相邻夫妻

Close Couples

专题: Probability / 概率
难度: L4
来源: QuantQuestion

个人笔记

题目详情

有 $n$ 对已婚夫妻（ $n$ 位丈夫与 $n$ 位妻子）随机坐在一个有 $2n$ 个座位的圆桌上，且座位安排必须男女交错。

令 $N$ 表示最终“相邻而坐（夫妻两人挨着坐）”的夫妻对数。求 $\mathbb{E}[N]$ （假设 $n>2$ ）。

英文原题

Let $n$ married couples ( $n$ husbands and $n$ wives) sit at a round table of $2n$ seats at random. The seating scheme is such that the seats must alternate between men and women. Find the number of couples that will sit together $\mathbb{E}[N]$ on average. Assume $n > 2$ .

解析

固定丈夫的座位：圆桌上有 $n$ 个“男座位”， $n$ 个“女座位”，且相邻关系固定。

对任意一位丈夫，他左右相邻的两个座位都是女座位（且 $n>2$ 时这两个座位不同）。他的妻子在 $n$ 个女座位中等概率落在任意一个，因此该夫妻相邻概率为 $\frac{2}{n}$ 。

令 $I_i$ 为第 $i$ 对夫妻相邻的指示变量，则

\mathbb{E}[N]=\sum_{i=1}^n \mathbb{E}[I_i]=n\cdot\frac{2}{n}=2.

所以 $\boxed{\mathbb{E}[N]=2}$ 。

英文解析

Fixed husband's seat: There are $n$ “men's seat” and $n$ “women's seat” on the round table, and the adjacency is fixed.

For any husband, the two seats adjacent to his left and right are female seats (and the two seats are different when $n>2$ ). His wife has a moderate probability of $n$ female seats, so the couple's adjacency probability is $\frac{2}{n}$ .

If $I_i$ is the $i$ pair of adjacent indicator variables, then

\mathbb{E}[N]=\sum_{i=1}^n \mathbb{E}[I_i]=n\cdot\frac{2}{n}=2.

So $\boxed{\mathbb{E}[N]=2}$ .