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第二次机会

Second Chance

专题
Probability / 概率
难度
L4

题目详情

掷一枚公平六面骰，点数是多少你就得到多少收益。

但你可以选择“再来一次”：重掷一遍，并且以第二次点数作为最终收益（覆盖第一次）。

在最优策略下，期望收益是多少？

提示：只有当“重掷后的期望”大于当前点数时才值得重掷。

Roll a die, and you get paid what the dice shows. But if you want, you can request a second chance & roll the die again; get paid what the second roll shows instead of the first. What is the expected value?

Hint

We only roll the die again if the expected value of the second roll is higher than the first.

解析

最优策略：当第一次掷到 1、2、3 时重掷；掷到 4、5、6 时不重掷。

期望收益：

E=36⋅3.5+16(4+5+6)=4.25.E=\frac{3}{6}\cdot 3.5 + \frac{1}{6}(4+5+6)=4.25.

Original Explanation

4.25

Solution

Let's compute the expected value of a single roll first.

A fair six-sided die has outcomes 1 through 6 each with probability 1/61/6. So the expected value of a single roll is:

16⋅1+16⋅2+16⋅3+16⋅4+16⋅5+16⋅6=3.5\dfrac{1}{6} \cdot 1 + \dfrac{1}{6} \cdot 2 + \dfrac{1}{6} \cdot 3 + \dfrac{1}{6} \cdot 4 + \dfrac{1}{6} \cdot 5 + \dfrac{1}{6} \cdot 6 = 3.5

Now, for the strategy where we can roll a second time, the optimal strategy is to roll again when the first roll is below the expected value (i.e., 1, 2, or 3) because on average, the second roll should be higher.

So, if we roll a 1, 2, or 3, we should roll again, and the expected value of that roll is again 3.5. If we roll a 4, 5, or 6, we should keep it.

Therefore, the expected value of the game under this optimal strategy is:

16⋅3.5+16⋅3.5+16⋅3.5+16⋅4+16⋅5+16⋅6\dfrac{1}{6} \cdot 3.5 + \dfrac{1}{6} \cdot 3.5 + \dfrac{1}{6} \cdot 3.5 + \dfrac{1}{6} \cdot 4 + \dfrac{1}{6} \cdot 5 + \dfrac{1}{6} \cdot 6

=25.56=4.25=\dfrac{25.5}{6} = 4.25

So, the expected payout under the optimal strategy of re-rolling when the first roll is 1, 2, or 3 is $4.25.

Follow up question: What if you are allowed to roll the dice as many times as you want? What is the expected value of the game?

Answer: 6

Solution: If the number is not 6, you can request a retry. Stop only when the value is equal to 6.