The three prisoners problem appeared in Martin Gardner's "Mathematical Games" column in Scientific American in 1959.
[1][2] It is mathematically equivalent to the Monty Hall problem with car and goat replaced respectively with freedom and execution.
[3] Three prisoners, A, B, and C, are in separate cells and sentenced to death.
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 two who are going to be executed.
And if I'm to be pardoned, secretly flip a coin to decide whether to give me name B or C." The warden gives him B's name.
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's chance of being pardoned is unchanged at 1/3, but he is pleased because his own chance has gone up to 2/3.
The answer is that prisoner A did not gain any information about his own fate, since he already knew that the warden would give him the name of someone else.
As the warden is asked by A, he can only answer B or C to be executed (or "not pardoned").
As the warden has answered that B will not be pardoned, the solution comes from the second column "not B".
the event that the warden tells A that prisoner B is to be executed, then, using Bayes' theorem, the posterior probability of A being pardoned, is:[4] The probability of C being pardoned, on the other hand, is: The crucial difference making A and C unequal is that
This adds up to the total of 1/3 of the time (1/6 + 1/6) A is being pardoned, which is accurate.
The key to this problem is that the warden may not reveal the name of a prisoner who will be pardoned.
If we eliminate this requirement, it can demonstrate the original problem in another way.
In this case, the warden flips a coin and chooses one of B and C to reveal the fate of.
However, the warden in the original case cannot reveal the fate of a pardoned prisoner.
For example, how the question is posed to the warden can affect the answer.
[4] Using Bayes' Theorem once again: However, if A simply asks if B will be executed, and the warden responds with "yes", the probability that A is pardoned becomes: A similar assumption is that A plans beforehand to ask the warden for this information.
[5] Another likely overlooked assumption is that the warden has a probabilistic choice.
as the conditional probability that the warden will name B given that C will be executed.
, that is, that we do not take into account that the warden is making a probabilistic choice, then
However, the reality of the problem is that the warden is flipping a coin (
[5] Judea Pearl (1988) used a variant of this example to demonstrate that belief updates must depend not merely on the facts observed but also on the experiment (i.e., query) that led to those facts.