HMMT 二月 2005 · COMB 赛 · 第 5 题
HMMT February 2005 — COMB Round — Problem 5
题目详情
英文原题
- Doug and Ryan are competing in the 2005 Wiffle Ball Home Run Derby. In each round,
each player takes a series of swings. Each swing results in either a home run or an out,
and an out ends the series. When Doug swings, the probability that he will hit a homerun is 1/3. When Ryan swings, the probability that he will hit a home run is 1/2. Inone round, what is the probability that Doug will hit more home runs than Ryan hits?
解析
英文解析
- Doug and Ryan are competing in the 2005 Wiffle Ball Home Run Derby. In each round,
each player takes a series of swings. Each swing results in either a home run or an out,
and an out ends the series. When Doug swings, the probability that he will hit a homerun is 1/3. When Ryan swings, the probability that he will hit a home run is 1/2. Inone round, what is the probability that Doug will hit more home runs than Ryan hits?
Solution: 1 / 5
Denote this probability by p . Doug hits more home runs if he hits a home run on hisfirst try when Ryan does not, or if they both hit home runs on their first try and Doug
1 1 1
hits more home runs thereafter. The probability of the first case occurring is · = ,
3 2 6
p p
1 1 1
and the probability of the second case occurring is · · p = . Therefore p = + ,
3 2 6 6 6
which we solve to find p = .1 5