Bayes’ theorem and rare disease

Bayes' Theorem is used to reverse the direction of conditioning. Suppose we want to ask what's the P(A|B) but we know it in terms of P(B|A). 

So we can write the P(A|B) = P(B|A) P(A) / P(B|A) P(A) + P(B| not A) P(not A)

This is same as P(A and B) / P(B)

This example is from an early test for HIV antibodies known as the ELISA test in North America.

Just for the example sake, I have replaced HIV with Covid19.

It's because this is a rare disease (see the probability of Covid19 in the screenshot ) and 

this is actually fairly common a problem for rare diseases. The number of false positives, 

greatly outnumbers the true positives because it's a rare disease. So even though the test is very accurate, we get more false positives than we get true positives. This obviously has important policy implications for things like mandatory testing. It makes much more sense to test in a sub population where the prevalence of Covid19 is higher, rather than in a general population where it's quite rare.