Three prisoners on death row are told that one of them has been chosen at random for execution the next...

decision/prisoners Added material

Problem

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?

Show solution

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.

Additional material

[ Editor: the rec-puzzles.org wiki version of this solution (2006)
extends the analysis above to other ways the warden might choose the
name. ]

The answer depends upon how the warden chooses the name given to the
prisoner.

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

Let A be the prisoner that is being told the name of the other prisoner
that will go free and B is the name that is given to the A.  Let's look
at three possibilities for the warden's method of choosing the name to
give to A.

1.  The warden chooses randomly between B and C when both will go free.
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

When the warden says 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.

2.  The warden always chooses B when B and C will go free.  We have:

Prisoner executed:  A    A    B    C
Name given to A:    B    C    C    B
Probability:       1/3   0   1/3  1/3

Again we eliminate the middle two choices above.  Now, among the two
remaining cases, C is just as likely as A to be the one executed.  Thus,
the probability that A will be executed jumps to 1/2, and C's chances
are 1/2.

3.  The warden always chooses C when B and C will go free.  We have:

Prisoner executed:  A    A    B    C
Name given to A:    B    C    C    B
Probability:        0   1/3  1/3  1/3

Again we eliminate the middle two choices above.  Now, among the two
remaining cases, C is sure to be the one executed.  Thus, the
probability that A will be executed drops to 0, and C's chances are
100%.

So we see that the prisoner needs to know something about the warden's
thinking in order to conclude anything.

see also Monty Hall [ Editor: entry decision/monty.hall ]

-- Chris Cole, rec-puzzles.org wiki