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连续两/三正面:一般公式

Two heads in a row

专题
Probability / 概率
难度
L4

题目详情

What is the expected number of tosses of an unfair coin needed to get two heads in a row (assume probability pp of a head)? Same question with three heads in a row.

解析

设正面概率为 pp

连续两正面期望: E2=1+pp2.E_2=\frac{1+p}{p^2}.

连续三正面期望: E3=1+p+p2p3.E_3=\frac{1+p+p^2}{p^3}.

一般连续 jj 个正面: Ej=1+p+p2++pj1pj.E_j=\frac{1+p+p^2+\cdots+p^{j-1}}{p^j}.