醉酒乘客

Drunk Passenger

专题: Probability / 概率
难度: L4
来源: QuantQuestion

题目详情

100 名乘客依次登机，每人票对应座位 1..100。

第 1 人喝醉了随机坐任意一个座位。

之后每人若自己座位空着就坐自己的；若被占，则随机坐剩余空座位之一。

你是第 100 人。问：你坐到 100 号座位的概率是多少？

There are 100 passengers lined up to board a plane, each holding a ticket for seat number 1 through 100. The first passenger is drunk and picks a seat at random. Each subsequent (sober) passenger goes to their seat if it is still free; otherwise, they pick any free seat at random. You are the 100th passenger. What is the probability that you sit in seat #100?

解析

答案为 $\frac{1}{2}$ 。

过程到最后只会在 1 号与 100 号之间决出归属：要么有人先占了 1 号，要么先占了 100 号。

由对称性，两者概率相同，所以最后一人坐到 100 号座位概率为 $1/2$ 。

Original Explanation

Only seat #1 and seat #100 matter in the end. By symmetry, the chance that the last passenger (you) ends up in seat #100 is 0.5.