连续三正面：期望抛掷次数

For a fair coin

个人笔记

概率题：连续三正面：期望抛掷次数。

For a fair coin, what is the expected number of tosses to get three heads in a row?

解析

设公平硬币正面概率 $p=1/2$ 。

连续 $k$ 个正面的期望抛掷次数为

E_k=\frac{1+p+p^2+\cdots+p^{k-1}}{p^k}.

取 $k=3,p=1/2$ 得

E_3=\frac{1+\frac12+\frac14}{(\frac12)^3}=\frac{7/4}{1/8}=\boxed{14}.

E_k=\frac{1+p+p^2+\cdots+p^{k-1}}{p^k}.

E_3=\frac{1+\frac12+\frac14}{(\frac12)^3}=\frac{7/4}{1/8}=\boxed{14}.