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Chapter 4: Q.4.56 (page 167)
How many people are needed so that the probability that at least one of them has the same birthday as you is greater than ?
The probability the same birthday found to be.
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A man claims to have extrasensory perception. As a test, a fair coin is flipped times and the man is asked to predict the outcome in advance. He gets out of correct. What is the probability that he would have done at least this well if he did not have ESP?
In Example , suppose that the department store incurs an additional cost of for each unit of unmet demand. (This type of cost is often referred to as a goodwill cost because the store loses the goodwill of those customers whose demands it cannot meet.) Compute the expected profit when the store stocks units, and determine the value of data-custom-editor="chemistry" that maximizes the expected profit.
There are types of coupons. Independently of the types of previously collected coupons, each new coupon collected is of typewith probability , . If n coupons are collected, find the expected number of distinct types that appear in this set. (That is, find the expected number of types of coupons that appear at least once in the set of coupons.)
An interviewer is given a list of people she can interview. If the interviewer needs to interview 5 people, and if each person (independently) agrees to be interviewed with probability 2 3 , what is the probability that her list of people will enable her to obtain her necessary number of interviews if the list consists of
(a) 5 people and
(b) 8 people? For part (b), what is the probability that the interviewer will speak to exactly
(c) 6 people and
(d) 7 people on the list?
In some military courts, judges are appointed. However, both the prosecution and the defense attorneys are entitled to a peremptory challenge of any judge, in which case that judge is removed from the case and is not replaced. A defendant is declared guilty if the majority of judges cast votes of guilty, and he or she is declared innocent otherwise. Suppose that when the defendant is, in fact, guilty, each judge will (independently) vote guilty with probability .whereas when the defendant is, in fact, innocent, this probability drops to .
(a) What is the probability that a guilty defendant is declared guilty when there are (i) , (ii) , and (iii) judges?
(b) Repeat part (a) for an innocent defendant.
(c) If the prosecuting attorney does not exercise the right to a peremptory challenge of a judge, and if the defense is limited to at most two such challenges, how many challenges should the defense attorney make if he or she is percent certain that the client is guilty?
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