Probability of three prisoner.

Question

Suppose that prisoners A, B, and X are on death row, and one of them has been
chosen at random (with equal probability) to be executed the next day. Prisoner
X asks the guard to tell him which of A and B will
not
be executed. The guard
gures that he gives no additional information about whether or not prisoner X will
be executed, so he decides to tell prisoner X that A will not be executed. (Assume
the guard is telling the truth.) Under each of the following scenarios, use Bayes
Theorem to calculate the conditional probability that X will be executed given that
the guard tells X that A will not be executed.
(a) Assume that if X is the one that will be executed, the guard will tell X either
A or B will not be executed with equal probability.
(b) Assume that if X is the one that will be executed, the guard will tell X that A
will not be executed with probability 1.
(c) Assume that if X is the one that will be executed, the guard will tell X that B
will not be executed with probability 1

Answer 1

P(X/~A)P(~A) = P(~A/X)P(X)

P(X) = P(X/~A)P(~A)/P(~A/X) = (1/2)(1)/(1) = 1/2

It's very similar to Monty Hall's 3 curtain paradox

though. If the guard is like Monty and selects only an inmate who is not being executed, it's like Monty selecting only a curtain with no prize. That means P(X) = 1/3, not 1/2. But if the guard just randomly checked and told X that A is not being executed, P(X) = 1/2

There's also a "Sleeping Beauty" paradox where you can use the same Bayes Theorem equations.

Oddly mathematicians seem certain Monty Hall's paradox's solution is switch curtains when Monty gives you a clue. While a virtual similar paradox with Sleeping Beauty there is no consensus, just a very heated argument with sides certain they're right.