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Chapter 4: Q.4.9 (page 170)
Show how the derivation of the binomial probabilities leads to a proof of the binomial theorem when and are nonnegative.
Hint: Let .
Assume the hint and the point that the totality of all possible probabilities must equal one.
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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?
and will take the same -question examination. Each question will be answered correctly by with probability, independently of her results on other questions. Each question will be answered correctly by B with probability , independently both of her results on the other questions and on the performance of
(a) Find the expected number of questions that are answered correctly by both A and B.(b) Find the variance of the number of questions that are answered correctly by either A or B
Four independent flips of a fair coin are made. Let denote the number of heads obtained. Plot the probability mass function of the random variable .
In Problem for if the coin is assumed fair, what are the probabilities associated with the values that X can take on?
A salesman has scheduled two appointments to sell encyclopedias. His first appointment will lead to a sale with probability and his second will lead independently to a sale with probability . Any sale made is equally likely to be either for the deluxe model, which costs , or the standard model, which costs Determine the probability mass function of , the total dollar value of all sales
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