网志好读版: (包括延伸讨论) http://jcyang1225.blogspot.com/2009/08/three-prisoners.html Three Prisoners 三个囚犯问题 Casella and Berger (2002) 的 Statistical Inference (2nd edition) 的 1.3 节提到了下面这个有趣的问题: Three prisoners Three prisoners, A, B, and C, are on death row. The governer decides to pardon one of the three and chooses at random the prisoner to pardon. He informs the warden of his choice but requests that the name be kept secret for a few days. The next day, A tries to get the warden to tell him who had been pardoned. The warden refuses. A then asks which of B or C will be executed. The warden thinks for a while, then tells A that B is to be executed. Warden's reasoning: Each prisoners has a 1/3 chance of being pardoned. Clearly, either B or C must be executed, so I have given A no information about whether A will be pardoned. A's reasoning: Given that B will be executed, then either A or C will be pardoned. My chance of being pardoned has risen to 1/2. 请问, 典狱长和囚犯 A 的想法, 谁才是对的? 解答: 分别定义事件 A, 事件 B, 和事件 C 为囚犯 A, B 或 C 被特赦的事件, 我们知道三个事件的机率都是 1/3. 定义事件 W 为典狱长告诉囚犯 A 说囚犯 B 将会被处死的事件, 利用 条件机率的定义, 囚犯 A 可以更新他被特赦的机率. P(A|W) = P(A and W) / P(W) 在囚犯 A 去询问典狱长谁会被特赦之前, 可能发生的情况有以下 4 种: 事件 a1 = { "囚犯 A 被特赦" and "典狱长告诉囚犯 A 说囚犯 B 被处死"} 事件 a2 = { "囚犯 A 被特赦" and "典狱长告诉囚犯 A 说囚犯 C 被处死"} 事件 b = { "囚犯 B 被特赦" and "典狱长告诉囚犯 A 说囚犯 C 被处死"} 事件 c = { "囚犯 C 被特赦" and "典狱长告诉囚犯 A 说囚犯 B 被处死"} 其中, 很显然的事件 b 和事件 c 发生的机率都是 1/3. 至於事件 a1 和事件 a2, 机率则分别为 1/6. 事件 W 发生的机率为 1/6 + 1/3 = 1/2, 所以: P(A|W) = P(A and W) / P(W) = (1/6) / (1/2) = 1/3 因此, 典狱长的想法才是正确的. 囚犯 A 犯的错误, 在於他将事件 W 误认为事件 B 的补集 (B'). 由於: P(A|B') = P(A and B') / P(B') = (1/3) / (2/3) = 1/2 所以囚犯 A 误认为 P(A|W) = 1/2. --

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