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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 jar contains chips. Suppose that a boy successively draws a chip from the jar, each time replacing the one drawn before drawing another. The process continues until the boy draws a chip that he has previously drawn. Let denote the number of draws, and compute its probability mass function.
A total of people, consisting of married couples, are randomly divided into pairs. Arbitrarily number the women, and let denote the event that woman is paired with her husband.
At time a coin that comes up heads with probability p is flipped and falls to the ground. Suppose it lands on heads. At times chosen according to a Poisson process with rate , the coin is picked up and flipped. (Between these times, the coin remains on the ground.) What is the probability that the coin is on its head side at time? Hint: What would be the conditional probability if there were no additional flips by time , and what would it be if there were additional flips by time ?
Let be a negative binomial random variable with parameters and , and let be a binomial random variable with parameters and . Show that
Hint: Either one could attempt an analytical proof of the preceding equation, which is equivalent to proving the identity
or one could attempt a proof that uses the probabilistic interpretation of these random variables. That is, in the latter case, start by considering a sequence of independent trials having a common probability p of success. Then try to express the events to express the events and in terms of the outcomes of this sequence.
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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