1. Problem 13.13 A fair coin is tossed no more than n times, where n is fixed in advance. You stop the coin-toss experiment as soon as the proportion of heads exceeds 1 2

or as soon as n tosses are done, whichever occurs first. Use the law of conditional expectations to calculate, for n = 5, 10, 25, and 50, the expected value of the proportion of heads at the moment the coin-toss experiment is stopped. Hint: define the random variable Xk(i) as the proportion of heads upon stopping given that k tosses are still possible and heads turned up i times so far. Set up a recursion equation for E[Xk(i)].