HMMT 二月 2003 · GEN2 赛 · 第 7 题

英文解析

7. Daniel and Scott are playing a game where a player wins as soon as he has two pointsmore than his opponent. Both players start at par, and points are earned one at atime. If Daniel has a 60% chance of winning each point, what is the probability thathe 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, werevert to the original situation. In particular, after every two points, either the gamereturns 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 winswith probability (9 / 25) / (9 / 25 + 4 / 25) = 9 / 13. Since this holds for any k , we concludethat Daniel wins the game with probability 9 / 13.