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草率的陪审员

The Flippant Juror

专题
Probability / 概率
难度
L3

题目详情

一个三人陪审团中，有两名成员各自独立地以概率 pp 作出正确判断，第三名成员每次都靠掷硬币决定（按多数票裁决）。另有一个单人陪审团，其作出正确判断的概率也是 pp。哪个陪审团作出正确判断的概率更高？

A three-man jury has two members each of whom independently has probability pp of making the correct decision and a third member who flips a coin for each decision (majority rules). A one-man jury has probability pp of making the correct decision. Which jury has the better probability of making the correct decision?

解析
两个陪审团作出正确判断的概率相同\boxed{\text{两个陪审团作出正确判断的概率相同}}

在三人陪审团中，两名认真陪审员都给出正确判断的概率是 p×p=p2p \times p = p^2，在这些情况下，那个靠掷硬币的人怎么投都不影响结果。

三人陪审团其余作出正确判断的情况是：两名认真陪审员意见相反，而掷硬币的陪审员恰好投给“正确”的一方。两名认真陪审员分裂的概率是 p(1−p)+(1−p)p=2p(1−p)p(1-p) + (1-p)p = 2p(1-p)。再乘以 12\frac{1}{2}，因为硬币只有一半概率站在正确一边。

因此，三人陪审团作出正确判断的总概率为 p2+p(1−p)=p2+p−p2=p,p^2 + p(1-p) = p^2 + p - p^2 = p, 这与单人陪审团的正确概率完全相同。

Original Explanation

The two juries have the same chance of a correct decision\boxed{\text{The two juries have the same chance of a correct decision}}

In the threeman jury, the two serious jurors agree on the correct decision in the fraction p×p=p2p \times p= p^2 of the cases, and for these cases the vote of the joker with the coin does not matter.

In the other correct decisions by the three-man jury, the serious jurors vote oppositely, and the joker votes with the "correct" juror. The chance that the serious jurors split is p(1−p)+(1−p)pp(1 - p) + (1 - p)p or 2p(1−p)2p(1- p). Halve this because the coin favors the correct side half the time.

Finally, the total probability of a correct decision by the three-man jury is p2+p(1−p)=p2+p−p2=pp^2 + p(1 - p) = p^2 + p - p^2 = p, which is identical with the probability given for the one-man jury.