# 三人成虎和贝叶斯公式

$$P(H|E)=\frac{P(E|H)P(H)}{P(E)}$$

$$P(E)=P(E|H)P(H)+P(E|¬H)P(¬H)$$

$$P(H|E)=\frac{P(E|H)P(H)}{P(E)}=\frac{0.9\times0.01}{ 0.9\times0.01 + 0.1\times0.99 }=\frac{1}{12}=0.08333…$$

$$P(H|E)=\frac{P(E|H)P(H)}{P(E)}=\frac{0.9\times\frac{1}{12}}{ 0.9\times \frac{1}{12} + 0.1\times \frac{11}{12}}=0.45$$

$$P(H|E)=\frac{P(E|H)P(H)}{P(E)}=\frac{0.9\times0.45}{ 0.9\times 0.45 + 0.1\times 0.55}=0.88…$$

$$P(H|E)P(E)=P(E|H)P(H)$$

100个人中有10个罪犯. 已知10个罪犯中，有5个是穿红衣服，有8个穿白袜子。这一百个人中，有10个人穿红衣服，有45个穿白袜子，请问一个概率，遇到一个红衣服白袜子的人，多大的可能这人是个罪犯？

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