> a correct analysis of this problem requires you to know the guard's exact policy for what to say in every possible situation.
Just as with Monty Hall, where one ought to specify that the host knows exactly what's where, and always opens a remaining door such that it contains a goat (choosing uniformly if there's more than one such choice).
The nice thing about the variant in the article is that it is fairly well and intuitively specified.
(By the way, if the guard chooses B with probability x (rather than 1/2) in case A is pardoned, then the probability that A is pardoned is x/(1+x), which is between 0 and 1/2, but indeed is 1/3 only for x = 1/2.).
A = A is pardoned
etc.
P(A) = P(B) = P(C) = 1/3
SA = guard says A still condemned
etc.
P(SB | A) = x
P(SB | B) = 0
P(SB | C) = 1
P(SC | A) = 1-x
P(SC | B) = 1
P(SC | C) = 0
P(SA) = 0
P(SB) = (1+x)/3
P(SC) = (2-x)/3
P(A | SB) = P(SB | A) P(A) / P(SB)
= x * 1/3 / (1+x) * 3 = x/(1+x)
Just as with Monty Hall, where one ought to specify that the host knows exactly what's where, and always opens a remaining door such that it contains a goat (choosing uniformly if there's more than one such choice).
The nice thing about the variant in the article is that it is fairly well and intuitively specified.
(By the way, if the guard chooses B with probability x (rather than 1/2) in case A is pardoned, then the probability that A is pardoned is x/(1+x), which is between 0 and 1/2, but indeed is 1/3 only for x = 1/2.).