三门问题

三门问题二

专题: Probability / 概率
难度: L4
来源: QuantQuestion

题目详情

Now we will ask you the same question as the previous one, except that when it comes time for the host to reveal an empty door, he instead selects someone from the audience who chooses randomly and by chance chooses a door that is revealed to be empty. Should you switch?

Note: There are two ways to interpret this question. You could assume that the game can be played repeatedly with an audience member always revealing a door to be empty, or you could assume a one- off game where the audience member (ignorant of the prize's location) just happens to have chosen an empty door. Try answering both.

解析

若“观众随机开门，且这次恰好开到空门”，则在观察到“开到的是空门”这一条件下，换与不换 胜率相同。

设你初选门 3，观众在门 1 与门 2 中等概率随机选一扇打开。观察到他打开门 2 且为空。

记事件 $E$ 为“打开门 2 且为空”。

若奖在门 3（先验 $1/3$ ），则门 1、2 都空， $\mathbb{P}(E\mid奖=3)=1/2$ 。
若奖在门 1（先验 $1/3$ ），门 2 为空， $\mathbb{P}(E\mid奖=1)=1/2$ 。
若奖在门 2（先验 $1/3$ ），门 2 不空， $\mathbb{P}(E\mid奖=2)=0$ 。

所以后验

\mathbb{P}(奖=3\mid E)=\mathbb{P}(奖=1\mid E)=\frac{1}{2}.

因此换到门 1 的胜率为 $1/2$ ，不换留在门 3 的胜率也为 $1/2$ ，换不换都一样。

（若不加“恰好开到空门”的条件，而是允许观众可能开到奖门，则游戏机制已不同，需另行建模。）