8 男 7 女排座：异性相邻对数的期望

Eight boys and seven girls

专题: Probability / 概率
难度: L4
来源: QuantQuestion

题目详情

Eight boys and seven girls went to the movies and sat in the same row of 15 seats. Assuming that all the 15 ! permutations of their seating arrangements are equally likely, compute the expected number of pairs of neighbors of different genders.

解析

对每个相邻位置 $k,k+1$ 定义指示变量 $D_k$ （异性则 1）。则期望为

\mathbb{E}[N]=\sum_{k=1}^{14}\mathbb{E}[D_k]=14\,\mathbb{P}(\text{位置 1 与 2 异性}).

而

\mathbb{P}(\text{异性})=2\cdot\frac{8}{15}\cdot\frac{7}{14}=\frac{8}{15}.

所以

\boxed{\mathbb{E}[N]=14\cdot\frac{8}{15}=\frac{112}{15}}\approx 7.467.