12.1 Suppose there are 1000,000 people in a given population and 1000 of these people carry a certain genetic mutation. Andsuppose there's test for this mutation that is 95% accurate in the sense that 95% of those who have the mutation will test positive, and only 5% of those who do not have the mutation will test positive. If a test for detection for mutation is carried out and result is positive, What is the probability that there is really mutation.

This is a question that can be solved using Bayes’ theorem, which is a formula that relates the conditional probabilities of two events. Bayes’ theorem can help us update our probabilities given new information1
Let us define some symbols for this problem:
Let A be the event that a person has the mutation.
Let B be the event that a person tests positive for the mutation.
We are given that P(A) = 1000/1000000 = 0.001, which is the probability of having the mutation in the population.
We are also given that P(B|A) = 0.95, which is the probability of testing positive given that the person has the mutation.
We are also given that P(B|A’) = 0.05, which is the probability of testing positive given that the person does not have the mutation.
We want to find P(A|B), which is the probability of having the mutation given that the person tests positive. Using Bayes’ theorem, we can write:
P(A∣B)=P(A)P(B|A) / P(B)

To find P(B), we can use the law of total probability, which states that:

P(B)=P(B∣A)P(A)+P(B∣A’)P(A)

Plugging in the values we have, we get:

P(B)=0.95×0.001+0.05×0.999=0.0509

Therefore, the probability of having the mutation given that the person tests positive is:

P(A∣B)= 0.001×0.95 / 0.0509 ≈0.0187

This means that there is only about a 1.87% chance that the person really has the mutation, even though the test is positive. This is because the mutation is very rare in the population, and the test has a relatively high false positive rate. I hope that helps