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A bin of 5 electrical components is known to contain 2 that are defective. If the components are to be tested one at a time, in random order, until the defectives are discovered, find the expected number of tests that are made.

If \(X\) is a geometric random variable, show analytically that $$ P\\{X=n+k \mid X>n\\}=P\\{X=k\\} $$ Give a verbal argument using the interpretation of a geometric random variable as to why the equation above is true.

An insurance company writes a policy to the effect that an amount of money \(A\) must be paid if some event \(E\) occurs within a year. If the company estimates that \(E\) will occur within a year with probability \(p\), what should it charge the customer in order that its expected profit will be 10 percent of \(A ?\)

Balls numbered 1 through \(N\) are in an urn. Suppose that \(n, n \leq N\), of them are randomly selected without replacement. Let \(Y\) denote the largest number selected. (a) Find the probability mass function of \(Y\). (b) Derive an expression for \(E[Y]\) and then use Fermat's combinatorial identity (see Theoretical Exercise 11 of Chapter 1) to simplify.

A person tosses a fair coin until a tail appears for the first time. If the tail appears on the \(n\)th flip, the person wins \(2^{n}\) dollars. Let \(X\) denote the player's winnings. Show that \(E[X]=+\infty\). This problem is known as the St. Petersburg paradox. (a) Would you be willing to pay \(\$ 1\) million to play this game once? (b) Would you be willing to pay \(\$ 1\) million for each game if you could play for as long as you liked and only had to settle up when you stopped playing?

Each night different meteorologists give us the probability that it will rain the next day. To judge how well these people predict, we will score each of them as follows: If a meteorologist says that it will rain with probability \(p\), then he or she will receive a score of $$ \begin{array}{ll} 1-(1-p)^{2} & \text { if it does rain } \\ 1-p^{2} & \text { if it does not rain } \end{array} $$ We will then keep track of scores over a certain time span and conclude that the meteorologist with the highest average score is the best predictor of weather. Suppose now that a given meteorologist is aware of this and so wants to maximize his or her expected score. If this person truly believes that it will rain tomorrow with probability \(p^{*}\), what value of \(p\) should he or she assert so as to maximize the expected score?

A newsboy purchases papers at 10 cents and sells them at 15 cents. However, he is not allowed to return unsold papers. If his daily demand is a binomial random variable with \(n=10, p=\frac{1}{3}\), approximately how many papers should he purchase so as to maximize his expected profit?

An urn initially contains one red and one blue ball. At each stage a ball is randomly chosen and then replaced along with another of the same color. Let \(X\) denote the selection number of the first chosen ball that is blue. For instance, if the first selection is red and the second blue, then \(X\) is equal to 2 . (a) Find \(P\\{X>i\\}, i \geq 1\). (b) Show that with probability 1 , a blue ball is eventually chosen. (That is, show that \(P\\{X<\infty\\}=1 .\) ) (c) Find \(E[X]\).

A box contains 5 red and 5 blue marbles. Two marbles are withdrawn randomly. If they are the same color, then you win \(\$ 1.10\); if they are different colors, then you win \(-\$ 1.00\) (that is, you lose \(\$ 1.00\) ). Calculate (a) the expected value of the amount you win; (b) the variance of the amount you win.

In some military courts, 9 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 \(.7\), whereas when the defendant is, in fact, innocent, this probability drops to \(.3\). (a) What is the probability that a guilty defendant is declared guilty when there are (i) 9 , (ii) 8 , and (iii) 7 judges? (b) Repeat part (a) for an innocent defendant. (c) If the prosecution 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 60 percent certain that the client is guilty?