HMMT 二月 2003 · GEN2 赛 · 第 7 题

HMMT February 2003 — GEN2 Round — Problem 7

Daniel and Scott are playing a game where a player wins as soon as he has two points more than his opponent. Both players start at par, and points are earned one at a time. If Daniel has a 60% chance of winning each point, what is the probability that he will win the game?

解析

Daniel and Scott are playing a game where a player wins as soon as he has two points more than his opponent. Both players start at par, and points are earned one at a time. If Daniel has a 60% chance of winning each point, what is the probability that he will win the game? Solution: 9 / 13 Consider the situation after two points. Daniel has a 9 / 25 chance of winning, Scott, 4 / 25, and there is a 12 / 25 chance that the players will be tied. In the latter case, we revert to the original situation. In particular, after every two points, either the game returns to the original situation, or one player wins. If it is given that the game lasts 2 k rounds, then the players must be at par after 2( k − 1) rounds, and then Daniel wins with probability (9 / 25) / (9 / 25 + 4 / 25) = 9 / 13. Since this holds for any k , we conclude that Daniel wins the game with probability 9 / 13.