HMMT 二月 2007 · GEN2 赛 · 第 5 题

英文解析

5. [ 4 ] A best of 9 series is to be played between two teams. That is, the first team to win 5 games is thewinner. One of the teams, the Mathletes, has a 2/3 chance of winning any given game. What is the probability that the winner is determined in the 7 th game?
   Answer: . If the Mathletes are the winners, they must win the 7 th game and have won exactly 20
   four of the previous 6 games. The probability of this occurring is 81
   ( ( ))
   4 26
   (2 / 3) · (1 / 3) · · (2 / 3) .
   ( )2
   ( )
   4 2
   Analogously, the other team wins with probability (1 / 3) · (2 / 3) · · (1 / 3). Summing, the prob-6
   ability is 2
   ( ) ( )
   2 3 2 56
   · 2 · 2 · 1 + 1
   5 · 4 20 = = .2
   7 4
   3 3 81