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| In the St. Petersburg Paradox, which is more of a gambling game than a paradox, a balanced coin is tossed fairly until the first head appears. The gambler's winnings are based on the number of tosses that are made before the game ends. If a head appears on the first toss, the player wins $2. If not, the "kitty" is doubled to $4 -- the reward if a head appers on the second toss. The pot is doubled after every coin toss that results in a tail. The winnings are $2 raised to the power of the number of tosses until and including the first head. This procedure is more interesting when you think about what amount you would be willing to pay for the privilege of playing the game. The probability that n+1 tosses will occur before payment is the probability that there is a run of n tails and that the (n+1)st toss is a head or (1/2)n+1. The payoff for n+1 tosses is 2n+1. We calculate the player's expected receipts from the sum $2(1/2)+$22(1/2)2+$23(1/2)3+...=$1+1+1+...=$infinite. Since the number of $1's in this sum is unlimited, the expected receipts from a play of this game are infinite! Whatever amount you were willing to play must have been a finite amount and therefore less than the expected receipts. What price would you be willing to pay to play this game? |