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: (1-L)0.75
Telling truth but fails test: (1-L)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
(1-L)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.
To determine the new probability that the suspect is still lying given the polygraph test results, we can use Bayes' theorem. Bayes' theorem allows us to update probabilities based on new evidence.
Let's assign some variables to make the calculation easier:
- A: The suspect is lying.
- B: The polygraph test indicates the suspect is telling the truth.
We want to find P(A|B), the probability that the suspect is lying given the polygraph test results.
According to the given information:
- P(A) = 0.80 (prior probability that the suspect is lying)
- P(B|A) = 0.25 (probability that the suspect passes the polygraph if he is lying)
- P(B|not A) = 0.75 (probability that the suspect passes the polygraph if he is telling the truth)
Bayes' theorem states:
P(A|B) = (P(B|A) * P(A)) / P(B)
To calculate P(B), we need to consider both cases:
- The suspect is lying (A)
- The suspect is telling the truth (not A)
P(B) = P(B|A) * P(A) + P(B|not A) * P(not A)
P(not A) represents the probability that the suspect is telling the truth and can be calculated as 1 - P(A) = 1 - 0.80 = 0.20.
P(B) = 0.25 * 0.80 + 0.75 * 0.20 = 0.20 + 0.15 = 0.35
Plugging the values into Bayes' theorem:
P(A|B) = (0.25 * 0.80) / 0.35 = 0.20 / 0.35 = 4/7 ≈ 0.57
Therefore, the probability that the suspect is still lying given the polygraph test results is between 25% and 50%. The correct answer is b. between 25% and 50%.