Three Prisoners 三個囚犯問題 - 經濟

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http://jcyang1225.blogspot.com/2009/08/three-prisoners.html


Three Prisoners 三個囚犯問題

Casella and Berger (2002) 的 Statistical Inference (2nd edition)
的 1.3 節提到了下面這個有趣的問題:

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.

請問, 典獄長和囚犯 A 的想法, 誰才是對的?


解答:

分別定義事件 A, 事件 B, 和事件 C 為囚犯 A, B 或 C 被特赦的事件,
我們知道三個事件的機率都是 1/3.

定義事件 W 為典獄長告訴囚犯 A 說囚犯 B 將會被處死的事件, 利用
條件機率的定義, 囚犯 A 可以更新他被特赦的機率.

P(A|W) = P(A and W) / P(W)

在囚犯 A 去詢問典獄長誰會被特赦之前, 可能發生的情況有以下 4 種:

事件 a1 = { "囚犯 A 被特赦" and "典獄長告訴囚犯 A 說囚犯 B 被處死"}
事件 a2 = { "囚犯 A 被特赦" and "典獄長告訴囚犯 A 說囚犯 C 被處死"}
事件 b = { "囚犯 B 被特赦" and "典獄長告訴囚犯 A 說囚犯 C 被處死"}
事件 c = { "囚犯 C 被特赦" and "典獄長告訴囚犯 A 說囚犯 B 被處死"}

其中, 很顯然的事件 b 和事件 c 發生的機率都是 1/3. 至於事件 a1
和事件 a2, 機率則分別為 1/6.

事件 W 發生的機率為 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, 典獄長的想法才是正確的.

囚犯 A 犯的錯誤, 在於他將事件 W 誤認為事件 B 的補集 (B'). 由於:

P(A|B') = P(A and B') / P(B') = (1/3) / (2/3) = 1/2

所以囚犯 A 誤認為 P(A|W) = 1/2.

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