The three prisoners problem

Question

Three prisoners, A, B, and C, are held; exactly one, chosen uniformly at random, will be pardoned. Prisoner A asks the warden to name one of the other two who will be executed. The warden, who knows the outcome, truthfully names B.

Does A's probability of being pardoned change, and what is C's probability now?

Accepted Answer

This is Monty Hall wearing a different costume. A starts with pardon probability 	frac13. The subtle part is the warden's protocol: if A is pardoned he may name either B or C (say each with probability 	frac12); if B is pardoned he must name C; if C is pardoned he must name B. Condition on "warden names B": - A pardoned (prob 	frac13): names B w.p. 	frac12. Contribution 	frac13\cdot	frac12 = 	frac16. - B pardoned (prob 	frac13): never names B (B is the one pardoned). Contribution 0. - C pardoned (prob 	frac13): must name B. Contribution 	frac13\cdot 1 = 	frac13. Total probability o…

The three prisoners problem

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