Lie detectors are sometimes used in criminal investigations, but they do not function perfectly. If a suspect is guilty, there is a 0.8 chance that the lie detector will detect guilt. If a suspect is not-guilty, there is a 0.2 chance that it will (wrongly) detect guilt. Assume that 20% of all suspects are guilty. Use Bayes Rule to answer the following questions:
(a) In the investigation of a single suspect, if the lie detector detects guilt, how likely is it that the suspect is guilty? Do you think that the evidence of the lie detector should be relied upon in the investigation?
(b) Now assume that a certain gang containing ten members is under suspicion of a crime. Assume that the probability of the gang being guilty of the crime is 20% (since other gangs are also under suspicion). Each member of the gang is independently subjected to the lie detector, and guilt is detected in five out of ten. How likely is it that the gang is guilty?
(c) How many of the ten gang members would need to "fail" the lie-detector in order for you to be 99% sure of the gang's guilt?