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Chapter 4: Q.4.9 (page 173)
If is a binomial random variable with expected value and variance, find
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In Example , what percentage of defective lots does the purchaser reject? Find it forGiven that a lot is rejected, what is the conditional probability that it contained defective components?
In the game of Two-Finger Morra, players show or fingers and simultaneously guess the number of fingers their opponent will show. If only one of the players guesses correctly, he wins an amount (in dollars) equal to the sum of the fingers shown by him and his opponent. If both players guess correctly or if neither guesses correctly, then no money is exchanged. Consider a specified player, and denote by X the amount of money he wins in a single game of Two-Finger Morra.
(a) If each player acts independently of the other, and if each player makes his choice of the number of fingers he will hold up and the number he will guess that his opponent will hold up in such a way that each of the possibilities is equally likely, what are the possible values of and what are their associated probabilities?
(b) Suppose that each player acts independently of the other. If each player decides to hold up the same number of fingers that he guesses his opponent will hold up, and if each player is equally likely to hold up or fingers, what are the possible values of and their associated probabilities?
A box contains red and blue marbles. Two marbles are withdrawn randomly. If they are the same color, then you win ; if they are different colors, then you win . (That is, you lose .) Calculate
(a) the expected value of the amount you win;
(b) the variance of the amount you win.
There are n components lined up in a linear arrangement. Suppose that each component independently functions with probability p. What is the probability that no 2 neighboring components are both nonfunctional?
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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