Suppose that you toss a fair coin 100 times, getting 38 heads and 62 tails, which is 24 more tails than heads.
a. Explain why, on your next toss, the difference in the numbers of heads and tails is as likely to grow to 25 as it is to shrink to 23.
b. Extend your explanation from part a to explain why, if you toss the coin 1,000 more times, the final difference in the numbers of heads and tails is as likely to be larger than 24 as it is to be smaller than 24.
c. Suppose that you continue tossing the coin. Explain why the following statement is true: If you stop at any random time, you always are more likely to have fewer heads than tails, in total.
d. Suppose that you are betting on heads with each coin toss. After the first 100 tosses, you are well on the losing side (having lost the bet 62 times while winning only 38 times). Explain why, if you continue to bet, you will most likely remain on the losing side. How is this answer related to the gambler's fallacy?