基础骰子游戏 I

Basic Dice Game I

专题: Probability / 概率
难度: L2
来源: QuantQuestion

题目详情

赌场提供一个六面骰游戏：你第一次掷骰得到点数即为收益。你可以选择立刻兑现，或者选择再掷一次并以第二次点数兑现（只能再掷一次）。

问：该游戏的公平价值是多少？

A casino offers you a game with a $6$ -sided die where you are paid the value of the roll. The casino lets you roll the first time. If you are happy with your roll, you can cash out. Else, you can choose to re-roll and cash out on this second value. What is the fair value of this game?

解析

第二次若掷，期望为 $3.5$ 。

因此当第一次点数小于 3.5（即 1、2、3）时应再掷；当第一次为 4、5、6 时应停。

期望为

\frac{1}{2}\cdot 5+\frac{1}{2}\cdot 3.5=4.25.

Original Explanation

The expected value of a roll on a fair die is the average of its faces, i.e $\frac{1+2+3+4+5+6}{6} = 3.5$ . Therefore, only re-roll if your first roll is under $3.5$ , since the better roll will do better on average. So, you will not re-roll if you obtain a 4, 5, or 6, which happens $\frac{1}{2}$ of the time with an expectation of 5. Hence, the fair value of this game is:

\frac{1}{2} \times 5 + \frac{1}{2} \times 3.5 = 4.25