Here is a famous problem called the St. P etersburg Paradox. Wikipedia states the problem as follows: "A casino offers a game of chance for a single player in which a fair coin is tossed at each stage. The pot starts at 1 dollar and is doubled every time a head appears. The first time a tail appears, the game ends and the player wins whatever is in the pot. Thus the player wins 1 dollar if a tail appears on the first toss, 2 dollars if a head appears on the first toss and a tail on the second, 4 dollars if a head appears on the first two tosses and a tail on the third, 8 dollars if a head appears on the first three tosses and a tail on the fourth, and so on. In short, the player wins 2^k−1 dollars if the coin is tossed k times until the first tail appears. What would be a fair price to pay the casino for entering the game?"

a. Let X be the amount of money (in dollars) that the player wins. Find EX.

b. What is the probability that the player wins more than 65 dollars?

c. Now suppose that the casino only has a finite amount of money. Specifically, suppose that the maximum amount of the money that the casion will pay you is 2³⁰ dollars (around 1.07 billion dollars). That is, if you win more than 2³⁰ dollars, the casino is going to pay you only 2³⁰ dollars. Let Y be the money that the player wins in this case. Find EY.