Murder case Conditional probability

[dede4metal]

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!

 

[dede4metal]

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!

 

[willem2]

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.)

 

[SW VandeCarr]

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

 

[blue_raver22]

I can explain the concept of conditional probability and how it applies to this murder case.

Conditional probability is the probability of an event occurring given that another event has already occurred. In this case, the event is failing the lie detector test and the condition is being guilty of murder.

(a) To determine the number of people expected to fail the lie detector test, we can use the given information that the probability of failing the test is 80% for the guilty person and 5% for an innocent person. Using the formula for expected value, we get:

Expected number of people failing the test = N * p = 100 * 0.80 = 80

Therefore, we can expect 80 people to fail the test.

(b) To calculate the probability that Professor Plum is guilty given that Mickey has failed the test, we can use Bayes' Theorem:

P(Professor Plum is guilty | Mickey fails the test) = P(Mickey fails the test | Professor Plum is guilty) * P(Professor Plum is guilty) / P(Mickey fails the test)

= 0.80 * 1/100 / (0.80 * 1/100 + 0.05 * 99/100)

= 0.80 / (0.80 + 4.95)

= 0.80 / 5.75

= 0.139

Therefore, the probability that Professor Plum is guilty given that Mickey has failed the test is 0.139 or approximately 14%.

(c) To design a more discriminating lie detector, we need to consider the ratio of q/p. This ratio determines the likelihood of an innocent person being falsely identified as guilty. In order to have a probability of at least 75% that an interviewee is guilty if they fail the test, the ratio q/p must be less than or equal to 0.25. This means that the probability of incorrectly identifying an innocent person as guilty (q) should be no more than one-fourth of the probability of correctly identifying a guilty person (p).

In terms of improving the parameters p and q, it is important to focus on both as they both play a crucial role in accurately identifying the guilty person. However, if we had to prioritize, it would be more important to improve the parameter p as it directly affects the probability of correctly identifying the guilty person. A lower value of q is also desirable, but it may not have as