aez-notes

Three prisoners, \(A\), \(B\), and \(C\), with apparently equally good records have applied for parole. The parole board has decided to release to of the three, and the prisoners know this but not which two. A warder friend of prisoner \(A\) knows who are to be released. Prisoner \(A\) realizes that it would be unethical to ask the warder if he, \(A\), is to be released, but thinks of asking for the name of one prisoner other than himself who is to be released. He thinks that before he asks, his chances of release are \(2/3\). He thinks that if the warder says "\(B\) will be released," his own chances have now gone down to \(1/2\), because either \(A\) and \(B\) or \(B\) and \(C\) are to be released. And so \(A\) decides not to reduce his chances by asking. However, \(A\) is mistaken in his calculations. Explain.

The warder would have said one one the names regardless of who was to be released, so it doesn't actually tell \(A\) anything about their chance of release, hence it remains \(2/3\) for \(A\). This problem has recently found fame as the Monty Hall problem.