1. 3.104 Which of the following pairs of events are mutually exclusive?

   1. $A=$ $\{$The Tampa Bay Rays win the World Series next year. $\}$

      $B=$ $\{$Evan Longoria, Rays infielder, hits 75 home runs runs next year. $\}$

   2. $A=$ $\{$Psychiatric patient Tony responds to a stimulus within 5 seconds. $\}$

      $B=$ $\{$Psychiatric patient Tony has the fastest stimulus response time of 2.3 seconds. $\}$

   3. $A=$ $\{$High school graduate Cindy enrolls at the University of South Florida next year. $\}$

      $B=$ $\{$High school graduate Cindy does not enroll in college next year. $\}$

2. 3.105 Use the symbols $\cap ,\cup ,|,$ and ${}^c$ to convert the following statements into compound events involving events A and B, where $A=\{$You purchase a notebook computer $\}$ and $B=\{$You vacation in Europe $\}$:

   1. You purchase a notebook computer or vacation in Europe.

   2. You will not vacation in Europe.

   3. You purchase a notebook computer and vacation in Europe.

   4. Given that you vacation in Europe, you will not purchase a notebook computer.

1. 3.106 A sample space consists of four sample points $S_1,S_2,S_3,$ and $S_4,$ where $P(S_1)=.2,P(S_2)=.1,P(S_3)=.3,$ and $P(S_4)=.4.$

   1. Show that the sample points obey the two probability rules for a sample space.

   2. If an event $A=\{S_1,S_4\},$ find P(A).

2. 3.107 A and B are mutually exclusive events, with $P(A)=.2$ and $P(B)=.3.$

   1. Find $P(A|B)$.

   2. Are A and B independent events?

3. 3.108 For two events A and B, suppose $P(A)=.7,P(B)=.5,$ and $P(A\cap B)=.4.$ Find $P(A\cup B).$

4. 3.109 Given that $P(A\cap B)=.4$ and $P(A|B)=.8,$ find P(B).

5. 3.110 The Venn diagram on the next page illustrates a sample space containing six sample points and three events, A, B, and C. The probabilities of the sample points are

   $P(1)=.3,P(2)=.2,P(3)=.1,P(4)=.1,P(5)=.1,$ and $P(6)=.2.$

   1. Find $P(A\cap B),$ $P(B\cap C),$ $P(A\cup C),$ $P(A\cup B\cup C),$ $P(B^c),$ $P(A^c\cap B),$ $P(B|C),$ and $P(B|A).$

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

   3. Are B and C independent? Mutually exclusive? Why?

      Venn Diagram For Exercise 3.110

      

6. 3.111 A fair die is tossed, and the up face is noted. If the number is even, the die is tossed again; if the number is odd, a fair coin is tossed. Consider the following events:

   [&A: |cbo|~rom~A head appears on the coin~normal~.|cbc| &]

   [&B: |cbo|~rom~The die is tossed only one time~normal~.|cbc| &]

   $$\begin{array}{l}A:\left\{\text{A head appears on the coin}\text{.}\right\}\hfill \\ B:\left\{\text{The die is tossed only one time}\text{.}\right\}\hfill \end{array}$$

   1. List the sample points in the sample space.

   2. Give the probability for each of the sample points.

   3. Find P(A) and P(B).

   4. Identify the sample points in $A^c,B^c,A\cap B,$ and $A\cup B.$

   5. Find $P(A^c),P(B^c),P(A\cap B),P(A\cup B),P(A|B),$ and $P(B|A)$.

   6. Are A and B mutually exclusive events? Independent events? Why?

7. 3.112 A balanced die is thrown once. If a 4 appears, a ball is drawn from urn 1; otherwise, a ball is drawn from urn 2. Urn 1 contains four red, three white, and three black balls. Urn 2 contains six red and four white balls.

   1. Find the probability that a red ball is drawn.

   2. Find the probability that urn 1 was used given that a red ball was drawn.

8. 3.113 Two events, A and B, are independent, with $P(A)=.3$ and $P(B)=.1.$

   1. Are A and B mutually exclusive? Why?

   2. Find $P(A|B)$ and $P(B|A)$.

   3. Find $P(A\cup B).$

9. 3.114 Find the numerical value of

   1. $6!$

   2. $\left(\begin{array}{c}10\\ 9\end{array}\right)$

   3. $\left(\begin{array}{c}10\\ 1\end{array}\right)$

   4. $\left(\begin{array}{c}6\\ 3\end{array}\right)$

   5. $0!$

Use the applet entitled Random Numbers to generate a list of 50 numbers between 1 and 100, inclusive. Use the list to find each of the following probabilities:

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

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

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

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

   5. Do your results from parts a–d support the conclusion that the events less than or equal to 50 and even are independent? Explain.

1. 3.115 Going online for health information. A cyberchondriac is defined as a person who regularly searches the Web for health care information. A Harris Poll surveyed 1,010 U.S. adults by telephone and asked each respondent how often (in the past month) he or she looked for health care information online. The results are summarized in the following table. Consider the response category of a randomly selected person who participated in the Harris poll.

   Response (# per month)	Percentage of Respondents
   None	25
   1 or 2	31
   3–5	25
   6–9	5
   10 or more	14
   Total	100%

   1. List the sample points for the experiment.

   2. Assign reasonable probabilities to the sample points.

   3. Find the probability that the respondent looks for health care information online more than two times per month.

2. 3.116 Study of ancient pottery. Refer to the Chance (Fall 2000) study of ancient pottery found at the Greek settlement of Phylakopi, presented in Exercise 2.186 (p. 106). Of the 837 pottery pieces uncovered at the excavation site, 183 were painted. These painted pieces included 14 painted in a curvilinear decoration, 165 painted in a geometric decoration, and 4 painted in a naturalistic decoration. Suppose 1 of the 837 pottery pieces is selected and examined.

   1. What is the probability that the pottery piece is painted?

   2. Given that the pottery piece is painted, what is the probability that it is painted in a curvilinear decoration?

3. 3.117 Post office violence. The Wall Street Journal (Sept. 1, 2000) reported on an independent study of postal workers and violence at post offices. In a sample of 12,000 postal workers, 600 were physically assaulted on the job in a recent year. Use this information to estimate the probability that a randomly selected postal worker will be physically assaulted on the job during the year.

4. 3.118 Sterile couples in Jordan. A sterile family is a couple that has no children by their deliberate choice or because they are biologically infertile. Researchers at Yarmouk University (in Jordan) estimated the proportion of sterile couples in that country to be .06 (Journal of Data Science, July 2003). Also, 64% of the sterile couples in Jordan are infertile. Find the probability that a Jordanian couple is both sterile and infertile.

5. CRASH 3.119 NHTSA new car crash testing. Refer to the National Highway Traffic Safety Administration (NHTSA) crash tests of new car models, presented in Exercise 2.190 (p. 107). Recall that the NHTSA has developed a “star” scoring system, with results ranging from one star (*) to five stars (*****). The more stars in the rating, the better the level of crash protection in a head-on collision. A summary of the driver-side star ratings for 98 cars is reproduced in the accompanying MINITAB printout. Assume that one of the 98 cars is selected at random. State whether each of the following is true or false:

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

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

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

   4. The car has a better chance of having a two-star rating than of having a five-star rating.

      

6. 3.120 Selecting a sample. A random sample of five students is to be selected from 50 sociology majors for participation in a special program. In how many different ways can the sample be drawn?

7. 3.121 Fungi in beech forest trees. The current status of the beech tree species in East Central Europe was evaluated by Hungarian university professors in Applied Ecology and Environmental Research (Vol. 1, 2003). Of 188 beech trees surveyed, 49 had been damaged by fungi. Depending on the species of fungus, damage will occur on either the trunk, branches, or leaves of the tree. In the damaged trees, the trunk was affected 85% of the time, the leaves 10% of the time, and the branches 5% of the time.

   1. Give a reasonable estimate of the probability of a beech tree in East Central Europe being damaged by fungi.

   2. A fungus-damaged beech tree is selected at random, and the area (trunk, leaf, or branch) affected is observed. List the sample points for this experiment, and assign a reasonable probability to each one.

8. 3.122 Do you have a library card? According to a Harris poll, 68% of all American adults have a public library card. The percentages do differ by gender—62% of males have library cards compared to 73% of females.

   1. Consider the three probabilities: $P(A)=.68,$ $P(A|B)=.62,$ $\text{and}P(A|C)=\mathrm{.73.}$ Define events A, B, and C.

   2. Assuming that half of all American adults are males and half are females, what is the probability that an American adult is a female who owns a library card?

9. 3.123 Beach erosional hot spots. Beaches that exhibit high erosion rates relative to the surrounding beach are defined as erosional hot spots. The U.S. Army Corps of Engineers is conducting a study of beach hot spots. Through an online questionnaire, data are collected on six beach hot spots. The data are listed in the next table.

   

   1. Suppose you record the nearshore bar condition of each beach hot spot. Give the sample space for this experiment.

   2. Find the probabilities of the sample points in the sample space you defined in part a.

   3. What is the probability that a beach hot spot has either a planar or single, shore-parallel nearshore bar condition?

   4. Now suppose you record the beach condition of each beach hot spot. Give the sample space for this experiment.

   5. Find the probabilities of the sample points in the sample space you defined in part d.

   6. What is the probability that the condition of the beach at a particular beach hot spot is not flat?

10. 3.124 Chemical insect attractant. An entomologist is studying the effect of a chemical sex attractant (pheromone) on insects. Several insects are released at a site equidistant from the pheromone under study and a control substance. If the pheromone has an effect, more insects will travel toward it rather than toward the control. Otherwise, the insects are equally likely to travel in either direction. Suppose the pheromone under study has no effect, so that it is equally likely that an insect will move toward the pheromone or toward the control. Suppose five insects are released.

    1. List or count the number of different ways the insects can travel.

    2. What is the chance that all five travel toward the pheromone?

    3. What is the chance that exactly four travel toward the pheromone?

    4. What inference would you make if the event in part c actually occurs? Explain.

11. 3.125 Toxic chemical incidents. Process Safety Progress (Sept. 2004) reported on an emergency response system for incidents involving toxic chemicals in Taiwan. The system has logged over 250 incidents since being implemented. The next table gives a breakdown of the locations where these incidents occurred. Consider the location of a toxic chemical incident in Taiwan.

    1. List the sample points for this experiment.

    2. Assign reasonable probabilities to the sample points.

    3. What is the probability that the incident occurs in a school laboratory?

    4. What is the probability that the incident occurs in either a chemical or a nonchemical plant?

    5. What is the probability that the incident does not occur in transit?

    6. What is the probability that in a sample of 3 independently selected incidents, all 3 occur in a school laboratory?

    7. Would you expect the event in part f to occur? Explain.

    Location	Percent of Incidents
    School laboratory	6%
    In transit	26%
    Chemical plant	21%
    Nonchemical plant	35%
    Other	12%
    Total	100%

1. 3.126 Federal civil trial appeals. The Journal of the American Law and Economics Association (Vol. 3, 2001) published the results of a study of appeals of federal civil trials. The accompanying table, extracted from the article, gives a breakdown of 2,143 civil cases that were appealed by either the plaintiff or the defendant. The outcome of the appeal, as well as the type of trial (judge or jury), was determined for each civil case. Suppose one of the 2,143 cases is selected at random and both the outcome of the appeal and the type of trial are observed.

   

   Alternate View

   	Jury	Judge	Totals
   Plaintiff trial win— reversed	194	71	265
   Plaintiff trial win— affirmed/dismissed	429	240	669
   Defendant trial win— reversed	111	68	179
   Defendant trial win— affirmed/dismissed	731	299	1,030
   Totals	1,465	678	2,143

   1. Find P(A), where $A=\left\{\text{jury}\text{trial}\right\}.$

   2. Find P(B), where $B=\{$plaintiff trial win is reversed $\}$.

   3. Are A and B mutually exclusive events?

   4. Find $P(A^c).$

   5. Find $P(A\cup B).$

   6. Find $P(A\cap B).$

2. 3.127 Winning at roulette. Roulette is a very popular game in many American casinos. In Roulette, a ball spins on a circular wheel that is divided into 38 arcs of equal length, bearing the numbers 00, 0, 1, 2, $\dots$, 35, 36. The number of the arc on which the ball stops is the outcome of one play of the game. The numbers are also colored in the manner shown in the following table.

   

   Alternate View

   Red: 1, 3, 5, 7, 9, 12, 14, 16, 18, 19, 21, 23, 25, 27, 30, 32, 34, 36 Black: 2, 4, 6, 8, 10, 11, 13, 15, 17, 20, 22, 24, 26, 28, 29, 31, 33, 35 Green: 00, 0

   Players may place bets on the table in a variety of ways, including bets on odd, even, red, black, high, low, etc. Consider the following events:

   $\begin{array}{ll}A:\hfill & \{\text{The outcome is an odd number }(\text{00 and 0 are}\hfill \\ \hfill & \text{considered neither odd nor even}\text{.})\}\hfill \\ B:\hfill & \left\{\text{The outcome is a black number}\text{.}\right\}\hfill \\ C:\hfill & \{\text{The outcome is a low number }(1-18).\}\hfill \end{array}$

   1. Define the event $A\cap B$ as a specific set of sample points.

   2. Define the event $A\cup B$ as a specific set of sample points.

   3. Find $P(A),P(B),P(A\cap B),P(A\cup B),$ and P(C) by summing the probabilities of the appropriate sample points.

   4. Define the event $A\cap B\cap C$ as a specific set of sample points.

   5. Use the additive rule to find $P(A\cup B).$ Are events A and B mutually exclusive? Why?

   6. Find $P(A\cap B\cap C)$ by summing the probabilities of the sample points given in part d.

   7. Define the event $(A\cup B\cup C)$ as a specific set of sample points.

   8. Find $P(A\cup B\cup C)$ by summing the probabilities of the sample points given in part g.

3. 3.128 Cigar smoking and cancer. The Journal of the National Cancer Institute (Feb. 16, 2000) published the results of a study that investigated the association between cigar smoking and death from tobacco-related cancers. Data were obtained for a national sample of 137,243 American men. The results are summarized in the table below. Each male in the study was classified according to his cigar-smoking status and whether or not he died from a tobacco-related cancer.

   1. Find the probability that a randomly selected man never smoked cigars and died from cancer.

   2. Find the probability that a randomly selected man was a former cigar smoker and died from cancer.

   3. Find the probability that a randomly selected man was a current cigar smoker and died from cancer.

   4. Given that a male was a current cigar smoker, find the probability that he died from cancer.

   5. Given that a male never smoked cigars, find the probability that he died from cancer.

      

4. 3.129 Elderly wheelchair user study. The American Journal of Public Health (Jan. 2002) reported on a study of elderly wheelchair users who live at home. A sample of 306 wheelchair users, age 65 or older, were surveyed about whether they had an injurious fall during the year and whether their home featured any one of five structural modifications: bathroom modifications, widened doorways/hallways, kitchen modifications, installed railings, and easy-open doors. The responses are summarized in the accompanying table. Suppose we select, at random, one of the 306 wheelchair users surveyed.

   

   1. Find the probability that the wheelchair user had an injurious fall.

   2. Find the probability that the wheelchair user had all five features installed in the home.

   3. Find the probability that the wheelchair user had no falls and none of the features installed in the home.

   4. Given no features installed in the home, find the probability of an injurious fall.

5. 3.130 Selecting new-car options. A company sells midlevel models of automobiles in five different styles. A buyer can get an automobile in one of eight colors and with either standard or automatic transmission. Would it be reasonable to expect a dealer to stock at least one automobile in every combination of style, color, and transmission? At a minimum, how many automobiles would the dealer have to stock?

6. 3.131 Shooting free throws. In college basketball games, a player may be afforded the opportunity to shoot two consecutive foul shots (free throws).

   1. Suppose a player who makes (i.e., scores on) 80% of his foul shots has been awarded two free throws. If the two throws are considered independent, what is the probability that the player makes both shots? exactly one? neither shot?

   2. Suppose a player who makes 80% of his first attempted foul shots has been awarded two free throws and the outcome on the second shot is dependent on the outcome of the first shot. In fact, if this player makes the first shot, he makes 90% of the second shots; and if he misses the first shot, he makes 70% of the second shots. In this case, what is the probability that the player makes both shots? exactly one? neither shot?

   3. In parts a and b, we considered two ways of modeling the probability that a basketball player makes two consecutive foul shots. Which model do you think gives a more realistic explanation of the outcome of shooting foul shots; that is, do you think two consecutive foul shots are independent or dependent? Explain.

7. 3.132 Lie detector test. Consider a lie detector called the Computerized Voice Stress Analyzer (CVSA). The manufacturer claims that the CVSA is 98% accurate and, unlike a polygraph machine, will not be thrown off by drugs and medical factors. However, laboratory studies by the U.S. Defense Department found that the CVSA had an accuracy rate of 49.8%, slightly less than pure chance. Suppose the CVSA is used to test the veracity of four suspects. Assume that the suspects’ responses are independent.

   1. If the manufacturer’s claim is true, what is the probability that the CVSA will correctly determine the veracity of all four suspects?

   2. If the manufacturer’s claim is true, what is the probability that the CVSA will yield an incorrect result for at least one of the four suspects?

   3. Suppose that in a laboratory experiment conducted by the U.S. Defense Department on four suspects, the CVSA yielded incorrect results for two of the suspects. Make an inference about the true accuracy rate of the new lie detector.

8. 3.133 Maize seeds. The genetic origin and properties of maize (modern-day corn) were investigated in Economic Botany. Seeds from maize ears carry either single spikelets or paired spikelets but not both. Progeny tests on approximately 600 maize ears revealed the following information: Forty percent of all seeds carry single spikelets, while 60% carry paired spikelets. A seed with single spikelets will produce maize ears with single spikelets 29% of the time and paired spikelets 71% of the time. A seed with paired spikelets will produce maize ears with single spikelets 26% of the time and paired spikelets 74% of the time.

   1. Find the probability that a randomly selected maize ear seed carries a single spikelet and produces ears with single spikelets.

   2. Find the probability that a randomly selected maize ear seed produces ears with paired spikelets.

9. 3.134 Monitoring the quality of power equipment. Mechanical Engineering (Feb. 2005) reported on the need for wireless networks to monitor the quality of industrial equipment. For example, consider Eaton Corp., a company that develops distribution products. Eaton estimates that 90% of the electrical switching devices it sells can monitor the quality of the power running through the device. Eaton further estimates that, of the buyers of electrical switching devices capable of monitoring quality, 90% do not wire the equipment up for that purpose. Use this information to estimate the probability that an Eaton electrical switching device is capable of monitoring power quality and is wired up for that purpose.

10. 3.135 Series and parallel systems. Consider the two systems shown in the schematic on page 163. System A operates properly only if all three components operate properly. (The three components are said to operate in series.) The probability of failure for system A components 1, 2, and 3 is .12, .09, and .11, respectively. Assume that the components operate independently of each other.

    

    

    System B comprises two subsystems said to operate in parallel. Each subsystem has two components that operate in series. System B will operate properly as long as at least one of the subsystems functions properly. The probability of failure for each component in the system is .1. Assume that the components operate independently of each other.

    1. Find the probability that System A operates properly.

    2. What is the probability that at least one of the components in System A will fail and therefore that the system will fail?

    3. Find the probability that System B operates properly.

    4. Find the probability that exactly one subsystem in System B fails.

    5. Find the probability that System B fails to operate properly.

    6. How many parallel subsystems like the two shown would be required to guarantee that the system would operate properly at least 99% of the time?

11. 3.136 Forest fragmentation study. Refer to the Conservation Ecology (Dec. 2003) study of the causes of forest fragmentation, presented in Exercise 2.166 (p. 97). Recall that the researchers used advanced high-resolution satellite imagery to develop fragmentation indexes for each forest. $\mathrm{A}3\times 3$ grid was superimposed over an aerial photo of the forest, and each square (pixel) of the grid was classified as forest (F), as earmarked for anthropogenic land use (A), or as natural land cover (N). An example of one such grid is shown here. The edges of the grid (where an “edge” is an imaginary line that separates any two adjacent pixels) are classified as F–A, F–N, A–A, A–N, N–N, or F–F edges.

    A	A	N
    N	F	F
    N	F	F

    1. Note that there are 12 edges inside the grid. Classify each edge as F–A, F–N, A–A, A–N, N–N, or F–F.

    2. The researchers calculated the fragmentation index by considering only the F-edges in the grid. Count the number of F-edges. (These edges represent the sample space for the experiment.)

    3. If an F-edge is selected at random, find the probability that it is an F–A edge. (This probability is proportional to the anthropogenic fragmentation index calculated by the researchers.)

    4. If an F-edge is selected at random, find the probability that it is an F–N edge. (This probability is proportional to the natural fragmentation index calculated by the researchers.)

12. 3.137 Sociology fieldwork methods. Refer to University of New Mexico professor Jane Hood’s study of the fieldwork methods used by qualitative sociologists, presented in Exercise 2.195 (p. 108). Recall that she discovered that fieldwork methods could be classified into four distinct categories: Interview, Observation plus Participation, Observation Only, and Grounded Theory. The table that follows, reproduced from Teaching Sociology (July 2006), gives the number of sociology field research papers in each category. Suppose we randomly select one of these research papers and determine the method used. Find the probability that the method used is either Interview or Grounded Theory.

    Fieldwork Method	Number of Papers
    Interview	5,079
    Observation plus Participation	1,042
    Observation Only	848
    Grounded Theory	537

1. 3.138 Sex composition patterns of children in families. In having children, is there a genetic factor that causes some families to favor one sex over the other? That is, does having boys or girls “run in the family”? This was the question of interest in Chance (Fall 2001). Using data collected on children’s sex for over 4,000 American families that had at least two children, the researchers compiled the accompanying table. Make an inference about whether having boys or girls “runs in the family.”

   Sex Composition of First Two Children	Frequency
   Boy–Boy	1,085
   Boy–Girl	1,086
   Girl–Boy	1,111
   Girl–Girl	926
   Total	4,208

2. 3.139 Odds of winning a horse race. Handicappers for horse races express their beliefs about the probability of each horse winning a race in terms of odds. If the probability of event E is P(E ), then the odds in favor of E are P(E ) to $1-P(E).$ Thus, if a handicapper assesses a probability of .25 that Smarty Jones will win the Belmont Stakes, the odds in favor of Smarty Jones are ${\displaystyle \raisebox{1ex}{$25$}\!\left/ \!\raisebox{-1ex}{$100$}\right.}$ to ${\displaystyle \raisebox{1ex}{$75$}\!\left/ \!\raisebox{-1ex}{$100$}\right.}$, or 1 to 3. It follows that the odds against E are $1-P(E)$ to P(E ), or 3 to 1 against a win by Smarty Jones. In general, if the odds in favor of event E are a to b, then $P(E)=a/(a+b).$

   1. A second handicapper assesses the probability of a win by Smarty Jones to be ${\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$. According to the second handicapper, what are the odds in favor of a Smarty Jones win?

   2. A third handicapper assesses the odds in favor of Smarty Jones to be 1 to 1. According to the third handicapper, what is the probability of a Smarty Jones win?

   3. A fourth handicapper assesses the odds against Smarty Jones winning to be 3 to 2. Find this handicapper’s assessment of the probability that Smarty Jones will win.

3. 3.140 Chance of winning at blackjack. Blackjack, a favorite game of gamblers, is played by a dealer and at least one opponent. At the outset of the game, 2 cards of a 52-card bridge deck are dealt to the player and 2 cards to the dealer. Drawing an ace and a face card is called blackjack. If the dealer does not draw a blackjack and the player does, the player wins. If both the dealer and player draw blackjack, a “push” (i.e., a tie) occurs.

   1. What is the probability that the dealer will draw a blackjack?

   2. What is the probability that the player wins with a blackjack?

4. 3.141 Finding an organ transplant match. One of the problems encountered with organ transplants is the body’s rejection of the transplanted tissue. If the antigens attached to the tissue cells of the donor and receiver match, the body will accept the transplanted tissue. Although the antigens in identical twins always match, the probability of a match in other siblings is .25, and that of a match in two people from the population at large is .001. Suppose you need a kidney and you have two brothers and a sister.

   1. If one of your three siblings offers a kidney, what is the probability that the antigens will match?

   2. If all three siblings offer a kidney, what is the probability that all three antigens will match?

   3. If all three siblings offer a kidney, what is the probability that none of the antigens will match?

   4. Repeat parts b and c, this time assuming that the three donors were obtained from the population at large.

5. 3.142 Chance of winning at “craps.” A version of the dice game “craps” is played in the following manner. A player starts by rolling two balanced dice. If the roll (the sum of the two numbers showing on the dice) results in a 7 or 11, the player wins. If the roll results in a 2 or a 3 (called craps), the player loses. For any other roll outcome, the player continues to throw the dice until the original roll outcome recurs (in which case the player wins) or until a 7 occurs (in which case the player loses).

   1. What is the probability that a player wins the game on the first roll of the dice?

   2. What is the probability that a player loses the game on the first roll of the dice?

   3. If the player throws a total of 4 on the first roll, what is the probability that the game ends (win or lose) on the next roll?

6. 3.143 Accuracy of pregnancy tests. Seventy-five percent of all women who submit to pregnancy tests are really pregnant. A certain pregnancy test gives a false positive result with probability .02 and a valid positive result with probability .99. If a particular woman’s test is positive, what is the probability that she really is pregnant? [Hint: If A is the event that a woman is pregnant and B is the event that the pregnancy test is positive, then B is the union of the two mutually exclusive events $A\cap B$ and $A^c\cap B.$ Also, the probability of a false positive result may be written as $P(B|A^c)=.02$.]

7. 3.144 Odd Man Out. Three people play a game called “Odd Man Out.” In this game, each player flips a fair coin until the outcome (heads or tails) for one of the players is not the same as that for the other two players. This player is then “the odd man out” and loses the game. Find the probability that the game ends (i.e., either exactly one of the coins will fall heads or exactly one of the coins will fall tails) after only one toss by each player. Suppose one of the players, hoping to reduce his chances of being the odd man out, uses a two-headed coin. Will this ploy be successful? Solve by listing the sample points in the sample space.

1. 3.145 “Let’s Make a Deal.” Marilyn vos Savant, who is listed in the Guinness Book of World Records Hall of Fame as having the “Highest IQ,” writes a weekly column in the Sunday newspaper supplement Parade Magazine. Her column, “Ask Marilyn,” is devoted to games of skill, puzzles, and mind-bending riddles. In one issue (Parade Magazine, Feb. 24, 1991), vos Savant posed the following question:

   Suppose you’re on a game show and you’re given a choice of three doors. Behind one door is a car; behind the others, goats. You pick a door—say, #1—and the host, who knows what’s behind the doors, opens another door—say #3—which has a goat. He then says to you, “Do you want to pick door #2?” Is it to your advantage to switch your choice?

   Marilyn’s answer: “Yes, you should switch. The first door has a ${\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$ chance of winning [the car], but the second has a ${\displaystyle \raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$ chance [of winning the car].” Predictably, vos Savant’s surprising answer elicited thousands of critical letters, many of them from Ph.D. mathematicians, that disagreed with her. Who is correct, the Ph.D.s or Marilyn?

2. 3.146 Most likely coin-toss sequence. In Parade Magazine’s (Nov. 26, 2000) column “Ask Marilyn,” the following question was posed: “I have just tossed a [balanced] coin 10 times, and I ask you to guess which of the following three sequences was the result. One (and only one) of the sequences is genuine.”

   1. H H H H H H H H H H

   2. H H T T H T T H H H

   3. T T T T T T T T T T

   Marilyn’s answer to the question posed was “Though the chances of the three specific sequences occurring randomly are equal … it’s reasonable for us to choose sequence (2) as the most likely genuine result.” Do you agree?

   Activity Exit Polls

   Exit polls are conducted in selected locations as voters leave their polling places after voting. In addition to being used to predict the outcome of elections before the votes are counted, these polls seek to gauge tendencies among voters. The results are usually stated in terms of conditional probabilities.

   The accompanying table shows the results of exit polling that suggest that men were move likely to vote for Mitt Romney, while women were more likely to vote for Barack Obama in the 2012 presidential election. In addition, the table suggests that more women than men voted in the election. The six percentages in the last three columns represent conditional probabilities, where the given event is gender.

   2012 Presidential Election, Vote by Gender

   

   1. Find similar exit poll results in which voters are categorized by race, income, education, or some other criterion for a recent national, state, or local election. Choose two different examples, and interpret the percentages given as probabilities, or conditional probabilities where appropriate.

   2. Use the multiplicative rule of probability to find the probabilities related to the percentages given. [For example, in the accompanying table, find P(Obama and Male).] Then interpret each of these probabilities and use them to determine the total percentage of the electorate that voted for each candidate.

   3. Describe a situation in which a political group might use a form of exit polling to gauge voters’ opinions on a “hot-­button” topic (e.g., global warming). Identify the political group, the “hot-button” topic, the criterion used to categorize voters, and how the voters’ opinions will be determined. Then describe how the results will be summarized as conditional probabilities. How might the results of the poll be used to support a particular agenda?