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两箱取球：另一箱剩余的期望

What is the expected number of remaining balls

专题
Probability / 概率
难度
L4

题目详情

Two boxes, AA and BB , contain nn balls each. In every step, you randomly choose a box and then draw one ball from it. Repeat this until the box you choose actually turns out to be empty. What is the expected number of remaining balls in the other box at the end of this process?

解析

设最终另一箱剩余球数为 XX（取空的那一步，另一箱未被选中的那箱）。可推出

P(X=k)=(2n−kn)(12)2n−k,0≤k≤n.\mathbb{P}(X=k)=\binom{2n-k}{n}\left(\frac12\right)^{2n-k},\quad 0\le k\le n.

由该分布可化简得到期望

E[X]=2n+122n(2nn)−1.\boxed{\mathbb{E}[X]=\frac{2n+1}{2^{2n}}\binom{2n}{n}-1}.