HMMT 二月 2004 · COMB 赛 · 第 5 题

英文解析

5. A best-of-9 series is to be played between two teams; that is, the first team to win 5
   games is the winner. The Mathletes have a chance of 2 / 3 of winning any given game.
   What is the probability that exactly 7 games will need to be played to determine awinner?
   Solution: 20 / 81
   If the Mathletes are to win, they must win exactly 5 out of the 7 games. One ofthe 5 games they win must be the 7 th game, because otherwise they would win thetournament before 7 games are completed. Thus, in the first 6 games, the Mathletesmust win 4 games and lose 2. The probability of this happening and the Mathleteswinning the last game is
   [ ]
   ( ) ( ) ( ) ( )
   4 2
   6 2 1 2
   · · · .
   2 3 3 3
   Likewise, the probability of the other team winning on the 7 th game is
   [ ]
   ( ) ( ) ( ) ( )
   4 2
   6 1 2 1
   · · · .
   2 3 3 3
   Summing these values, we obtain 160 / 729 + 20 / 729 = 20 / 81 .