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# stats question: The Prisoner’s Dilemma watch

1. Hi,

I really need help with this question

Three prisoners A, B, C are in solitary confinement under sentence of death, but each knows that one of them,
chosen at random with equal probability, is to be pardoned. Prisoner A begs the governor to tell him whether he,
A, is to be pardoned or executed. The governor refuses to answer this, but he does say that B is to be executed.
The governor thinks that he isn’t giving useful information, as A knows that at least one of B and C must die.
A suddenly feels much happier, as he believes his chances of being pardoned have risen from 1/3 to 1/2. The
governor, who, if A were actually to be pardoned, would be equally likely to give C’s name rather than B’s, is
mystified by A’s euphoria. Who is correct?

Let A, B, C be the events that A, B or C respectively are pardoned. Then A, B, C partition Ω. Now
let be the event that the governor tells A that B is to be executed. I want P(A | ), so I need to consider the
three conditional probabilities of given A, B and C respctively, and then use Bayes Theorem.

i know P(A)=P(B)=P(C)=

And P()=P(|A)P(A)+P(|B)P(B)+P(|C)P(C)

Now here is the problem i have no idea how to calculate

P(|A), P(|B), P(|C)

i assume P(|A)= 0.5
and P(|B)=0
and here is were i need help. I am unsure as to what probability i should assaign to P(|C). I know its either 1 or 0.5 but i have no justifcation for using either of them.

Can some one please explain which one i should use

Thanks

Reeper

Sorry for the long post
2. If C was going to be pardoned, the governor wouldn't be able to tell A that 'C will be executed' as he would be lying. The governor also would not be able to say that 'A will be executed' as he is refusing to answer A's question of whether or not A would be executed. This means that the governor would have to say 'B will be executed'.

Therefore, if C will be pardoned, the governor will certainly say that B will be executed, so P(|C) = 1.
3. This isn't the prisoner's dilemma, the prisoner's dilemma is a problem in which two players both choose an option (A) which is worse for both of them than option (B) which they could also choose.
4. (Original post by ttoby)
If C was going to be pardoned, the governor wouldn't be able to tell A that 'C will be executed' as he would be lying. The governor also would not be able to say that 'A will be executed' as he is refusing to answer A's question of whether or not A would be executed. This means that the governor would have to say 'B will be executed'.

Therefore, if C will be pardoned, the governor will certainly say that B will be executed, so P(|C) = 1.
thanks mate thats perfect
5. Consider this scenario:

The govenor is a bit of a sadist, and basically wants to confuse A. So,

If A and B will be executed, he tells him that B will be executed.
In all other scenarios, he says nothing. (Note that this satisfies the "A is pardoned" case: the probability of him naming B or C is 0).

So (unbeknownst to A), it's actually terrible news for him to hear that B will be executed!

I can't see how this is inconsistent with what we're told in the problem, but perhaps I'm missing something.
6. This is the Monty Hall problem, not the prisoner's dilemna.

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