HMMT 十一月 2014 · THM 赛 · 第 4 题
HMMT November 2014 — THM Round — Problem 4
题目详情
英文原题
- Suppose there are initially 1001 townspeople and two goons. What is the probability that, when thegame ends, there are exactly 1000 people in jail?
解析
英文解析
- Suppose there are initially 1001 townspeople and two goons. What is the probability that, when thegame ends, there are exactly 1000 people in jail?
Answer: By considering the parity of the number of people in jail, we see that this situation 3
1003
arises if and only if the goons win after the 500 th night. That means that at this point we must haveexactly one townsperson and two goons remaining. In other words, this situation arises if and only ifno goon is ever sent to jail. The probability that this occurs is
1001 999 997 3 3
· · · . . . = .
1003 1001 999 5 1003
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