A new lie-detector test has been devised and must be tested before it is put int

Question

A new lie-detector test has been devised and must be tested before it is put into use. Two hundred people are selected at random, and each person draws and keeps a card from a box of 200 cards. Half the cards instruct the person to lie and the others instruct the person to tell the truth. Of those who lied, 85% fail the new lie-detector test (that is the test indicated lying). Of those who told the truth, 5% failed the test.

A) What is the probability that a randomly chosen subject will have lied given that the subject failed the test?

B) What is the probability that the subject will not have lied given that the subject failed the test?

(Round to four decimal places for questions A and B)

Explanation / Answer

probability that subject failed the test =P(cards instruct the person to lie &  fail the new lie-detector test+ instruct the person to tell the truth & failed the test) =(1/2)*0.85+(1/2)*0.05=0.45

therefore

a) probability that a randomly chosen subject will have lied given that the subject failed the test

=P(cards instruct the person to lie &  fail the new lie-detector test)/P(faled the test)=(1/2)*0.85/0.45=0.9444

b)

probability that the subject will not have lied given that the subject failed the test

=P(instruct the person to tell the truth & failed the test)/P(faled the test)=(1/2)*0.05/0.45 =0.0556

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