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The prisoner's dilemma. The release of two out of three prisoners has been announced. but their identity is kept secret. One of the prisoners considers asking a friendly guard to tell him who is the prisoner other than himself that will be released, but hesitates based on the following rationale: at the prisoner's present state of knowledge, the probability of being released is 2/3, but after he knows the answer, the probability of being released will become 1 /2, since there will be two prisoners (including himself) whose fate is unknown and exactly one of the two will be released. What is wrong with this line of reasoning

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Answer:

we conclude that there are no changes in the conditional probability of being released

Step-by-step explanation:

With the given exercise we know that we nominate as A, B and C as the probability that 3 prisoners are released and A has a particular factor which is the friendly guard

the probability of one being released is 1/3 by the following pairs: AB, BC, AC

we know that the guard tells B that he is released

P (B) = P (A and B are being released and the guard has to tell him that B is released) + P (B and C are being released and the guard can tell that one of B or C is being released)

Let's get the following equation

= P (AB) * P (B | AB) + P (BC) * P (B | BC)

we replace the data defining that

= (1/3) * (1) + (1/3) * (1/2) = 1/2

we focus on finding the result

therefore P (A is released since B is released) = P (A | B) = P (AB) / P (B) = (1/3) / (1/2) = 2/3

we conclude that there are no changes in the conditional probability of being released.

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