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等待序列 11 vs 12：谁更快 & 期望次数

Mean Waiting Time

专题
Probability / 概率
难度
L4

题目详情

Peter and Paula play a simple game of dice, as follows. Peter keeps throwing the (unbiased) die until he obtains the sequence 1- 1 in two successivethrows. For Paula, the rules are similar, but she throws the die until she obtains the sequence 1 - 2 in two successive throws.

a. On average, will both have to throw the die the same number of times? If not, whose expected waiting time is shorter (no explicit calculations are required)?

b. Derive the actual expected waiting times for Peter and Paula.

解析

(a) Paula（等到 1-2）更快。因为 1-2 没有自重叠，而 1-1 有重叠结构导致等待更久。

(b) 设每次掷出 1 的概率 p=1/6p=1/6。

  • 等到 1-1：两连“成功”期望为
1+pp2=1+1/6(1/6)2=42.\frac{1+p}{p^2}=\frac{1+1/6}{(1/6)^2}=\boxed{42}.
  • 等到 1-2：模式无自重叠，期望为 1/P(1,2)=1/(1/36)=361/\mathbb{P}(1,2)=1/(1/36)=\boxed{36}。