掷骰累积分：停在 4/5 得分，停在 6 得 0

Fair die

专题: Strategy / 策略
难度: L4
来源: QuantQuestion

题目详情

You will roll a fair die until the game stops. The game stops when you get a 4,5 , or 6 . For every number 1,2 , or 3 you have thrown your score increases by $+1$ . If the game stops with a 4 or 5 , you get paid the accumulated score. If the game stops with a 6 you get nothing. What is the expected payoff of this game?

解析

令 $N$ 为在首次掷出 $4/5/6$ 之前掷出 $1/2/3$ 的次数。每次继续的概率为 $1/2$ ，停止概率为 $1/2$ ，所以 $N$ 是“成功前失败次数”的几何分布：

\mathbb{E}[N]=\frac{1-p}{p}=\frac{1/2}{1/2}=1.

一旦停止，停止点数在 $\{4,5,6\}$ 中均匀，因此被支付（停在 4 或 5）的条件概率为 $2/3$ 。

期望奖金

\boxed{\mathbb{E}[\text{payoff}]=\frac{2}{3}\,\mathbb{E}[N]=\frac{2}{3}}.