Three prisoners A, B, and C on death row know that exactly

Chapter 2, Problem 34

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Three prisoners A, B, and C on death row know that exactly two of them are going to be executed, but they do not know which two. Prisoner A knows that the jailer will not tell him whether or not he is going to be executed. He therefore asks the jailer to tell him the name of one prisoner other than A himself who will be executed. The jailer responds that B will be executed. Upon receiving this response, Prisoner A reasons as follows: Before he spoke to the jailer, the probability was 2/3 that he would be one of the two prisoners executed. After speaking to the jailer, he knows that either he or prisoner C will be the other one to be executed. Hence, the probability that he will be executed is now only 1/2. Thus, merely by asking the jailer his question, the prisoner reduced the probability that he would be executed from 2/3 to 1/2, because he could go through exactly this same reasoning regardless of which answer the jailer gave. Discuss what is wrong with prisoner As reasoning.