There are three prisoners, A, B, and C and one guard. One of the
prisoners will be executed, and only the guard knows who the prisoner
is. The night before the execution, the guard brings food to the
prisoner A. Prisoner A is anxious about the upcoming execution and
asks the guard whether he is the unlucky inmate. The guard replies
that he is not allowed to tell the prisoner if it is he that will be
executed, but he informs the prisoner that prisoner B will be set
free. Prsioner A is greatly alrmed by this response because he
immedietly infers that the probability of his death has increased from
1/3 to 1/2. Is prisoner A mistaken or not in is calculation? Please
explain analytically and/or by direct simulation.

I would prefer simulation to prove the point but if the analytical
answer is very detailed that will be fine as well. But please explain
carefully... Thank you

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Request for Question Clarification by pinkfreud-ga on 21 Jan 2005 18:40 PST

Here is a similar problem (see problem 1.2):

Problems:
http://www.dam.brown.edu/people/geman/am171/HW1.pdf

Solutions:
http://www.dam.brown.edu/people/geman/am171/S_HW1.pdf

Is this helpful?

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Clarification of Question by asadmu-ga on 22 Jan 2005 11:49 PST

Well first thank you all for the response..
the response by pinkfrued is similar and has the same logic but the
parameters are different so it changes the problem by quite a a bit.
However, the question does has a solution as pinkfrued has indicated
through the relevant website which can be derived analytically and as
well by simulation. So the comments by mathtalk are not valid due to
1) we have assumed that the guard is telliing the truth 2) the
question is NOT a paradox 3) mathtalk is reading too deep into the
question regarding what if scenarios... The question can be solved
mathemtically albiet there are several ways to approach it with
different logic and agreed that the answers will be different. As long
as the answer makes a good case of why it is correct and it is
justified in its logic that answer can be appreciated to be valid in
its logic.

Analytically this is the way i thought about it:

Pr{a will survive}=2/3
Pr{b will survive}=2/3
Pr{A and B will survive}=(2/3)(1/2)=(1/3)
Pr(B will survive}=Pr{B will survive|C survives}Pr{C survives}+Pr{B
will survive|A survives}Pr{A will survive}=(1/2)(2/3)+(1/2)(2/3)=2/3
Pr{A will survive|B will survive}=Pr{A and B survive}/Pr{B
survives}=(1/3)/(2/3)= (1/2)

I dont know if this logic is correct and if this answer is justified.
Now i need someone to give a more indepth analysis in terms of the
logic that i have dictated. I need a more analytical answer and one
that is more "justified" in its respeonse than i have stated above. So
i need an answer that has analytical component to it and most of all a
simulation to prove that in the long run indeed that is what the case
is...If you think that my answer above is correct give me reasons why
and simulate the problem..

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Request for Question Clarification by elmarto-ga on 22 Jan 2005 16:32 PST

Hi asadmu,
Actually, your problem is equivalent to the one posted in pinkfreud's solution.

In the problem provided by pinkfreud, two of the prisoners are set
executed, and the guard is allowed to say which one of B or C is going
to be executed; whereas in your problem, two of the prisoners are to
be set free, and the guard is allowed to say which one of B or C is
going to set free. As you see, the problem is exactly the same, only
changing the words "set free" by "executed" every time they appear.

If you still have doubts, here is a "translation" of the answer
supplied by pinkfreud to fit your problem:

We want to calculate the following probability:

 Prob(A is set free | guard says B is free)
=Prob(A is set free AND guard says B is free)/Prob(guard says B is free)

Notice that setting the condition to "guard says B is free" rather
than "B is set free" is crucial to get the solution to this problem
right.

The three possible cases are that:

AB are free
BC are free
AC are free
(each of these cases has a 1/3 probability of ocurring)

If AB are to be set free, then the probability that the guard will say
that B is free is 1. The guard cannot tell A that he is to be set
free, so he will surely tell him that B is going to be set free.

If BC are to be set free, the guard could tell A that any of them will
be set free. We'll make the additional assumption that in this case
the guards chooses each one with a 1/2 probability. Therefore, the
probability that guard will tell A that B is to be set free is, in
this case, 1/2.

If AC are to be set free, then the probability that the guard says
that B is to be set free is clearly 0.

Now we can calculate the conditional probability required by the
problem. First we have to find:

Prob(A is set free AND guard says B is free)

This event can only happen in the first case (where AB are to be set
free). In the second case A is not set free and in the third case the
guard can't say that B will be set free. Since the first case has a
1/3 probability of happening, and given that in this case the guard
will say with probability 1 that B is being set free, then

Prob(A is set free AND guard says B is free) = (1/3) * 1 = 1/3

Now we must find

Prob(guard says B is free)

We've already said that in the first case the probability of this
event (guard says B is free) is 1; in the second case is 1/2 and in
the third case is 0. Since each of the cases has a 1/3 probability of
happening:

Prob(guard says B is free) = (1/3)*1 + (1/3)*(1/2) + (1/3)*0 = 1/2

Therefore,

Prob(A is set free | guard says B is free) = (1/3)/(1/2) = 2/3

So his probability is, as one supposes intuitively, unchanged by what
the guard says.

The mistake in the solution you provide is to set the condition to "B
survives" rather than "the guard says B survives". While the
probability that B survives is 2/3 (as you correctly stated), the
probability that the guard says that B survives is actually 1/2, as
I've shown above, and also as pinkfreud's solution states.

Do you find this helpful?

Regards,
elmarto

Looks like it's a variant of the Monty Hall problem. You can read a
detailed explanation here:

http://en.wikipedia.org/wiki/Monty_Hall_problem

Vladimir

I don't think the wording of the problem is tight enough that one can
"prove" anything, whether by simulation or otherwise.  Does the guard
always tell the truth?  Even if for the sake of a puzzle we assume the
guard is truthful, what latitude if any was allowed to the guard in
his responses?  If literally the prisoner who will be executed is
known to the guard, then in reality there is no "chance" involved in
whether prisoner A is that person.

I suspect the problem is designed to echo the well known Monty Hall
problem, but to present as it were the other side of the issue, since
there is no possiblity of "trading places".  The prisoner who will be
executed cannot escape their fate.

[Monty Hall Dilemma]
http://www.cut-the-knot.org/hall.shtml

However the parallel is not exact.  No issue arises as to the
likelihood of truth telling in the Monty Hall version, because the
host visually discloses one of the less desirable outcomes.  Moreover
an element of chance adheres due to the game show participant's own
"random" choices, despite Monty's knowledge of all the prize
placements.

In order to make a correct analysis of the situation with regards to
the prisoners and the guard, we need to clarify the source of our
information.  What has been stated about prisoner A's talk with the
guard MIGHT be truthfully asserted without essential modification
about the guard's interaction with each of the three prisoners the
night before the execution.  [Such an outcome would be consistent with
the hypothesis of a truthful but malicious guard, who intends to
maximize the anxiety of all three prisoners.]

That is, one might consistently suppose that the guard brings food to
each prisoner, hears each ask the same question about whether they
will be executed, and in each case replies that he's not allowed to
answer that, but instead names one of the other two prisoners as
someone who will not be executed.  [The extra assertion that prisoner
B is to "be set free" seems a bit of distraction; nothing had earlier
been stated about any of the prisoners being freed, only that one will
be executed.]  Of course, knowing that all three conversations took
place (but without knowing the specific names involved) one could
scarcely alter a uniform estimation of chances for any particular
prisoner to be executed.

Again, to me it seems incorrect to say prisoner A "infers the
probability of his death has increased".  Prisoner A may be alarmed at
his own revised estimate of the probability of his death, on the basis
of information that (unknown to him) has been biased or censored.  In
such a case the prisoner may be doing the best that can be done with
the information available (so no blame for a "mistaken...
calculation"), but observers with access to more information need not
draw the same conclusion.

regards, mathtalk-ga

Seems to be a popular question.

http://www.redhotpawn.com/board/showthread.php?threadid=11542&page=5