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

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.