Consider the following game. A person flips a coin repeatedly until a head comes up. This person receives a payment of 2n dollars if the first head comes up at the nth flip.
a) Let X be a random variable equal to the amount of money the person wins. Show that the expected value of X does not exist (that is, it is infinite). Show that a rational gambler, that is, someone willing to pay to play the game as long as the price to play is not more than the expected payoff, should be willing to wager any amount of money to play this game. (This is known as the St. Petersburg paradox. Why do you suppose it is called a paradox?)
b) Suppose that the person receives 2n dollars if the first head comes up on the nth flip where n < 8 and 28 = 256 dollars if the first head comes up on or after the eighth flip. What is the expected value of the amount of money the person wins? How much money should a person be willing to pay to play this game?