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Probability puzzle 1

November 10, 2011

Here is an interesting puzzle in probability theory (again from reading Judea Pearl’s book):  Three prisoners A, B, and C, are awaiting news about the outcome of a trial.  They are told that on the morrow one will be executed, and two will walk free.  However, only the warden is told which one of the prisoner is to go to the gallows.  The warden is also instructed not to reveal this information to the prisoners.  During the night prisoner A asks the warden to give a letter to whomever of the pair B or C will not be executed so that he can deliver it to his family.  The warden does so, and when he returns prisoner A asks who the letter was delivered to.  He explains to the warden that this will not provide any new information, since he already knows that either one or both of B, C will walk free.  Revealing one of them will not provide any new information to prisoner A.

The warden considers, and tells A that the letter was delivered to B. Prisoner A now thinks: “Well, it’s either C or me who will die tomorrow – my chance of dying has just increased to 50%! But I shouldn’t have received any more information from the warden.  What did I do wrong?”

This puzzle is actually equivalent to the Monty Hall problem (Wikipedia has an excellent discussion of that problem – perhaps one of the best entries I have seen so far).  Think of the prisoners as doors, being executed as “the prize”, and the warden as Monty Hall.  The probability of being chosen at the start is exactly 1/3 for prisoners A, B, and C – and it is 2/3 for the pair B, C.  The probability of the pair B, C is not altered by revealing that B was not chosen.  Thus A’s reasoning that his chance of dying increased to 50% is wrong – his chance is still 1/3.  However, he does know that the chances of C dying are now 2/3 (since the pair B, C has probability 2/3, but we know that B will walk free, leaving only C).

But here is a subtle point:  If prisoner A had asked outright “Will prisoner B die?” the reasoning above would have been correct.  In that case prisoner A receives actual information, if the warden answers the question.  To see that the situations are different think about this: In the case of the letter the warden had the choice of delivering the letter to B or C.  Thus when asked which one of the two will be released he had the choice of answering B or C, if both were innocent.  When faced with a direct question, such as “Will B walk free?” the warden does not have this choice in answering.   Thus the context in which the answer was obtained is of essential importance here.

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