Math
posted by Cami on .
You are a policeman on a case in which you believe there is a strong suspect. Based on your overall experience, your prior knowledge of this “person of interest,” and the evidence you’ve seen against him thus far, you believe that when he claims to be innocent there is a 80% chance that he is lying. He’s given a polygraph test which you believe to be “75% reliable [1] .” The results of the polygraph test indicate, to your surprise, that he is telling the truth! Given this new information, what should you say is the probability that he is still lying?
[1] You can take this to mean that when the subject is really telling the truth there is a 75% chance that he will pass the polygraph, and when he is lying there is a 75% chance that he will fail.
Choose one answer.
a. 25% or less
b. between 25% and 50%
c. exactly 50%.
d. between 50% and 80%
e. 80% or higher

Let L be the probability he lies, on the average. Then the possible situations are:
Lying but passes test: 0.25L
Lying and fails test: 0.75 L
Telling truth and passes test: (1L)0.75
Telling truth but fails test: (1L)0.25
If reliable evidence suggests he lies 80% of the time, assume L = 0.80. Since he passed the test, the probability he is telling the truth is
(1L)0.75 = 0.375 and that he is lying is 0.625. That would satisfy answer d.
In any case it looks like there is a reasonable doubt. This is not how guilt and innocence should be established.