连续两次相同：游戏结束期望

Either two heads/tails

专题: Probability / 概率
难度: L4
来源: QuantQuestion

题目详情

We play a game, with a fair coin. The game stops when either two heads (H) or tails (T) appear consecutively. What is the expected time until the game stops?

解析

设 $N$ 为停止抛掷次数。对 $k\ge 2$ ，序列长度为 $k$ 且首次在末尾形成 HH 或 TT 的概率为 $2\cdot(1/2)^k=2^{1-k}$ ，因此

\mathbb{E}[N]=\sum_{k=2}^{\infty}k\cdot 2^{1-k}=\boxed{3}.