Question

3. Suppose that a particular polygraph test has 80% accuracy when taken by a guilty party, and 90% accuracy when taken by an innocent party. The police produce 10 suspects and it is knowns that two guilty parties are among them. An individual is chosen at random from among the suspects. (a) What is the probability that this individual is guilty? (b) What is the probability that this individual will “test guilty”? (c) What is the probability of true guilt, given that the individual tests guilty? (d) What is the probability that this individual will test correctly?

Answer 1

Let G be the event that a randomly selected individual is guilty.

Let I be the event that a randomly selected individual is innocent.

Let Tg be the event that a random individual taking the test is tested guilty

Let Ti be the event that a random individual taking the test is tested innovent

The polygraph test has 80% occuracy when taken by a guilty party. That is the probability that the individual will test guilty given that the person is guilty is 0.80

This also means that the probability that the individual will test innovent given that the person is guilty is 0.20

Next we know that the test has 90% accuracy when taken by an innocent party. This means the probability that the individual will test innocent given that the person is innocent is 0.90

This also means that the probability that the individual will test guilt given that the person is innocent is 0.10

a) It is known that 2 guilty parties among the 10 suspects.

Hence the probaility that a randomly selected individual from the 10 suspects is guilty is

The probability that a randomly selected individual from the 10 suspect is innocent is

b) A randomly selected individual can be tested guilty, both when the person is guilty or innocent.

The probability that a randomly selected individual will test guilty, can be written as

Using the conditional probability formula we get

c) the probability of true guilt given that the individual tests guilty is given by

d) The probability that a randomly selected individual will test correctly is the probabilty that the guilty individual is tested guilty and innocent individual is tested innovent.