袋中 10 球：抽样全白后袋内全白的后验概率

Ten balls are put in a bag

专题: Probability / 概率
难度: L4
来源: QuantQuestion

题目详情

Ten balls are put in a bag based on the resultof the tosses of an unbiased coin. If the coin turns up heads, put in a black ball, if tails, put in a white ball. When the bag contains ten balls hand it to someone who hasn't seen the colours selected. Ask them to take out ten balls, one at a time with replacement. If all ten examined balls turn out to be white, what is the probability that all ten balls in the bag are white?

解析

设袋里白球数为 $N\in\{0,1,\ldots,10\}$ ，先验 $\mathbb{P}(N=n)=\binom{10}{n}2^{-10}$ 。

观察到“放回抽 10 次全白”事件 $B$ 的似然为

\mathbb{P}(B\mid N=n)=\left(\frac{n}{10}\right)^{10}.

所求为

\mathbb{P}(N=10\mid B)=\frac{\mathbb{P}(B\mid N=10)\mathbb{P}(N=10)}{\sum_{n=0}^{10}\mathbb{P}(B\mid N=n)\mathbb{P}(N=n)} =\frac{1}{\sum_{n=0}^{10}\binom{10}{n}\left(\frac{n}{10}\right)^{10}}.

数值约为 $\boxed{0.0702}$ （约 7.0%）。