at the University of Florida?

Problem 50E Probability played a role in the rigging of the April 24, 1980, Pennsylvania state lottery (Los Angeles Times, September 8, 1980). To determine each digit of the three-digit winning number, each of the numbers 0, 1, 2, . . . , 9 is placed on a Ping-Pong ball, the ten balls are blown into a compartment, and the number selected for the digit is the one on the ball that floats to the top of the machine. To alter the odds, the conspirators injected a liquid into all balls used in the game except those numbered 4 and 6, making it almost certain that the lighter balls would be selected and determine the digits in the winning number. Then they bought lottery tickets bearing the potential winning numbers. How many potential winning numbers were there (666 was the eventual winner)?

Problem 48E If we wish to expand (x + y)8, what is the coefficient of x5 y3? What is the coefficient of x3 y5?

A local fraternity is conducting a raffle where 50 tickets are to be sold—one per customer. There are three prizes to be awarded. If the four organizers of the raffle each buy one ticket, what is the probability that the four organizers win a. all of the prizes? b. exactly two of the prizes? c. exactly one of the prizes? d. none of the prizes?

Problem 52E An experimenter wishes to investigate the effect of three variables—pressure, temperature, and the type of catalyst—on the yield in a refining process. If the experimenter intends to use three settings each for temperature and pressure and two types of catalysts, how many experimental runs will have to be conducted if he wishes to run all possible combinations of pressure, temperature, and types of catalysts?

Problem 53E Five firms, F1, F2, . . . , F5, each offer bids on three separate contracts, C1, C2, and C3. Any one firm will be awarded at most one contract. The contracts are quite different, so an assignment of C1 to F1, say, is to be distinguished from an assignment of C2 to F1. a How many sample points are there altogether in this experiment involving assignment of contracts to the firms? (No need to list them all.) b Under the assumption of equally likely sample points, find the probability that F3 is awarded a contract.

Problem 54E A group of three undergraduate and five graduate students are available to fill certain student government posts. If four students are to be randomly selected from this group, find the probability that exactly two undergraduates will be among the four chosen.

Problem 56E A student prepares for an exam by studying a list of ten problems. She can solve six of them. For the exam, the instructor selects five problems at random from the ten on the list given to the students. What is the probability that the student can solve all five problems on the exam?

Problem 57E Two cards are drawn from a standard 52-card playing deck. What is the probability that the draw will yield an ace and a face card?

Problem 55E A study is to be conducted in a hospital to determine the attitudes of nurses toward various administrative procedures. A sample of 10 nurses is to be selected from a total of the 90 nurses employed by the hospital. a How many different samples of 10 nurses can be selected? b Twenty of the 90 nurses are male. If 10 nurses are randomly selected from those employed by the hospital, what is the probability that the sample of ten will include exactly 4 male (and 6 female) nurses?

Problem 58E Five cards are dealt from a standard 52-card deck. What is the probability that we draw a 3 aces and 2 kings? b a “full house” (3 cards of one kind, 2 cards of another kind)?

Problem 59E Five cards are dealt from a standard 52-card deck. What is the probability that we draw a 1 ace, 1 two, 1 three, 1 four, and 1 five (this is one way to get a “straight”)? b any straight?

Problem 61E Suppose that we ask n randomly selected people whether they share your birthday. a Give an expression for the probability that no one shares your birthday (ignore leap years). b How many people do we need to select so that the probability is at least .5 that at least one shares your birthday?

Problem 63E The eight-member Human Relations Advisory Board of Gainesville, Florida, considered the complaint of a woman who claimed discrimination, based on sex, on the part of a local company. The board, composed of five women and three men, voted5–3 in favor of the plaintiff, the five women voting in favor of the plaintiff, the three men against. The attorney representing the company appealed the board’s decision by claiming sex bias on the part of the board members. If there was no sex bias among the board members, it might be reasonable to conjecture that any group of five board members would be as likely to vote for the complainant as any other group of five. If this were the case, what is the probability that the vote would split along sex lines (five women for, three men against)?

Problem 60E Refer to Example 2.7. Suppose that we record the birthday for each of n randomly selected persons. a Give an expression for the probability that none share the same birthday. b What is the smallest value of n so that the probability is at least .5 that at least two people share a birthday? Reference Consider an experiment that consists of recording the birth day for each of 20randomly selected persons. Ignoring leap years and assuming that there are only 365 possible distinct birthdays, find the number of points in the sample space S for this experiment. If we assume that each of the possible sets of birthdays is equiprobable, what is the probability that each person in the 20 has a different birthday?

Problem 62E A manufacturer has nine distinct motors in stock, two of which came from a particular supplier. The motors must be divided among three production lines, with three motors going to each line. If the assignment of motors to lines is random, find the probability that both motors from the particular supplier are assigned to the first line.

Problem 64E A balanced die is tossed six times, and the number on the uppermost face is recorded each time. What is the probability that the numbers recorded are 1, 2, 3, 4, 5, and 6 in any order?

Problem 65E Refer to Exercise 2.64. Suppose that the die has been altered so that the faces are 1, 2, 3, 4, 5, and 5. If the die is tossed five times, what is the probability that the numbers recorded are 1, 2, 3, 4, and 5 in any order? Reference A balanced die is tossed six times, and the number on the uppermost face is recorded each time. What is the probability that the numbers recorded are 1, 2, 3, 4, 5, and 6 in any order?

Refer to Example 2.10. What is the probability that a an ethnic group member is assigned to each type of job? b no ethnic group member is assigned to a type 4 job? A labor dispute has arisen concerning the distribution of 20 laborers to four different construction jobs. The first job (considered to be very undesirable) required 6 laborers; the second, third, and fourth utilized 4, 5, and 5 laborers, respectively. The dispute arose over an alleged random distribution of the laborers to the jobs that placed all 4 members of a particular ethnic group on job 1. In considering whether the assignment represented injustice, a mediation panel desired the probability of the observed event. Determine the number of sample points in the sample space S for this experiment. That is, determine the number of ways the 20 laborers can be divided into groups of the appropriate sizes to fill all of the jobs. Find the probability of the observed event if it is assumed that the laborers are randomly assigned to jobs.

Prove that \(\left(\begin{array}{c} n+1 \\ k \end{array}\right)=\left(\begin{array}{l} n \\ k \end{array}\right)+\left(\begin{array}{c} n \\ k-1 \end{array}\right) \). Equation Transcription: Text Transcription: (_k^n+1)=(_k^n)+(_k-1^n)

Refer to Example 2.13. Suppose that the number of distributors is \(M=10\) and that there are \(n=7\) orders to be placed. What is the probability that a all of the orders go to different distributors? *b distributor I gets exactly two orders and distributor II gets exactly three orders? *c distributors I, II, and III get exactly two, three, and one order(s), respectively? Equation Transcription: Text Transcription: M=10 n=7

Consider the situation where items are to be partitioned into \(k<n\) distinct subsets. The multinomial coefficients \(\left(\begin{array}{c} n \\ n_{1} n_{2} \ldots n_{k} \end{array}\right) \) provide the number of distinct partitions where \(n_{1}\) items are in group 1, \(n_{2}\) are in group 2,..., \(n_{k}\) are in group . Prove that the total number of distinct partitions equals \(k^{n}\). [Hint Recall Exercise Equation Transcription: Text Transcription: k<n (n_1n_2...n_k^n) n_1 n_2 n_k k^n

Show that, for any integer \(n \geq 1\), a\( (n n =1)\). Interpret this result. b \( (n 0 =1)\). Interpret this result. c \((n r =n n-r)\). Interpret this result. d \(\sum_{i=0}^{n}(n i)=2^{n}\) . [Hint: Consider the binomial expansion of \((x+y)^{n}\) with \(x=y=1\).] Equation Transcription: Text Transcription: n geq1 (n n =1) (n 0 =1) (n r =n n-r) sum_i=0^n (n i) =2^n (x+y)^n x=y=1

Problem 71E If two events, A and B, are such that P( A) = .5, P( B) = .3, and P( A ? B) = .1, find the following: a P ( A|B) b P ( B| A) c P ( A| A ? B) d P ( A| A ? B) e P ( A ? B| A ? B)

For a certain population of employees, the percentage passing or failing a job competency exam, listed according to sex, were as shown in the accompanying table. That is, of all the people taking the exam, 24% were in the male-pass category, 16% were in the male-fail category, and so forth. An employee is to be selected randomly from this population. Let A be the event that the employee scores a passing grade on the exam and let M be the event that a male is selected. Sex Outcome Male (M) Female (F) Total Pass (A) 24 36 60 Fail \(\bar{A}\) 16 24 40 Total 40 60 100 a Are the events A and M independent? b Are the events \(\bar{A}\) and F independent? Equation Transcription: Text Transcription: A bar A bar

Problem 75E Cards are dealt, one at a time, from a standard 52-card deck. a If the first 2 cards are both spades, what is the probability that the next 3 cards are also spades? b If the first 3 cards are all spades, what is the probability that the next 2 cards are also spades? c If the first 4 cards are all spades, what is the probability that the next card is also a spade?

Gregor Mendel was a monk who, in 1865, suggested a theory of inheritance based on the science of genetics. He identified heterozygous individuals for flower color that had two alleles (one r = recessive white color allele and one R = dominant red color allele). When these individuals were mated, \(3 / 4\) of the offspring were observed to have red flowers, and \(1 / 4\) had white flowers. The following table summarizes this mating; each parent gives one of its alleles to form the gene of the offspring. We assume that each parent is equally likely to give either of the two alleles and that, if either one or two of the alleles in a pair is dominant (R), the offspring will have red flowers. What is the probability that an offspring has a at least one dominant allele? b at least one recessive allele? c one recessive allele, given that the offspring has red flowers? Parent 2 Parent 1 r R r rr rR R Rr RR Equation Transcription: Text Transcription: 3/4 1/4

One hundred adults were interviewed in a telephone survey. Of interest was their opinions regarding the loan burdens of college students and whether the respondent had a child currently in college. Their responses are summarized in the table below: Loan Burden Child in College Too High (A) About Right (B) Too Little (C) Total Yes (D) .20 .09 .01 .30 No (E) .41 .21 .08 .70 Total .61 .30 .09 1.00 Which of the following are independent events? a A and D b B and D c C and D

Problem 79E Suppose that A and B are mutually exclusive events, with P( A) > 0 and P( B) < 1. Are A and B independent? Prove your answer.

Problem 78E In the definition of the independence of two events, you were given three equalities to check: P( A|B) = P( A) or P( B| A) = P( B) or P( A? B) = P( A) P( B). If any one of these equalities holds, A and B are independent. Show that if any of these equalities hold, the other two also hold.

Problem 76E A survey of consumers in a particular community showed that 10% were dissatisfied with plumbing jobs done in their homes. Half the complaints dealt with plumber A, who does 40% of the plumbing jobs in the town. Find the probability that a consumer will obtain a an unsatisfactory plumbing job, given that the plumber was A. b a satisfactory plumbing job, given that the plumber was A.

A study of the posttreatment behavior of a large number of drug abusers suggests that the likelihood of conviction within a two-year period after treatment may depend upon the offenders education. The proportions of the total number of cases falling in four education–conviction categories are shown in the following table: Status within 2 Years after Treatment Education Convicted Not Convicted Total 10 years or more .10 .30 .40 9 years or less .27 .33 .60 Total .37 .63 1.00 Suppose that a single offender is selected from the treatment program. Define the events: A: The offender has 10 or more years of education. B: The offender is convicted within two years after completion of treatment. Find the following: a \(P(A)\). b \(P(B)\). c \(P(A \cap B)\). d \(P(A \cup B)\). e \(P(\bar{A})\). f \(P(\overline{A \cup B})\). g \(P(\overline{A \cap B})\). h \(P(A \mid B)\). i \(P(B \mid A)\). Equation Transcription: Text Transcription: P(A) P(B) P(A cap B) P(A U B) P(A bar) P(overline AUB) P(overline A cap B) P(A|B) P(B|A)

Problem 81E If P( A) > 0, P( B) > 0, and P( A) < P( A|B), show that P( B) < P( B| A).

Problem 80E Suppose that A ? B and that P( A) > 0 and P( B) > 0. Are A and B independent? Prove your answer.

Problem 82E Suppose that A ? B and that P( A) > 0 and P( B) > 0. Show that P( B| A) = 1 and P( A|B) = P( A)/P( B).

If and are mutually exclusive events and \(P(B)>0\), show that \(P(A \mid A \cup B)=\frac{P(A)}{P(A)+P(B)}\) Equation Transcription: Text Transcription: P(B)>0 P(A|AUB)=P(A) over P(A)+P(B)

If and are independent events, show that and \(\bar{B}\) are also independent. Are \(\bar{A}\) and \(\bar{B}\) independent? Equation Transcription: Text Transcription: B bar A bar B bar

Problem 84E If A1, A2, and A3 are three events and P( A1 ? A2) = P( A1 ? A3) 7= 0 but P( A2 ? A3) = 0, show that P (at least one Ai ) = P( A1) + P( A2) + P( A3) ? 2P( A1 ? A2).

Suppose that A and B are two events such that P( A) = .8 and P( B) = .7. a.) Is it possible that \(P(A \cap B)\) = .1? Why or why not? b.) What is the smallest possible value for \(P(A \cap B)\)? c.) Is it possible that \(P(A \cap B)\) = .77? Why or why not? d.) What is the largest possible value for \(P(A \cap B)\)?

Problem 88E Suppose that A and B are two events such that P( A) = .6 and P( B) = .3. a Is it possible that P( A ? B) = .1? Why or why not? b What is the smallest possible value for P( A ? B)? c Is it possible that P( A ? B) = .7? Why or why not? d What is the largest possible value for P( A ? B)?

Problem 87E Suppose that A and B are two events such that P( A) + P( B) > 1. a What is the smallest possible value for P( A ? B)? b What is the largest possible value for P( A ? B)?

Problem 89E Suppose that A and B are two events such that P( A) + P( B) < 1. a What is the smallest possible value for P( A ? B)? b What is the largest possible value for P( A ? B)?

Problem 90E Suppose that there is a 1 in 50 chance of injury on a single skydiving attempt. a If we assume that the outcomes of different jumps are independent, what is the probability that a skydiver is injured if she jumps twice? b A friend claims if there is a 1 in 50 chance of injury on a single jump then there is a 100% chance of injury if a skydiver jumps 50 times. Is your friend correct? Why?

Problem 92E A policy requiring all hospital employees to take lie detector tests may reduce losses due to theft, but some employees regard such tests as a violation of their rights. Past experience indicates that lie detectors have accuracy rates that vary from 92% to 99%.2 To gain some insight into the risks that employees face when taking a lie detector test, suppose that the probability is .05 that a lie detector concludes that a person is lying who, in fact, is telling the truth and suppose that any pair of tests are independent. What is the probability that a machine will conclude that a each of three employees is lying when all are telling the truth? b at least one of the three employees is lying when all are telling the truth?

Problem 91E Can A an B be mutually exclusive if P( A) = .4 and P( B) = .7? If P( A) = .4 and P( B) = .3? Why?

Problem 93E In a game, a participant is given three attempts to hit a ball. On each try, she either scores a hit, H, or a miss, M. The game requires that the player must alternate which hand she uses in successive attempts. That is, if she makes her first attempt with her right hand, she must use her left hand for the second attempt and her right hand for the third. Her chance of scoring a hit with her right hand is .7 and with her left hand is .4. Assume that the results of successive attempts are independent and that she wins the game if she scores at least two hits in a row. If she makes her first attempt with her right hand, what is the probability that she wins the game?

Problem 94E A smoke detector system uses two devices, A and B. If smoke is present, the probability that it will be detected by device A is .95; by device B, .90; and by both devices, .88. a If smoke is present, find the probability that the smoke will be detected by either device A or B or both devices. b Find the probability that the smoke will be undetected.

With relays operating as in Exercise 2.97, compare the probability of current flowing from a to b in the series system shown

If A and B are independent events with \(P(A)=.5\) and \(P(B)=.2\), find the following: a \(P(A \cup B)\) b \(P(\bar{A} \cap \bar{B})\) c \(P(\bar{A} \cup \bar{B})\) Equation Transcription: Text Transcription: P(A)=.5 P(B)=.2 P(AUB) P(A bar cap B bar) P(A bar U B bar)

Consider the following portion of an electric circuit with three relays. Current will flow from point a to point b if there is at least one closed path when the relays are activated. The relays may malfunction and not close when activated. Suppose that the relays act independently of one another and close properly when activated, with a probability of .9. a What is the probability that current will flow when the relays are activated? b Given that current flowed when the relays were activated, what is the probability that relay 1 functioned?

Suppose that and are independent events such that the probability that neither occurs is and the probability of is . Show that \(P(A)=\frac{1-b-a}{1-b}\). Equation Transcription: Text Transcription: P(A)=1-b-a over 1-b

In a game, a participant is given three attempts to hit a ball. On each try, she either scores a hit, H, or a miss, M. The game requires that the player must alternate which hand she uses in successive attempts. That is, if she makes her first attempt with her right hand, she must use her left hand for the second attempt and her right hand for the third. Her chance of scoring a hit with her right hand is .7 and with her left hand is .4. Assume that the results of successive attempts are independent and that she wins the game if she scores at least two hits in a row. If she makes her first attempt with her right hand, what is the probability that she wins the game?

Problem 101E Articles coming through an inspection line are visually inspected by two successive inspectors. When a defective article comes through the inspection line, the probability that it gets by the first inspector is .1. The second inspector will “miss” five out of ten of the defective items that get past the first inspector. What is the probability that a defective item gets by both inspectors?

Problem 102E Diseases I and II are prevalent among people in a certain population. It is assumed that 10%of the population will contract disease I sometime during their lifetime, 15% will contract disease II eventually, and 3%will contract both diseases. a Find the probability that a randomly chosen person from this population will contract at least one disease. b Find the conditional probability that a randomly chosen person from this population will contract both diseases, given that he or she has contracted at least one disease.

Show that Theorem , the additive law of probability, holds for conditional probabilities. That is, if , and are events such that \(P(C)>0\), prove that \(P(A \cup B \mid C)=P(A \mid C)+P(B \mid C)-P(A \cap B \mid C)\). Hint: Make use of the distributive law \((A \cup B) C=(A \cap C) \cup(B \cap C)\).] Equation Transcription: Text Transcription: P(C)>0 P(AUB|C)=P(A|C)+P(B|C)-P(A cap B|C) (AUB)C=(A cap C)U(B cap C)

Problem 103E Refer to Exercise 2.50. Hours after the rigging of the Pennsylvania state lottery was announced, Connecticut state lottery officials were stunned to learn that their winning number for the day was 666 (Los Angeles Times, September 21, 1980). a All evidence indicates that the Connecticut selection of 666 was due to pure chance. What is the probability that a 666 would be drawn in Connecticut, given that a 666 had been selected in the April 24, 1980, Pennsylvania lottery? b What is the probability of drawing a 666 in the April 24, 1980, Pennsylvania lottery (remember, this drawing was rigged) and a 666 in the September 19, 1980, Connecticut lottery? Reference Probability played a role in the rigging of the April 24, 1980, Pennsylvania state lottery (Los Angeles Times, September 8, 1980). To determine each digit of the three-digit winning number, each of the numbers 0, 1, 2, . . . , 9 is placed on a Ping-Pong ball, the ten balls are blown into a compartment, and the number selected for the digit is the one on the ball that floats to the top of the machine. To alter the odds, the conspirators injected a liquid into all balls used in the game except those numbered 4 and 6, making it almost certain that the lighter balls would be selected and determine the digits in the winning number. Then they bought lottery tickets bearing the potential winning numbers. How many potential winning numbers were there (666 was the eventual winner)?

If A and B are two events, prove that \(P(A \cap B) \geq 1-P(\bar{A})-P(\bar{B})\). [Note: This is a simplified version of the Bonferroni inequality.] Equation Transcription: Text Transcription: P(A cap B) >/=1-P(A bar)-P(B bar)

If the probability of injury on each individual parachute jump is .05, use the result in Exer- cise 2.104 to provide a lower bound for the probability of landing safely on both of two jumps.

Problem 106E If A and B are equally likely events and we require that the probability of their intersection be at least .98, what is P( A)?

Problem 107E Let A, B, and C be events such that P( A) > P( B) and P(C) > 0. Construct an example to demonstrate that it is possible that P( A|C) < P( B|C).

Problem 109E If A, B, and C are three equally likely events, what is the smallest value for P( A) such that P( A ? B ? C) always exceeds 0.95?

If A, B, and C are three events, use two applications of the result in Exercise 2.104 to prove that \(P(A \cap B \cap C) \geq 1-P(\bar{A})-P(\bar{B})-P(\bar{C})\). Equation Transcription: Text Transcription: P(A cap B cap C)>/=1-P(A bar)-P(B bar)-P(C bar)

Problem 111E An advertising agency notices that approximately 1 in 50 potential buyers of a product sees a given magazine ad, and 1 in 5 sees a corresponding ad on television. One in 100 sees both. One in 3 actually purchases the product after seeing the ad, 1 in 10 without seeing it. What is the probability that a randomly selected potential customer will purchase the product?

Problem 110E Of the items produced daily by a factory, 40% come from line I and 60% from line II. Line I has a defect rate of 8%, whereas line II has a defect rate of 10%. If an item is chosen at random from the day’s production, find the probability that it will not be defective.

Problem 112E Three radar sets, operating independently, are set to detect any aircraft flying through a certain area. Each set has a probability of .02 of failing to detect a plane in its area. If an aircraft enters the area, what is the probability that it a goes undetected? b is detected by all three radar sets?

Problem 114E A lie detector will show a positive reading (indicate a lie) 10% of the time when a person is telling the truth and 95% of the time when the person is lying. Suppose two people are suspects in a one-person crime and (for certain) one is guilty and will lie. Assume further that the lie detector operates independently for the truthful person and the liar. What is the probability that the detector a shows a positive reading for both suspects? b shows a positive reading for the guilty suspect and a negative reading for the innocent suspect? c is completely wrong—that is, that it gives a positive reading for the innocent suspect and a negative reading for the guilty? d gives a positive reading for either or both of the two suspects?

Problem 113E Consider one of the radar sets of Exercise 2.112. What is the probability that it will correctly detect exactly three aircraft before it fails to detect one, if aircraft arrivals are independent single events occurring at different times? Reference Three radar sets, operating independently, are set to detect any aircraft flying through a certain area. Each set has a probability of .02 of failing to detect a plane in its area. If an aircraft enters the area, what is the probability that it a goes undetected? b is detected by all three radar sets?

Problem 116E A communications network has a built-in safeguard system against failures. In this system if line I fails, it is bypassed and line II is used. If line II also fails, it is bypassed and line III is used. The probability of failure of any one of these three lines is .01, and the failures of these lines are independent events. What is the probability that this system of three lines does not completely fail?

Problem 115E A football team has a probability of .75 of winning when playing any of the other four teams in its conference. If the games are independent, what is the probability the team wins all its conference games?

Problem 117E A state auto-inspection station has two inspection teams. Team 1 is lenient and passes all automobiles of a recent vintage; team 2 rejects all autos on a first inspection because their “headlights are not properly adjusted.” Four unsuspecting drivers take their autos to the station for inspection on four different days and randomly select one of the two teams. a If all four cars are new and in excellent condition, what is the probability that three of the four will be rejected? b What is the probability that all four will pass?

Problem 118E An accident victim will die unless in the next 10 minutes he receives some type A, Rh-positive blood, which can be supplied by a single donor. The hospital requires 2 minutes to type a prospective donor’s blood and 2 minutes to complete the transfer of blood. Many untyped donors are available, and 40%of them have type A, Rh-positive blood. What is the probability that the accident victim will be saved if only one blood-typing kit is available? Assume that the typing kit is reusable but can process only one donor at a time.

Problem 119E Suppose that two balanced dice are tossed repeatedly and the sum of the two uppermost faces is determined on each toss. What is the probability that we obtain a a sum of 3 before we obtain a sum of 7? b a sum of 4 before we obtain a sum of 7?

Problem 120E Suppose that two defective refrigerators have been included in a shipment of six refrigerators. The buyer begins to test the six refrigerators one at a time. a What is the probability that the last defective refrigerator is found on the fourth test? b What is the probability that no more than four refrigerators need to be tested to locate both of the defective refrigerators? c When given that exactly one of the two defective refrigerators has been located in the first two tests, what is the probability that the remaining defective refrigerator is found in the third or fourth test?

Problem 121E A new secretary has been given n computer passwords, only one of which will permit access to a computer file. Because the secretary has no idea which password is correct, he chooses one of the passwords at random and tries it. If the password is incorrect, he discards it and randomly selects another password from among those remaining, proceeding in this manner until he finds the correct password. a What is the probability that he obtains the correct password on the first try? b What is the probability that he obtains the correct password on the second try? The third try? c A security system has been set up so that if three incorrect passwords are tried before the correct one, the computer file is locked and access to it denied. If n = 7, what is the probability that the secretary will gain access to the file?

Applet Exercise Use the applet Bayes’ Rule as a Tree to obtain the results given in Figure 2.13.

Applet Exercise Refer to Exercise 2.122 and Example 2.23. Suppose that lines 2 through 5 remained the same, but line 1 was partially repaired and produced a smaller percentage of defects. a What impact would this have on \(P(A \mid B)\)? b Suppose that \(P(A \mid B)\) decreased to and all other probabilities remained unchanged. Use the applet Bayes' Rule as a Tree to re-evaluate \(P(B \mid A)\). c How does the answer you obtained in part (b) compare to that obtained in Exercise 2.122? Are you surprised by this result? d Assume that all probabilities remain the same except \(P(A \mid B)\). Use the applet and trial and error to find the value of \(P(A \mid B)\) for which \(P(B \mid A)=.3000\). e If line 1 produces only defective items but all other probabilities remain unchanged, what is \(P(B \mid A)\)? f A friend expected the answer to part (e) to be 1. Explain why, under the conditions of part (e), \(P(B \mid A) \neq 1\). Equation Transcription: Text Transcription: P(A|B) P(A|B) P(B|A) P(A|B) P(A|B) P(B|A)=.3000 P(B|A) P(B|A)not = 1

Problem 124E A population of voters contains 40% Republicans and 60% Democrats. It is reported that 30% of the Republicans and 70% of the Democrats favor an election issue. A person chosen at random from this population is found to favor the issue in question. Find the conditional probability that this person is a Democrat.

Problem 125E A diagnostic test for a disease is such that it (correctly) detects the disease in 90% of the individuals who actually have the disease. Also, if a person does not have the disease, the test will report that he or she does not have it with probability .9. Only 1% of the population has the disease in question. If a person is chosen at random from the population and the diagnostic test indicates that she has the disease, what is the conditional probability that she does, in fact, have the disease? Are you surprised by the answer? Would you call this diagnostic test reliable?

Problem 126E Applet Exercise Refer to Exercise 2.125. The probability that the test detects the disease given that the patient has the disease is called the sensitivity of the test. The specificity of the test is the probability that the test indicates no disease given that the patient is disease free. The positive predictive value of the test is the probability that the patient has the disease given that the test indicates that the disease is present. In Exercise 2.125, the disease in question was relatively rare, occurring with probability .01, and the test described has sensitivity = specificity = .90 and positive predictive value = .0833. a In an effort to increase the positive predictive value of the test, the sensitivity was increased to .95 and the specificity remained at .90, what is the positive predictive value of the “improved” test? b Still not satisfied with the positive predictive value of the procedure, the sensitivity of the test is increased to .999. What is the positive predictive value of the (now twice) modified test if the specificity stays at .90? c Look carefully at the various numbers that were used to compute the positive predictive value of the tests. Why are all of the positive predictive values so small? [Hint: Compare the size of the numerator and the denominator used in the fraction that yields the value of the positive predictive value. Why is the denominator so (relatively) large?] d The proportion of individuals with the disease is not subject to our control. If the sensitivity of the test is .90, is it possible that the positive predictive value of the test can be increased to a value above .5? How? [Hint: Consider improving the specificity of the test.] e Based on the results of your calculations in the previous parts, if the disease in question is relatively rare, how can the positive predictive value of a diagnostic test be significantly increased? Reference A diagnostic test for a disease is such that it (correctly) detects the disease in 90% of the individuals who actually have the disease. Also, if a person does not have the disease, the test will report that he or she does not have it with probability .9. Only 1% of the population has the disease in question. If a person is chosen at random from the population and the diagnostic test indicates that she has the disease, what is the conditional probability that she does, in fact, have the disease? Are you surprised by the answer? Would you call this diagnostic test reliable?

Problem 129E Males and females are observed to react differently to a given set of circumstances. It has been observed that 70% of the females react positively to these circumstances, whereas only 40% of males react positively. A group of 20 people, 15 female and 5 male, was subjected to these circumstances, and the subjects were asked to describe their reactions on a written questionnaire. A response picked at random from the 20 was negative. What is the probability that it was that of a male?

A plane is missing and is presumed to have equal probability of going down in any of three regions. If a plane is actually down in region i, let \(1-\alpha_{i}\) denote the probability that the plane will be found upon a search of the ith region, \(i=1\), 2, 3. What is the conditional probability that the plane is in a region 1, given that the search of region 1 was unsuccessful? b region 2, given that the search of region 1 was unsuccessful? c region 3, given that the search of region 1 was unsuccessful? Equation Transcription: Text Transcription: 1-alpha_i i=1,2,3

Problem 130E A study of Georgia residents suggests that those who worked in shipyards during World War II were subjected to a significantly higher risk of lung cancer (Wall Street Journal, September 21, 1978).3 It was found that approximately 22% of those persons who had lung cancer worked at some prior time in a shipyard. In contrast, only 14% of those who had no lung cancer worked at some prior time in a shipyard. Suppose that the proportion of all Georgians living during World War II who have or will have contracted lung cancer is .04%. Find the percentage of Georgians living during the same period who will contract (or have contracted) lung cancer, given that they have at some prior time worked in a shipyard.

Problem 127E Applet Exercise Refer to Exercises 2.125 and 2.126. Suppose now that the disease is not particularly rare and occurs with probability .4 . a If, as in Exercise 2.125, a test has sensitivity = specificity = .90, what is the positive predictive value of the test? b Why is the value of the positive predictive value of the test so much higher that the value obtained in Exercise 2.125? [Hint: Compare the size of the numerator and the denominator used in the fraction that yields the value of the positive predictive value.] c If the specificity of the test remains .90, can the sensitivity of the test be adjusted to obtain a positive predictive value above .87? d If the sensitivity remains at .90, can the specificity be adjusted to obtain a positive predictive value above .95? How? e The developers of a diagnostic test want the test to have a high positive predictive value. Based on your calculations in previous parts of this problem and in Exercise 2.126, is the value of the specificity more or less critical when developing a test for a rarer disease? Reference Applet Exercise Refer to Exercise 2.125. The probability that the test detects the disease given that the patient has the disease is called the sensitivity of the test. The specificity of the test is the probability that the test indicates no disease given that the patient is disease free. The positive predictive value of the test is the probability that the patient has the disease given that the test indicates that the disease is present. In Exercise 2.125, the disease in question was relatively rare, occurring with probability .01, and the test described has sensitivity = specificity = .90 and positive predictive value = .0833. a In an effort to increase the positive predictive value of the test, the sensitivity was increased to .95 and the specificity remained at .90, what is the positive predictive value of the “improved” test? b Still not satisfied with the positive predictive value of the procedure, the sensitivity of the test is increased to .999. What is the positive predictive value of the (now twice) modified test if the specificity stays at .90? c Look carefully at the various numbers that were used to compute the positive predictive value of the tests. Why are all of the positive predictive values so small? [Hint: Compare the size of the numerator and the denominator used in the fraction that yields the value of the positive predictive value. Why is the denominator so (relatively) large?] d The proportion of individuals with the disease is not subject to our control. If the sensitivity of the test is .90, is it possible that the positive predictive value of the test can be increased to a value above .5? How? [Hint: Consider improving the specificity of the test.] e Based on the results of your calculations in the previous parts, if the disease in question is relatively rare, how can the positive predictive value of a diagnostic test be significantly increased? Reference A diagnostic test for a disease is such that it (correctly) detects the disease in 90% of the individuals who actually have the disease. Also, if a person does not have the disease, the test will report that he or she does not have it with probability .9. Only 1% of the population has the disease in question. If a person is chosen at random from the population and the diagnostic test indicates that she has the disease, what is the conditional probability that she does, in fact, have the disease? Are you surprised by the answer? Would you call this diagnostic test reliable?

The symmetric difference between two events and is the set of all sample points that are in exactly one of the sets and is often denoted A \(\Delta\) B. Note that A \(\Delta\) \(B=(A \cap \bar{B}) \cup(\bar{A} \cap B)\). Prove that \(P(A\) \(\Delta\) \(B)=P(A)+P(B)-2 P(A \cap B)\). Equation Transcription: Text Transcription: Delta Delta B=(A cap B bar)U(A bar cap B) P(A Delta B)=P(A)+P(B)-2P(A cap B)

Problem 132E A plane is missing and is presumed to have equal probability of going down in any of three regions. If a plane is actually down in region i, let 1 ? ?i denote the probability that the plane will be found upon a search of the ith region, i = 1, 2, 3. What is the conditional probability that the plane is in a region 1, given that the search of region 1 was unsuccessful? b region 2, given that the search of region 1 was unsuccessful? c region 3, given that the search of region 1 was unsuccessful?

Problem 133E A student answers a multiple-choice examination question that offers four possible answers. Suppose the probability that the student knows the answer to the question is .8 and the probability that the student will guess is .2. Assume that if the student guesses, the probability of selecting the correct answer is .25. If the student correctly answers a question, what is the probability that the student really knew the correct answer?

Problem 134E Two methods, A and B, are available for teaching a certain industrial skill. The failure rate is 20% for A and 10% for B. However, B is more expensive and hence is used only 30% of the time. ( A is used the other 70%.) A worker was taught the skill by one of the methods but failed to learn it correctly. What is the probability that she was taught by method A?

Of the travelers arriving at a small airport, 60% fly on major airlines, 30% fly on privately owned planes, and the remainder fly on commercially owned planes not belonging to a major airline. Of those traveling on major airlines, 50% are traveling for business reasons, whereas 60% of those arriving on private planes and 90% of those arriving on other commercially owned planes are traveling for business reasons. Suppose that we randomly select one person arriving at this airport. What is the probability that the person a is traveling on business? b is traveling for business on a privately owned plane? c arrived on a privately owned plane, given that the person is traveling for business reasons? d is traveling on business, given that the person is flying on a commercially owned plane?

Problem 136E A personnel director has two lists of applicants for jobs. List 1 contains the names of five women and two men, whereas list 2 contains the names of two women and six men. A name is randomly selected from list 1 and added to list 2. A name is then randomly selected from the augmented list 2. Given that the name selected is that of a man, what is the probability that a woman’s name was originally selected from list 1?

Problem 138E Following is a description of the game of craps. A player rolls two dice and computes the total of the spots showing. If the player’s first toss is a 7 or an 11, the player wins the game. If the first toss is a 2, 3, or 12, the player loses the game. If the player rolls anything else (4, 5, 6, 8, 9 or 10) on the first toss, that value becomes the player’s point. If the player does not win or lose on the first toss, he tosses the dice repeatedly until he obtains either his point or a 7. He wins if he tosses his point before tossing a 7 and loses if he tosses a 7 before his point. What is the probability that the player wins a game of craps? [Hint: Recall Exercise 2.119.] Reference Suppose that two balanced dice are tossed repeatedly and the sum of the two uppermost faces is determined on each toss. What is the probability that we obtain a a sum of 3 before we obtain a sum of 7? b a sum of 4 before we obtain a sum of 7?

Problem 137E Five identical bowls are labeled 1, 2, 3, 4, and 5. Bowl i contains i white and 5 ? i black balls, with i = 1, 2, . . . , 5. A bowl is randomly selected and two balls are randomly selected (without replacement) from the contents of the bowl. a What is the probability that both balls selected are white? b Given that both balls selected are white, what is the probability that bowl 3 was selected?

Problem 139E Refer to Exercise 2.112. Let the random variable Y represent the number of radar sets that detect a particular aircraft. Compute the probabilities associated with each value of Y . Reference Three radar sets, operating independently, are set to detect any aircraft flying through a certain area. Each set has a probability of .02 of failing to detect a plane in its area. If an aircraft enters the area, what is the probability that it a goes undetected? b is detected by all three radar sets?

Problem 140E Refer to Exercise 2.120. Let the random variable Y represent the number of defective refrigerators found after three refrigerators have been tested. Compute the probabilities for each value of Y . Reference Suppose that two defective refrigerators have been included in a shipment of six refrigerators. The buyer begins to test the six refrigerators one at a time. a What is the probability that the last defective refrigerator is found on the fourth test? b What is the probability that no more than four refrigerators need to be tested to locate both of the defective refrigerators? c When given that exactly one of the two defective refrigerators has been located in the first two tests, what is the probability that the remaining defective refrigerator is found in the third or fourth test?

Problem 141E Refer again to Exercise 2.120. Let the random variable Y represent the number of the test in which the last defective refrigerator is identified. Compute the probabilities for each value of Y . Reference Suppose that two defective refrigerators have been included in a shipment of six refrigerators. The buyer begins to test the six refrigerators one at a time. a What is the probability that the last defective refrigerator is found on the fourth test? b What is the probability that no more than four refrigerators need to be tested to locate both of the defective refrigerators? c When given that exactly one of the two defective refrigerators has been located in the first two tests, what is the probability that the remaining defective refrigerator is found in the third or fourth test?

A spinner can land in any of four positions, A, B, C, and D, with equal probability. The spinner is used twice, and the position is noted each time. Let the random variable Y denote the number of positions on which the spinner did not land. Compute the probabilities for each value of Y .

Show that Theorem holds for conditional probabilities. That is, if \(P(B)>0\), then \(P(A \mid B)=1-P(\bar{A} \mid B)\). Equation Transcription: Text Transcription: P(B)>0 P(A|B)=1-P(A bar|B)

Let contain four sample points, \(E_{1}, E_{2}, E_{3}\), and \(E_{4}\). a List all possible events in \(S\) (include the null event). b In Exercise \(2.68( d)\), you showed that \(\sum_{i=1}^{n}(n i)=2^{n}\) . Use this result to give the total number of events in \(S\). c Let and be the events \(E_{4}\) and \(\left\{E_{2}, E_{4}\right\}\),respectively. Give the sample points in the following events:\(A \cup B, A \cap B, A^{-} \cap B^{-}\), and \(A^{-} \cup B\). Equation Transcription: Text Transcription: E_1,E_2,E_3 E_4 S 2.68( d) sum_i=1^n( n i) =2^n {E_1,E_2,E_3} {E_2,E_4} A cup B,A cup B,A^- cup B^- A^- cup B

Problem 146SE Five cards are drawn from a standard 52-card playing deck. What is the probability that all 5 cards will be of the same suit?

Problem 147SE Refer to Exercise 2.146. A gambler has been dealt five cards: two aces, one king, one five, and one 9. He discards the 5 and the 9 and is dealt two more cards. What is the probability that he ends up with a full house? Reference Five cards are drawn from a standard 52-card playing deck. What is the probability that all 5 cards will be of the same suit?

Problem 148SE A bin contains three components from supplier A, four from supplier B, and five from supplier C. If four of the components are randomly selected for testing, what is the probability that each supplier would have at least one component tested?

Problem 145SE A patient receiving a yearly physical examination must have 18 checks or tests performed. The sequence in which the tests are conducted is important because the time lost between tests will vary depending on the sequence. If an efficiency expert were to study the sequences to find the one that required the minimum length of time, how many sequences would be included in her study if all possible sequences were admissible?

Problem 149SE A large group of people is to be checked for two common symptoms of a certain disease. It is thought that 20% of the people possess symptom A alone, 30% possess symptom B alone, 10%possess both symptoms, and the remainder have neither symptom. For one person chosen at random from this group, find these probabilities: a The person has neither symptom. b The person has at least one symptom. c The person has both symptoms, given that he has symptom B.

Problem 151SE A Model for the World Series Two teams A and B play a series of games until one team wins four games. We assume that the games are played independently and that the probability that A wins any game is p. What is the probability that the series lasts exactly five games?

Problem 150SE Refer to Exercise 2.149. Let the random variable Y represent the number of symptoms possessed by a person chosen at random from the group. Compute the probabilities associated with each value of Y . Reference A large group of people is to be checked for two common symptoms of a certain disease. It is thought that 20% of the people possess symptom A alone, 30% possess symptom B alone, 10%possess both symptoms, and the remainder have neither symptom. For one person chosen at random from this group, find these probabilities: a The person has neither symptom. b The person has at least one symptom. c The person has both symptoms, given that he has symptom B.

Problem 152SE We know the following about a color metric method used to test lake water for nitrates. If water specimens contain nitrates, a solution dropped into the water will cause the specimen to turn red 95% of the time. When used on water specimens without nitrates, the solution causes the water to turn red 10% of the time (because chemicals other than nitrates are sometimes present and they also react to the agent). Past experience in a lab indicates that nitrates are contained in 30%of the water specimens that are sent to the lab for testing. If a water specimen is randomly selected a from among those sent to the lab, what is the probability that it will turn red when tested? b and turns red when tested, what is the probability that it actually contains nitrates?

Medical case histories indicate that different illnesses may produce identical symptoms. Suppose that a particular set of symptoms, denoted , occurs only when any one of three illnesses, \(I_{1}\), \(I_{2}\), or \(I_{3}\), occurs. Assume that the simultaneous occurrence of more that one of these illnesses is impossible and that \(P\left(I_{1}\right)=.01\), \(P\left(I_{2}\right)=.005\), \(P\left(I_{3}\right)=.02\). The probabilities of developing the set of symptoms , given each of these illnesses, are known to be \(P\left(H \mid I_{1}\right)=.90\), \(P\left(H \mid I_{2}\right)=.95\), \(P\left(H \mid I_{3}\right)=.75\). Assuming that an ill person exhibits the symptoms, , what is the probability that the person has illness \(I_{1}\)? Equation Transcription: Text Transcription: I_1 I_2 I_3 P(I_1)=.01 P(I_2)=.005 P(I_3)=.02 P(H|I_1)=.90 P(H|I_2)=.95 P(H|I_3)=.75 I_1

A group of men possesses the three characteristics of being married (A), having a college degree (B), and being a citizen of a specified state (C), according to the fractions given in the accompanying Venn diagram. That is, 5% of the men possess all three characteristics, whereas 20% have a college education but are not married and are not citizens of the specified state. One man is chosen at random from this group. Find the probability that he a is married. b has a college degree and is married. c is not from the specified state but is married and has a college degree. d is not married or does not have a college degree, given that he is from the specified state.

Problem 154SE a A drawer contains n = 5 different and distinguishable pairs of socks (a total of ten socks). If a person (perhaps in the dark) randomly selects four socks, what is the probability that there is no matching pair in the sample? *b A drawer contains n different and distinguishable pairs of socks (a total of 2n socks). A person randomly selects 2r of the socks, where 2r < n. In terms of n and r, what is the probability that there is no matching pair in the sample?

Problem 157SE A study of the residents of a region showed that 20%were smokers. The probability of death due to lung cancer, given that a person smoked, was ten times the probability of death due to lung cancer, given that the person did not smoke. If the probability of death due to lung cancer in the region is .006, what is the probability of death due to lung cancer given that the person is a smoker?

A bowl contains w white balls and b black balls. One ball is selected at random from the bowl, its color is noted, and it is returned to the bowl along with n additional balls of the same color. Another single ball is randomly selected from the bowl (now containing \(w+b+n\) balls) and it is observed that the ball is black. Show that the (conditional) probability that the first ball selected was white is \(\frac{w}{w+b+n}\). Equation Transcription: Text Transcription: w+b+n w over w+b+n

The accompanying table lists accidental deaths by age and certain specific types for the United States in 2002. a A randomly selected person from the United States was known to have an accidental death in 2002. Find the probability that i he was over the age of 15 years. ii the cause of death was a motor vehicle accident. iii the cause of death was a motor vehicle accident, given that the person was between 15 and 24 years old. iv the cause of death was a drowning accident, given that it was not a motor vehicle accident and the person was 34 years old or younger. b From these figures can you determine the probability that a person selected at random from the U.S. population had a fatal motor vehicle accident in 2002? Type of Accident Age All Types Motor Vehicle Falls Drowning Under 5 2,707 819 44 568 5-14 2,979 1,772 37 375 15-24 14,113 10,560 237 646 25-34 11,769 6,884 303 419 35-44 15,413 6,927 608 480 45-54 12,278 5,361 871 354 55-64 7,505 3,506 949 217 65-74 7,698 3,038 1,660 179 75 and over 23,438 4,487 8,613 244 Total 97,900 43,354 13,322 3,482

Problem 159SE It seems obvious that P(?) = 0. Show that this result follows from the axioms in Definition 2.6.

Problem 162SE Assume that there are nine parking spaces next to one another in a parking lot. Nine cars need to be parked by an attendant. Three of the cars are expensive sports cars, three are large domestic cars, and three are imported compacts. Assuming that the attendant parks the cars at random, what is the probability that the three expensive sports cars are parked adjacent to one another?

Relays used in the construction of electric circuits function properly with probability .9. Assuming that the circuits operate independently, which of the following circuit designs yields the higher probability that current will flow when the relays are activated?

Problem 160SE A machine for producing a new experimental electronic component generates defectives from time to time in a random manner. The supervising engineer for a particular machine has noticed that defectives seem to be grouping (hence appearing in a nonrandom manner), thereby suggesting a malfunction in some part of the machine. One test for nonrandomness is based on the number of runs of defectives and nondefectives (a run is an unbroken sequence of either defectives or nondefectives). The smaller the number of runs, the greater will be the amount of evidence indicating nonrandomness. Of 12 components drawn from the machine, the first 10were not defective, and the last 2 were defective (NNNNNNNNNNDD). Assume randomness. What is the probability of observing a this arrangement (resulting in two runs) given that 10 of the 12 components are not defective? b two runs?

Refer to Exercise 2.163 and consider circuit A. If we know that current is flowing, what is the probability that switches 1 and 4 are functioning properly?

Refer to Exercise 2.163 and consider circuit B. If we know that current is flowing, what is the probability that switches 1 and 4 are functioning properly?

Problem 161SE Refer to Exercise 2.160. What is the probability that the number of runs, R, is less than or equal to 3? Reference A machine for producing a new experimental electronic component generates defectives from time to time in a random manner. The supervising engineer for a particular machine has noticed that defectives seem to be grouping (hence appearing in a nonrandom manner), thereby suggesting a malfunction in some part of the machine. One test for nonrandomness is based on the number of runs of defectives and nondefectives (a run is an unbroken sequence of either defectives or nondefectives). The smaller the number of runs, the greater will be the amount of evidence indicating nonrandomness. Of 12 components drawn from the machine, the first 10were not defective, and the last 2 were defective (NNNNNNNNNNDD). Assume randomness. What is the probability of observing a this arrangement (resulting in two runs) given that 10 of the 12 components are not defective? b two runs?

Problem 166SE Eight tires of different brands are ranked from 1 to 8 (best to worst) according to mileage performance. If four of these tires are chosen at random by a customer, find the probability that the best tire among those selected by the customer is actually ranked third among the original eight.

Problem 167SE Refer to Exercise 2.166. Let Y denote the actual quality rank of the best tire selected by the customer. In Exercise 2.166, you computed P(Y = 3). Give the possible values of Y and the probabilities associated with all of these values. Reference Eight tires of different brands are ranked from 1 to 8 (best to worst) according to mileage performance. If four of these tires are chosen at random by a customer, find the probability that the best tire among those selected by the customer is actually ranked third among the original eight.

Problem 170SE Three names are to be selected from a list of seven names for a public opinion survey. Find the probability that the first name on the list is selected for the survey.

Problem 169SE Three beer drinkers (say I, II, and III) are to rank four different brands of beer (say A, B, C, and D) in a blindfold test. Each drinker ranks the four beers as 1 (for the beer that he or she liked best), 2 (for the next best), 3, or 4. a Carefully describe a sample space for this experiment (note that we need to specify the ranking of all four beers for all three drinkers). How many sample points are in this sample space? b Assume that the drinkers cannot discriminate between the beers so that each assignment of ranks to the beers is equally likely. After all the beers are ranked by all three drinkers, the ranks of each brand of beer are summed. What is the probability that some beer will receive a total rank of 4 or less?

Let A and B be any two events. Which of the following statements, in general, are false? a \(P(A \mid B)+P(\bar{A} \mid \bar{B})=1\). b \(P(A \mid B)+P(A \mid \bar{B})=1\). c \(P(A \mid B)+P(\bar{A} \mid B)=1\). Equation Transcription: Text Transcription: P(A|B)+P(A bar|B bar)=1 P(A|B)+P(A|B bar)=1 P(A|B)+P(A bar|B)=1

Problem 173SE As items come to the end of a production line, an inspector chooses which items are to go through a complete inspection. Ten percent of all items produced are defective. Sixty percent of all defective items go through a complete inspection, and 20% of all good items go through a complete inspection. Given that an item is completely inspected, what is the probability it is defective?

Problem 168SE As in Exercises 2.166 and 2.167, eight tires of different brands are ranked from 1 to 8 (best to worst) according to mileage performance. a If four of these tires are chosen at random by a customer, what is the probability that the best tire selected is ranked 3 and the worst is ranked 7? b In part (a) you computed the probability that the best tire selected is ranked 3 and the worst is ranked 7. If that is the case, the range of the ranks, R = largest rank ? smallest rank = 7 ? 3 = 4. What is P( R = 4)? c Give all possible values for R and the probabilities associated with all of these values. Reference Refer to Exercise 2.166. Let Y denote the actual quality rank of the best tire selected by the customer. In Exercise 2.166, you computed P(Y = 3). Give the possible values of Y and the probabilities associated with all of these values. Reference Eight tires of different brands are ranked from 1 to 8 (best to worst) according to mileage performance. If four of these tires are chosen at random by a customer, find the probability that the best tire among those selected by the customer is actually ranked third among the original eight.

Problem 174SE Many public schools are implementing a “no-pass, no-play” rule for athletes. Under this system, a student who fails a course is disqualified from participating in extracurricular activities during the next grading period. Suppose that the probability is .15 that an athlete who has not previously been disqualified will be disqualified next term. For athletes who have been previously disqualified, the probability of disqualification next term is .5. If 30%of the athletes have been disqualified in previous terms, what is the probability that a randomly selected athlete will be disqualified during the next grading period?

Problem 171SE An AP news service story, printed in the Gainesville Sun on May 20, 1979, states the following with regard to debris from Skylab striking someone on the ground: “The odds are 1 in 150 that a piece of Skylab will hit someone. But 4 billion people . . . live in the zone in which pieces could fall. So any one person’s chances of being struck are one in 150 times 4 billion—or one in 600 billion.” Do you see any inaccuracies in this reasoning?

Problem 175SE Three events, A, B, and C, are said to be mutually independent if P ( A ? B) = P( A) × P( B), P( B ? C) = P( B) × P(C), P ( A ? C) = P( A) × P(C), P( A ? B ? C) = P( A) × P( B) × P(C). Suppose that a balanced coin is independently tossed two times. Define the following events: A: Head appears on the first toss. B: Head appears on the second toss. C: Both tosses yield the same outcome. Are A, B, and C mutually independent?

Problem 178SE Suppose that the probability of exposure to the flu during an epidemic is .6. Experience has shown that a serum is 80% successful in preventing an inoculated person from acquiring the flu, if exposed to it. A person not inoculated faces a probability of .90 of acquiring the flu if exposed to it. Two persons, one inoculated and one not, perform a highly specialized task in a business. Assume that they are not at the same location, are not in contact with the same people, and cannot expose each other to the flu. What is the probability that at least one will get the flu?

Problem 176SE Refer to Exercise 2.175 and suppose that events A, B, and C are mutually independent. a Show that ( A ? B) and C are independent. b Show that A and ( B ? C) are independent. Reference Three events, A, B, and C, are said to be mutually independent if P ( A ? B) = P( A) × P( B), P( B ? C) = P( B) × P(C), P ( A ? C) = P( A) × P(C), P( A ? B ? C) = P( A) × P( B) × P(C). Suppose that a balanced coin is independently tossed two times. Define the following events: A: Head appears on the first toss. B: Head appears on the second toss. C: Both tosses yield the same outcome. Are A, B, and C mutually independent?

Problem 179SE Two gamblers bet $1 each on the successive tosses of a coin. Each has a bank of $6. What is the probability that a they break even after six tosses of the coin? b one player—say, Jones—wins all the money on the tenth toss of the coin?

Problem 177SE Refer to Exercise 2.90(b) where a friend claimed that if there is a 1 in 50 chance of injury on a single jump then there is a 100% chance of injury if a skydiver jumps 50 times. Assume that the results of repeated jumps are mutually independent. a What is the probability that 50 jumps will be completed without an injury? b What is the probability that at least one injury will occur in 50 jumps? c What is the maximum number of jumps, n, the skydiver can make if the probability is at least .60 that all n jumps will be completed without injury? Reference Suppose that there is a 1 in 50 chance of injury on a single skydiving attempt. a If we assume that the outcomes of different jumps are independent, what is the probability that a skydiver is injured if she jumps twice? b A friend claims if there is a 1 in 50 chance of injury on a single jump then there is a 100% chance of injury if a skydiver jumps 50 times. Is your friend correct? Why?

Problem 180SE Suppose that the streets of a city are laid out in a grid with streets running north–south and east–west. Consider the following scheme for patrolling an area of 16 blocks by 16 blocks. An officer commences walking at the intersection in the center of the area. At the corner of each block the officer randomly elects to go north, south, east, or west. What is the probability that the officer will a reach the boundary of the patrol area after walking the first 8 blocks? b return to the starting point after walking exactly 4 blocks?