HMMT 二月 2007 · GEN2 赛 · 第 5 题

HMMT February 2007 — GEN2 Round — Problem 5

[ 4 ] A best of 9 series is to be played between two teams. That is, the first team to win 5 games is the winner. 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 7th game?

解析

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