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先验 p∼U(0,1)p\sim U(0,1)：连续 n 次正面后再正面概率

An unfair coin

专题
Probability / 概率
难度
L4

题目详情

An unfair coin is tossed nn times and all nn tosses resulted in heads. What is the probability that the next toss will be head? Assume that the probability pp of getting heads is a random variable with uniform distribution on [0, 1].

解析

后验为 p∣Hn∼Beta(n+1,1)p\mid H^n\sim \mathrm{Beta}(n+1,1)。

因此

P(下次正面∣Hn)=E[p∣Hn]=n+1n+2.\mathbb{P}(\text{下次正面}\mid H^n)=\mathbb{E}[p\mid H^n]=\boxed{\frac{n+1}{n+2}}.