Question

A new lie-detector machine is invented and tested. When test subjects lie, the machine catches 86% of them in a lie. When test subjects tell the truth, the machine thinks that 6% of them lie. A local police department begins to use this lie-detector machine in interrogations. Suppose that 15% of people arrested lie in their interrogations. The police interrogated a subject and the machine indicated that the person was telling the truth. What is the probability that the subject was really lying?

Answer 1

P(machine detects lie I person lies ) = 0.86

P(machine detects lie  I person does not lie ) = 0.0.06

P(person lies) =0.15

To find

P(person lies I machine does not detect lie ) = ?

Using Bayes' theorem

P(person lies I machine does not detect lie )

= P(machine does not detect lie I Person lies ) .P(person lies)/ P(machine does not detect lie )

P (machine does not detect lie I Person lies ) = 1- P(machine detects lie I person lies )

= 1-0.86

= 0.14

P(machine does not detect lie ) = 1 - P(machine detects a lie )

=1-{ P(machine detects lie I person lies ).P(person lies)+P(machine detect lie I Person does not lie).P(Person does not lie)}

= 1- (0.86 *0.15 + 0.06 *0.85 ) = 0.82

  

Therefore ,

P(person lies I machine does not detect lie ) = 0.14 * 0.15 /0.82

= 0.0256 or 2.56%

Probability that the subject was lying given that the machine indicated that the person was telling truth is 0.01256 or 2.56%