Test predictability with Bayes' Theorem

Question

Say we have a disease and a test for it. P(A :=a person has the disease)= 0.01 ( example) P( B:=test is positive | A )=0.95 Is this enough information to calculate the probability that a person has the disease given that his test came back positive? I've seen quite a lot of solved exercises taking Baye's theorem and assuming (without even mentioning it) that P(B|A') is 0.05.Is it correct? Is there any other way to find the probability or do we need to know more?

Answer 1

You need to know more.

For problems like these, it is always good to take particular examples, even extreme examples.

In what follows assume that, in Reality, $1\%$ of the population has the disease.

Example A: A person without the disease NEVER tests positive. In that case, a positive test result is absolute proof that the person actually has the disease.

Example B: A person without the disease has a $5\%$ chance of testing positive. Given our assumptions that means that $.95\%$ of the population has the disease AND tests positive for it. It also means that $4.95\%$ of the population does not have the disease BUT tests positive for it. In this case a person who tests positive will actually have the disease with probability $$\frac {.95}{4.95+.95}\sim 16.1\%$$

Example C: (caricature example) A person without the disease ALWAYS tests positive for it. In that case we still have that $.95\%$ of the population has the disease AND tests positive for it, but now we have that $99\%$ of the population does not have the disease BUT tests positive for it. In that (admittedly extreme) situation the probability that a person who tests positive will actually have the disease is $$\frac {.95}{99+.95}\sim .95\%$$