B How to calculate this conditional probability?

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Tony Hau
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So this question arises from an online course on Bayes Theorem and Total Probability.

The question says that
  1. there are three prisoners A, B and C. The King decides to release two of them and sentence the remaining one.
  2. Supposed that you were A and so your chance of being released is ## \frac{1}{3} ##
  3. Suppose you know the guard well. You can ask him about who will be one of the released prisoners, except yourself, i.e. B and C.
  4. Let ## G_B = ## guard says B is released, and ## X_A = ## sentence A.
  5. Now calculate ## P(G_B \mid X_A)##.

I know the answer should be ## \frac{1}{2} ##, because given that ## X_A ## is sentenced, it means that both B and C will be released and so the chance of the guard telling you B is released is one half.

However, when I try to apply ## P(G_B \mid X_A) = \frac{ P(G_B \cap X_A)}{ P(X_A)} ##, the answer I get is ##\frac{3}{4}##. I get the answer ##\frac{3}{4}## because:

Event of being sentencedEvent of guard's telling
## X_A ####G_B##
## X_A ####G_C##
## X_B ####G_C##
## X_C ####G_B##

So ## P(G_B \cap X_A) = \frac{1}{4} ## and ## P(G_B \mid X_A) = \frac{3}{4} ##. Why am I wrong?


P.S. The lecture video is here:
 
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PeroK said:
## P(G_B \cap X_A) = \frac{1}{3} ##
Can you explain why it is so? There are four events as listed in the table and so the denominator should be 4. Plus if ## P(G_B \cap X_A) = \frac{1}{3} ##, then the resulting probability of ## P(G_B \mid X_A) = 1 ##
 
Tony Hau said:
Can you explain why it is so? There are four events as listed in the table and so the denominator should be 4. Plus if ## P(G_B \cap X_A) = \frac{1}{3} ##, then the resulting probability of ## P(G_B \mid X_A) = 1 ##
Okay, I got confused. I thought two were condemed, but I see two are released. Let me look at it again.
 
This is tricky. The equally likely events are ##X_A, X_B, X_C##, each with a probability ##1/3##. If ##X_A##, then the guard tells A about B or C with equally probability, so:
$$P(X_A \cap G_B) = P(X_A \cap G_C) = 1/6$$In the other cases, the guard has no choice, so:
$$P(X_B \cap G_C) = P(X_C \cap G_B) = 1/3$$So, the four events you listed are not equally likely.
 
PS you are asked to calculate ##P(G_B|X_A)##, which is fairly meaningless. Note that this is not ##P(X_A|G_B)##, which is the probability A is condemed, given that the guard says that B is to be released. Which is not the same as ##P(X_A|\neg X_B)##, which is the probability that A is condemed given that B is released.

It's early Sunday morning. I should have had another cup of coffee before looking at this.
 
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PPS You can't really calculate ##P(G_B|X_A)##, as it's an assumption of the problem that this is 1/2. I.e. in order to calculate anything we have to assume this. You can, however, verify Bayes' theorem in this case.

I'm starting to wake up!
 
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Tony Hau said:
There are four events as listed in the table
The probability of those four events are not equal. Listing a table and counting is only a viable strategy if all of the listed events are equiprobable.
 
Thank you for all your responses. Let me think about it more carefully, to be honest this is my first time learning about the term "equiprobable".
 
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