看一张纸判断大小：能否超过 1/2

Alice & Bob guess

专题: Probability / 概率
难度: L4
来源: QuantQuestion

题目详情

Alice writes two distinct real numbers between 0 and 1 on two sheets of paper. Bob selects one of the sheets randomly to inspect it.He then has to declare whether the number he sees is the bigger or smaller of the two.Is there any way Bob can expect to be correct more than half the times Alice plays this game with him?

解析

可以。

策略：看到纸上数为 $A$ 后，独立再生成一个随机阈值 $B$ （例如 $B\sim\mathrm{Unif}(0,1)$ ）。若 $B<A$ 则猜“我看到的是较大数”，否则猜“我看到的是较小数”。

设两数为 $0<a_1<a_2<1$ ，且你随机看到其中之一（各概率 $1/2$ ）。则

\mathbb{P}(\text{猜对})=\tfrac12\mathbb{P}(B>a_1)+\tfrac12\mathbb{P}(B<a_2) =\tfrac12(1-a_1)+\tfrac12 a_2 =\tfrac12+\tfrac12(a_2-a_1)\;>\;\tfrac12.