HMMT 二月 2008 · 冲刺赛 · 第 5 题

HMMT February 2008 — Guts Round — Problem 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, the moderator 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?

解析

[ 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, the moderator 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? 416 Answer: Consider the error that was introduced on day 1. The probability that the Moderator 729 1 3 misses this error on all three checks is 1 / 3 , so the probability that this error gets removed is 1 − . 3 3 1 1 Similarly, the probability that the moderator misses the other two errors are 1 − and 1 − . So the 2 3 3 probability that the article is error-free is ( ) ( ) ( ) 1 1 1 416 1 − 1 − 1 − = . 3 2 3 3 3 729