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www.downloadslide.com Supplementary Exercises 83 2.157 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 2.158 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 w selected was white is w + b + n . 2.159 It seems obvious that P(∅) = 0. Show that this result follows from the axioms in Definition 2.6. 2.160 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 10 were not defective, and the last 2 were defective (NNNNNNNNNNDD). Assume randomness. What is the probability of observing a b this arrangement (resulting in two runs) given that 10 of the 12 components are not defective? two runs? 2.161 Refer to Exercise 2.160. What is the probability that the number of runs, R, is less than or equal to 3? 2.162 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? 2.163 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? 1 3 1 3 2 4 B A B 2 4 A B 2.164 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? 2.165 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? 2.166 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.

www.downloadslide.com 84 Chapter 2 Probability 2.167 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. 2.168 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 b c 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? 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)? Give all possible values for R and the probabilities associated **with** all of these values. *2.169 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 b 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? 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? 2.170 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. 2.171 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? 2.172 Let A and B be any two events. Which of the following statements, in general, are false? a P(A|B) + P(A|B) = 1. b P(A|B) + P(A|B) = 1. c P(A|B) + P(A|B) = 1. 2.173 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? 2.174 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?

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