婚礼座位：最后一人坐到自己座位的概率

wedding table

专题: Probability / 概率
难度: L4
来源: QuantQuestion

题目详情

N guests are queuing at the entrance to get seated at a wedding table.Every guest has an assigned seat number but the first guest to choose his seat is too drunk and takes a random seat.The remaining guests choose their seat according to the following rule:

if their assigned seat is available they take it

if their assigned seat is taken they choose randomly an available seat

What is the probability that the last person gets his assigned seat?

解析

该问题等价于经典“飞机随机入座”问题。

无论 $N$ 多大（ $N\ge 2$ ），最后一人坐到自己座位的概率为

\boxed{\frac{1}{2}}.

直观解释：过程最终只会在“1 号座位”和“最后一人的座位”之间决出谁先被随机占用；两者对称，因此各占一半概率。