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## Main Question or Discussion Point

Guys my question should be easy but I cannot understand how it works, here is how it goes.

A murder case involves the death of a guest at a large party in a country house. The police are certain that there is only one murderer who is among the N = 100 remaining people in the house. However, they have no evidence at all so that they suspect everyone equally. They decide to use a lie detector test on everyone. Only the murderer lies. If an interviewee lies during the test, the lie detector will return a positive result (i.e. correctly identify the interviewee as a liar) with a probability of p = 80%. However, the lie detector also has a q = 5% chance of incorrectly identifying an honest interviewee as a liar.

(a) How many people are expected to fail the lie detector test?

(b) Mickey is the first to be interviewed. He fails the test. Show that the probability that Professor Plum is guilty is given by p/[p+(N−1)q]. Evaluate this quantity using the parameters given.

(c) One can attempt to design a more discriminating lie detector. What should be the condition for the ratio q/p if the goal is a probability of at least 75% that an interviewee is guilty if he/she fails the test. Comment on whether it is more important to improve the parameter p, q or both.

cheers!

A murder case involves the death of a guest at a large party in a country house. The police are certain that there is only one murderer who is among the N = 100 remaining people in the house. However, they have no evidence at all so that they suspect everyone equally. They decide to use a lie detector test on everyone. Only the murderer lies. If an interviewee lies during the test, the lie detector will return a positive result (i.e. correctly identify the interviewee as a liar) with a probability of p = 80%. However, the lie detector also has a q = 5% chance of incorrectly identifying an honest interviewee as a liar.

(a) How many people are expected to fail the lie detector test?

(b) Mickey is the first to be interviewed. He fails the test. Show that the probability that Professor Plum is guilty is given by p/[p+(N−1)q]. Evaluate this quantity using the parameters given.

(c) One can attempt to design a more discriminating lie detector. What should be the condition for the ratio q/p if the goal is a probability of at least 75% that an interviewee is guilty if he/she fails the test. Comment on whether it is more important to improve the parameter p, q or both.

cheers!