Murder case Conditional probability

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!

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WHAT I GET IS P(positive)=p+q(N-1)

P(murderer given positive)/P(positive given murderer)=P(meruderer)/P(positive)

but this gives me P(murderer given positive)=p/[N[p+(N−1)q]] therefore an extra factor of N!

WHAT I GET IS P(positive)=p+q(N-1)
This is wrong. you have to divide by N here. someone gets a positive test if
(they are the murderer (prob. 1/N) AND they then test positive (prob p). OR
(they are not the murderer (prob. (N-1)/N and they test positive (prob q.)

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!
The true positive probability is 0.8 and the false negative probability is 1 - 0.8 = 0.2 given you are guilty (P=0.01)

The true negative probability is 0.95 and the false positive probability is 1 - .95 = 0.05 given that you are innocent (P=0.99).

This is all you need to solve for the probability of finding the guilty person after k tests (or any related question). Note the probability of a false positive: (0.05)(.99)= 0.0495 is much larger than a true positive: (0.8)(.01)= 0.008

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