1. 2.32 Solve the following problems for the coin-tossing experiment:

(a) Use computer simulation to find the probability that the number of heads ever exceeds twice the number of tails if a fair coin is tossed 5 times. What is the probability if the coin is tossed 25 times. What is the probability if the coin is tossed 50 times? Verify experimentally that the probability approaches the value 1 2

(√

5 −1)if the number of tosses increases.

(b) A fair coin is tossed no more than n times, where n is fixed in advance. After eachtoss,youcandecidetostopthecoin-tossexperiment.Yourpayoffis1,000 dollars multiplied by the proportion of heads at the moment the experiment is stopped.Yourstrategyistostopassoonastheproportionofheadsexceeds 1 2

or as soon as n tosses are done, whichever occurs first. Use computer simulation to find your expected payoff for n = 5, 10, and 25. Verify experimentally that your expected payoff approaches the value 1 4

π times $1,000 if n becomes large. Can you devise a better strategy than the one proposed?