连续两个正面：期望次数

What is the expected number 2

专题: Probability / 概率
难度: L4
来源: QuantQuestion

题目详情

What is the expected number of coin tosses of a fair coin in order to get two heads in a row? What if the coin is biased with $25\%$ probability of getting heads?

解析

设正面概率为 $p$ 。两连正期望满足递推，解得

\boxed{E_2=\frac{1+p}{p^2}}.

公平硬币 $p=1/2$ 时 $E_2=6$ 。偏币 $p=1/4$ 时 $E_2=20$ 。