Many organizations (e.g., the CIA) hiring employees for sensitive positions give

Question

Many organizations (e.g., the CIA) hiring employees for sensitive positions give lie-detector to prospective employees. Most people don't have a reason to lie, so that the probability that a subject is telling the truth on the test is 80%. If a subject is lying then the test is fairly effective, and detects the lie (returns a "positive") 88% of the time, according to studies If a person is not lying, the studies show that the test correctly returns a "negative" with probability 86%. What is the probability that a subject is lying given the lie-detector test is positive (if we take the results of the studies as facts)? For notational ease, in your answer please let L be the event that the subject is lying and L be the event that the subject is telling the truth. Let N be the event that the lie-detector test comes up Negative (indicates that the subject is telling the truth) and N be the event that the test comes up positive (indicates that the subject is lying)

Explanation / Answer

Ans:

Given that

P(telling truth)=P(Lc)=0.8

P(telling lie)=P(L)=1-0.8=0.2

P(detect lie/telling lie)=P(Nc/L)=0.88

P(detect truth/telling truth)=P(N/Lc)=0.86

P(detect lie)=P(detect lie/telling lie)*P(telling lie)+P(detect lie/telling truth)*P(telling truth)

=0.88*0.2+(1-0.86)*0.8=0.288

We have to find:

P(telling lie/detect lie)=P(L/Nc)=P(detect lie/telling lie)*P(telling lie)/P(detect lie)

=(0.88*0.2)/0.288=0.6111

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