Tony Hau
- 107
- 30
So this question arises from an online course on Bayes Theorem and Total Probability.
The question says that
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:
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:
The question says that
- there are three prisoners A, B and C. The King decides to release two of them and sentence the remaining one.
- Supposed that you were A and so your chance of being released is ## \frac{1}{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.
- Let ## G_B = ## guard says B is released, and ## X_A = ## sentence A.
- 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 sentenced | Event 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:
Last edited: