91Ó°ÊÓ

Two fair dice are rolled. Let \(X\) equal the product of the 2 dice. Compute \(P\\{X=i\\}\) for \(i=1, \ldots, 36\)

Suppose that a die is rolled twice. What are the possible values that the following random variables can take on: (a) the maximum value to appear in the two rolls; (b) the minimum value to appear in the two rolls; (c) the sum of the two rolls; (d) the value of the first roll minus the value of the second roll?

A gambling book recommends the following "winning strategy" for the game of roulette: Bet \(\$ 1\) on red. If red appears (which has probability \(\frac{18}{38}\) ), then take the \(\$ 1\) profit and quit. If red does not appear and you lose this bet (which has probability \(\frac{20}{38}\) of occurring), make additional \(\$ 1\) bets on red on each of the next two spins of the roulette wheel and then quit. Let \(X\) denote your winnings when you quit. (a) Find \(P\\{X > 0\\}\) (b) Are you convinced that the strategy is indeed a "winning" strategy? Explain your answer! (c) Find \(E[X]\)

Suppose that two teams play a series of games that ends when one of them has won \(i\) games. Suppose that each game played is, independently, won by team \(A\) with probability \(p .\) Find the expected number of games that are played when (a) \(i=2\) and (b) \(i=3 .\) Also, show in both cases that this number is maximized when \(p=\frac{1}{2}\).

\(A\) and \(B\) play the following game: \(A\) writes down either number 1 or number \(2,\) and \(B\) must guess which one. If the number that \(A\) has written down is \(i\) and \(B\) has guessed correctly, \(B\) receives \(i\) units from \(A\). If \(B\) makes a wrong guess, \(B\) pays \(\frac{3}{4}\) unit to \(A .\) If \(B\) randomizes his decision by guessing 1 with probability \(p\) and 2 with probability \(1-p,\) determine his expected gain if (a) \(A\) has written down number 1 and (b) \(A\) has written down number 2 What value of \(p\) maximizes the minimum possible value of \(B\) 's expected gain, and what is this maximin value? (Note that \(B\) 's expected gain depends not only on \(p,\) but also on what \(A\) does.) Consider now player \(A\). Suppose that she also randomizes her decision, writing down number 1 with probability q. What is \(A\) 's expected loss if (c) \(B\) chooses number 1 and (d) \(B\) chooses number \(2 ?\) What value of \(q\) minimizes \(A\) 's maximum expected loss? Show that the minimum of \(A\) 's maximum expected loss is equal to the maximum of \(B\) 's minimum expected gain. This result, known as the minimax theorem, was first established in generality by the mathematician John von Neumann and is the fundamental result in the mathematical discipline known as the theory of games. The common value is called the value of the game to player \(B\).

Two coins are to be flipped. The first coin will land on heads with probability \(.6,\) the second with probability \(.7 .\) Assume that the results of the flips are independent, and let \(X\) equal the total number of heads that result. (a) Find \(P\\{X=1\\}\) (b) Determine \(E[X]\)

One of the numbers 1 through 10 is randomly chosen. You are to try to guess the number chosen by asking questions with "yes-no" answers. Compute the expected number of questions you will need to ask in each of the following two cases: (a) Your \(i\) th question is to be "Is it i?" \(i=\) 1,2,3,4,5,6,7,8,9,10 (b) With each question you try to eliminate one-half of the remaining numbers, as nearly as possible.

A sample of 3 items is selected at random from a box containing 20 items of which 4 are defective. Find the expected number of defective items in the sample.

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 \(1-(1-p)^{2} \quad\) if it does rain \(1-p^{2}\) if it does not rain 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 our scoring mechanism and 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?