Question

A test is used to determine if people have a predisposition for the formation of a blood clot inside a blood vessel that obstructs the flow of blood through the circulatory system. It is believed that 3% of people actually have this predisposition. The test is 99% accurate if a person actually has the predisposition, meaning that the probability of a positive test result when a person actually has the predisposition is 0.99. The test is 98% accurate if a person does not have the predisposition. What is the probability that a randomly selected person who tests positive for the predisposition by the test actually has the predisposition

Answer 1

The probability that a randomly selected person who tests positive for the predisposition by the test actually has the predisposition is 0.6049.

Explanation:

We are given that it is believed that 3% of people actually have this predisposition. The test is 99% accurate if a person actually has a predisposition.

The test is 98% accurate if a person does not have a predisposition.

Let the probability that people actually have predisposition = P(PD) = 0.03

The probability that people do not have a predisposition = P(PD') = 1 - P(PD) = 1 - 0.03 = 0.97

Let A = event that the test is accurate

So, the probability that the test is accurate if a person actually has a predisposition = P(A/PD) = 0.99

The probability that the test is correct if a person actually has a predisposition = P(A/PD') = 1 - 0.98 = 0.02

Now, the probability that a randomly selected person who tests positive for the predisposition by the test actually has the predisposition = P(PD/A)

We will use Bayes' theorem to calculate the above probability.

P(PD/A) =
`(P(PD) * P(A/PD))/(P(PD) * P(A/PD) + P(PD') * P(A/PD'))`

=
`(0.03 * 0.99)/(0.03 * 0.99+0.97 * 0.02)`

=
`(0.0297)/(0.0491)` = 0.6049.