A new lie detector test has been devised and must be tested beforeit is put into

Question

A new lie detector test has been devised and must be tested beforeit is put into use. One hundred people are selected at random, andeach person draws and keeps a card from a box of 100 cards. Halfthe cards instruct the to person lie and the others instruct theperson to tell the truth. The test indicates lying in 80% of thosewho lied and in 5% of those who did not. What is the probabilitythat a randomly chosen subject will have lied given that the testindicates lying? That the subject will not have lied given that thetest indicates lying?

Explanation / Answer

we'll use conditional probability: the probability of event Bgiven event A has already happened is: (probability of A and B happening) / (probability of Ahappening) In this case, we want to find the probability that a random subjectlied given that the test says he lied. Probability of subject lying and test showing he lied:(50%)*(80%)=40% Probability of test showing he lied:         - Probability of subjectlying and test showing he lied: (50%)*(80%)=40%         - Probability of subjecttelling truth but test shows he lied: (50%)*(5%)=2.5%        Total probability of testshowing he lied: 40%+2.5%=42.5% So, probability of test showing a lie and the person actually DIDlie: 40% / 42.5%=94.1176% That means probability of test showing a lie but the person DIDN'Tlie: 100%-94.1176%=5.8823%

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