On September 26, 1980, the New York Times reported that a mysterious stranger strode into a Las Vegas casino, placed a single bet of 777,00 dollars on the “don't pass” line at the crap table, and walked away with more than 1.5 million dollars. In the “don't pass” bet, the bettor is essentially betting with the house. An exception occurs if the roller rolls a 12 on the first roll. In this case, the roller loses and the “don't pass” better just gets back the money bet instead of winning. Show that the “don't pass” bettor has a more favorable bet than the roller.

Recall that in the martingale doubling system (see Exercise), the player doubles his bet each time he loses. Suppose that you are playing roulette in a fair casino where there are no 0's, and you bet on red each time. You then win with probability 1/2 each time. Assume that you enter the casino with 100 dollars, start with a 1-dollar bet and employ the martingale system. You stop as soon as you have won one bet, or in the unlikely event that black turns up six times in a row so that you are down 63 dollars and cannot make the required 64-dollar bet. Find your expected winnings under this system of play.

You have 80 dollars and play the following game. An urn contains two white balls and two black balls. You draw the balls out one at a time without replacement until all the balls are gone. On each draw, you bet half of your present fortune that you will draw a white ball. What is your expected final fortune?

In the hat check problem (see Example), it was assumed that [math]N[/math] people check their hats and the hats are handed back at random. Let [math]X_j = 1[/math] if the [math]j[/math]th person gets his or her hat and 0 otherwise. Find [math]E(X_j)[/math] and [math]E(X_j \cdot X_k)[/math] for [math]j[/math] not equal to [math]k[/math]. Are [math]X_j[/math] and [math]X_k[/math] independent?

A box contains two gold balls and three silver balls. You are allowed to choose successively balls from the box at random. You win 1 dollar each time you draw a gold ball and lose 1 dollar each time you draw a silver ball. After a draw, the ball is not replaced. Show that, if you draw until you are ahead by 1 dollar or until there are no more gold balls, this is a favorable game.

Gerolamo Cardano in his book, The Gambling Scholar, written in the early 1500s, considers the following carnival game. There are six dice. Each of the dice has five blank sides. The sixth side has a number between 1 and 6---a different number on each die. The six dice are rolled and the player wins a prize depending on the total of the numbers which turn up.

• Find, as Cardano did, the expected total without finding its distribution.
• Large prizes were given for large totals with a modest fee to play the game. Explain why this could be done.

Let [math]X[/math] be the first time that a failure occurs in an infinite sequence of Bernoulli trials with probability [math]p[/math] for success. Let [math]p_k= P(X = k)[/math] for [math]k = 1[/math], 2, .... Show that [math]p_k = p^{k - 1}q[/math] where [math]q = 1 - p[/math]. Show that [math]\sum_k p_k = 1[/math]. Show that [math]E(X) = 1/q[/math]. What is the expected number of tosses of a coin required to obtain the first tail?

Exactly one of six similar keys opens a certain door. If you try the keys, one after another, what is the expected number of keys that you will have to try before success?

A multiple choice exam is given. A problem has four possible answers, and exactly one answer is correct. The student is allowed to choose a subset of the four possible answers as his answer. If his chosen subset contains the correct answer, the student receives three points, but he loses one point for each wrong answer in his chosen subset. Show that if he just guesses a subset uniformly and randomly his expected score is zero.

You are offered the following game to play: a fair coin is tossed until heads turns up for the first time (see Example). If this occurs on the first toss you receive 2 dollars, if it occurs on the second toss you receive [math]2^2 = 4[/math] dollars and, in general, if heads turns up for the first time on the [math]n[/math]th toss you receive [math]2^n[/math] dollars.

• Show that the expected value of your winnings does not exist (i.e., is given by a divergent sum) for this game. Does this mean that this game is favorable no matter how much you pay to play it?
• Assume that you only receive [math]2^{10}[/math] dollars if any number greater than or equal to ten tosses are required to obtain the first head. Show that your expected value for this modified game is finite and find its value.
• Assume that you pay 10 dollars for each play of the original game. Write a program to simulate 100 plays of the game and see how you do.
• Now assume that the utility of [math]n[/math] dollars is [math]\sqrt n[/math]. Write an expression for the expected utility of the payment, and show that this expression has a finite value. Estimate this value. Repeat this exercise for the case that the utility function is [math]\log(n)[/math].