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 ½( - 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 each toss, you can decide to stop the coin-toss experiment. Your payoff is 1,000 dollars multiplied by the proportion of heads at the moment the experiment is stopped. Your strategy is to stop as soon as the proportion of heads exceeds ½ 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 4π times $1,000 if n becomes large. Can you devise a better strategy than the one proposed?