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157 decision/prisoners.p




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This article is from the Puzzles FAQ, by Chris Cole chris@questrel.questrel.com and Matthew Daly mwdaly@pobox.com with numerous contributions by others.

157 decision/prisoners.p


Three prisoners on death row are told that one of them has been chosen
at random for execution the next day, but the other two are to be
freed. One privately begs the warden to at least tell him the name of
one other prisoner who will be freed. The warden relents: 'Susie will
go free.' Horrified, the first prisoner says that because he is now
one of only two remaining prisoners at risk, his chances of execution
have risen from one-third to one-half! Should the warden have kept his
mouth shut?

decision/prisoners.s

Each prisoner had an equal chance of being the one chosen to be
executed. So we have three cases:

Prisoner executed:         A    B    C
Probability of this case: 1/3  1/3  1/3

Now, if A is to be executed, the warden will randomly choose either B or C,
and tell A that name. When B or C is the one to be executed, there is only
one prisoner other than A who will not be executed, and the warden will always
give that name. So now we have:

Prisoner executed:  A    A    B    C
Name given to A:    B    C    C    B
Probability:       1/6  1/6  1/3  1/3

We can calculate all this without knowing the warden's answer.
When he tells us B will not be executed, we eliminate the middle two
choices above. Now, among the two remaining cases, C is twice
as likely as A to be the one executed. Thus, the probability that
A will be executed is still 1/3, and C's chances are 2/3.

 

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