1000 枚硬币含 1 枚双头：10 连正后的后验

999 pennies

专题: Probability / 概率
难度: L4
来源: QuantQuestion

题目详情

You have a large jar containing 999 fair pennies and one twoheaded penny. Suppose you pick one coin out of the jar and flip it 10 times and get all heads. What is the probability that the coin you chose is the two- headed one?

解析

设事件 $TH$ 表示“拿到双头硬币”，事件 $10H$ 表示“10 次都是正面”。

先验 $\mathbb{P}(TH)=1/1000$ 。

似然： $\mathbb{P}(10H\mid TH)=1$ ， $\mathbb{P}(10H\mid TH^c)=(1/2)^{10}$ 。

贝叶斯公式：

\mathbb{P}(TH\mid 10H)=\frac{\frac{1}{1000}}{\frac{1}{1000}+\frac{999}{1000}\cdot 2^{-10}} =\frac{1}{1+\frac{999}{1024}}=\boxed{\frac{1024}{2023}}\approx 0.506.