Problem 44 Solution

The probability of getting x consecutive heads followed by a tails is (1/2)^x+1 = (1/2)*(1/2)^x.

The payoff for getting x heads followed by a tails is $2^x.

The expected return is the sum of the products of the probabilities and the payoffs is the sum for x=0 to infinity of (1/2) * (1/2)^x * 2^x.

(1/2)^x * 2^x = 1. So the answer is an infinite sum of 1/2's, or infinity.

This problem is a well known paradox in mathematics. The final result must be finite yet the expected value is infinite.

Michael Shackleford, A.S.A.