HMMT 二月 2008 · 冲刺赛 · 第 5 题

英文解析

5. [ 6 ] A Vandal and a Moderator are editing a Wikipedia article. The article originally is error-free.
   Each day, the Vandal introduces one new error into the Wikipedia article. At the end of the day, themoderator checks the article and has a 2/3 chance of catching each individual error still in the article.
   After 3 days, what is the probability that the article is error-free?
   Answer: Consider the error that was introduced on day 1. The probability that the Moderator 416
   1729
   misses this error on all three checks is 1 / 3 , so the probability that this error gets removed is 1 − .3
   1 133
   Similarly, the probability that the moderator misses the other two errors are 1 − and 1 − . So the
   3 32
   probability that the article is error-free is
   ( ) ( ) ( )
   1 1 1 416
   1 − 1 − 1 − = .
   3 2
   3 3 3 729