Question

A rare genetic disease is discovered. Although only one in a million people carry it, you consider getting screened. You are told that the genetic test is extremely good; it is 100% sensitive (it is always correct if you have the disease) and 99.99% specific (it gives a false positive result only 0.01% of the time). Having recently learned Bayes' theorem, you decide not to take the test. Why?

Answer 1

Because the probability of not having the disease, even when we get a positive test, is very close to 100%, so the test doesn't give enough information and it is useless to take it.

Explanation:

We want to justify why it is not convenient to take the genetic test to know if we have the disease or not.

According to the Bayes theorem, the probability of not having the disease (h), given that we get a positive test (P) can be expressed as:

`P(h|P)=(P(P|h)*P(h))/(P(PT))`

To calculate the probability of having a positive test we have to add the probability of a positive test (100% accuracy) on the proportion of the population that has the disease (1 in 1,000,000) and the proportion of false positives (0.01%) on the healthy population (999,999 in 1,000,000).

It can be written as:

`P(P)=P(P|not\,h)*P(not\,h)+P(P|h)*P(h)\\\\P(P)=1*0.000001+0.01*0.999999=0.000001+0.00999999=0.01`

We can calculate now the probability of getting a positive test even when we are healthy:

`P(h|P)=(P(P|h)*P(h))/(P(PT))=(0.01*0.999999)/(0.01)=0.999999\approx1`

Then, the probability of being healthy even when we get a positive test is very close to 100%, so the test doesn't give enough information and it is useless to take it.

We could have conclude that earlier looking at the ratio between false positives and correct tests (the former being much greater).