Problem: A person starting with cents and making bets, wins three times and loses three times, the wins and losses occurring in random order. The chance for a win is equal to the chance for a loss. If each wager is for half the money remaining at the time of the bet, then the final result is:
Answer Choices:
A. a loss of
B. a gain of
C. a loss of
D. neither a gain nor a loss
E. a gain or a loss depending upon the order in which the wins and losses occur.
Solution:
Let represent the number of cents at the start. Suppose the first bet results in a win; then the amount at the end of the first bet is . Let . Suppose the second bet results in a win; then the amount at the end of the second bet is . Hence, a win at any stage results in an amount times the amount at hand at that stage.
If the first bet results in a loss, the amount at the end of the first bet is . Let . Suppose the second bet resuits in a loss, then the amount at the end of the second bet is . Hence, a loss at any stage results in an amount times the amount at hand at that stage.
For three wins and three losses, in any order, the amount left is . Consequently, the amount lost is . Since cents, the amount lost is cents.