Solution
Slide 1 of 2
SOLUTION TO PRISONER PARADOX
Let A denote that A is to be executed.
Let B denote that B is to be executed.
Let C denote that C is to be executed.
A priori:
P(AB) = 1/3
P(BC) = 1/3
P(AC) = 1/3
Now consider the probability of the jailer saying C for each
possible scenario:
P(SaysC | AB) = 0
P(SaysC | BC) = 1/2
P(SaysC | AC) = 1
Hence
P(SaysC) = P(SaysC | AB)  P(AB)
+ P(SaysC | BC)  P(BC)
+ P(SaysC | AC)  P(AC)
= 01/3 + 1/21/3 + 11/3 = 1/2
P.D.Scott
University of Essex
Solution
Slide 2 of 2
Prisoner wants to know his chances of a pardon after the
jailer has said C
i.e. P(BC | SaysC)
By Bayes Theorem
P(BC | SaysC) = P(SaysC | BC)  ( P(BC)/ P(SaysC) )
= (1/2  1/3) / 1/2) = 1/3
So the prisoner’s chances of a pardon have not changed.
P.D.Scott
University of Essex