مسألة السجناء الثلاثة

The Three Prisoners problem appeared in Martin Gardner's Mathematical Games column in Scientific American in 1959. It is mathematically equivalent to the Monty Hall problem with car and goat replaced with freedom and execution respectively, and also equivalent to, and assumedly based on, Bertrand's box paradox.

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 Problem

Three prisoners, A, B and C, are in separate cells and sentenced to death. The governor has selected one of them at random to be pardoned. The warden knows which one is pardoned, but is not allowed to tell. Prisoner A begs the warden to let him know the identity of one of the others who is going to be executed. "If B is to be pardoned, give me C's name. If C is to be pardoned, give me B's name. And if I'm to be pardoned, flip a coin to decide whether to name B or C."

The warden tells A that B is to be executed. Prisoner A is pleased because he believes that his probability of surviving has gone up from 1/3 to 1/2, as it is now between him and C. Prisoner A secretly tells C the news, who reasons that A still has a chance of 1/3 to be the pardoned one, but his chance has gone up to 2/3.

What is the correct answer? Prisoner C is right, A's probability of surviving is still 1/3, but prisoner C's probability of receiving the pardon is 2/3.

 Solution

The answer is he didn't gain information about his own fate. Prisoner A, prior to hearing from the warden, estimates his chances of being pardoned as 1/3, the same as both B and C. As the warden says B will be executed, it's either because C will be pardoned (1/3 chance) or A will be pardoned (1/3 chance) and the B/C coin the warden flipped came up B (1/2 chance; for a total of a 1/6 chance B was named because A will be pardoned). Hence, after hearing that B will be executed, the estimate of A's chance of being pardoned is half that of C. This means his chances of being pardoned, now knowing B isn't, again are 1/3, but C has a 2/3 chance of being pardoned.

 انظر أيضاً

• Prisoner's dilemma, a game theory problem

 الهامش

• Frederick Mosteller: Fifty Challenging Problems in Probability. Dover 1987 (reprint), ISBN 0-486-65355-2, p. 28-29 (restricted online version, p. 28, في كتب گوگل)
• Richard Isaac: Pleasures of Probability. Springer 1995, ISBN 9780387944159, p. 24-27 (restricted online version, p. 24, في كتب گوگل)