A polygraph (lie detector) is said to be 90% reliable in the following sense: There is a 90% chance that a person who is telling the truth will pass the polygraph test, and there is a 90% chance that a person telling a lie will fail the polygraph test.
(a) Suppose a population consists of 5% liars. A random person takes a polygraph test, which concludes that they are lying. What is the probability that they are actually lying?
(b) Consider the probability that a person is actually lying given that the polygraph says that they are. Using the definition of reliability, how reliable must the polygraph test be in order that this probability is at least 80%?