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Chuck-a-Luck：单位下注的期望损失

Chuck-a-Luck

专题
Probability / 概率
难度
L4

题目详情

Chuck- a- Luck is a gambling game often played at carnivals and gambling houses. A player may bet on any one the numbers 1, 2, 3, 4, 5, 6. Three dice are rolled. If the player's number appears on one, two, or three of the dice, he receives respectively one, two, or three times his original stake plus his own money back; otherwise he loses his stake. What is the player's expected loss per unit stake? (Actually the player may distribute stakes on several numbers, but each such stake can be regarded as a separate bet.)

解析

押定一个点数，三骰独立。设该点数出现次数 K∼Bin(3,1/6)K\sim\mathrm{Bin}(3,1/6)。

若 K=0K=0 则亏 1；若 K=k≥1K=k\ge 1 则净赚 kk（拿回本金不计入收益）。所以期望净收益

E[profit]=E[K]−P(K=0)=36−(56)3=12−125216=−17216.\mathbb{E}[\text{profit}]=\mathbb{E}[K]-\mathbb{P}(K=0)=\frac{3}{6}-\left(\frac{5}{6}\right)^3 =\frac12-\frac{125}{216}=-\frac{17}{216}.

因此单位下注的期望损失为

17216≈0.0787.\boxed{\frac{17}{216}}\approx 0.0787.