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抛到第一个正面：期望次数

What is the expected number

专题
Probability / 概率
难度
L4

题目详情

What is the expected number of times you must flip a fair coin until it lands on head? What if the coin is biased and lands on head with probability pp ?

解析

设期望为 EE。对公平硬币： E=12⋅1+12(1+E)⇒E=2.E=\frac12\cdot1+\frac12(1+E)\Rightarrow E=2.

对正面概率为 pp 的偏币： E=p⋅1+(1−p)(1+E)⇒E=1p.E=p\cdot1+(1-p)(1+E)\Rightarrow E=\frac{1}{p}.