I was reading this example about finding the probability of one actually getting the disease, after he is diagnosed positive by a test, as he is determining whether or not to proceed with treatment as the treatment has great side effects.
Let's say P(A) = Probability of getting the disease and P(B) = Probability of getting tested positive
In the example, in order to find the probability of one actually getting the disease when he is tested positive is a conditional probability of P(A|B). However, in my opinion, this situation can also be represented by P(A and B). To me, P(A and B) means the probability of getting the disease AND ALSO getting tested positive. Isn't this also an accurate representation on what the question wants?
P(A|B) means the chance of him really infected by the disease given that he has been tested positive. But P(A and B) means that he is both infected by the disease as well as being tested positive. However, the calculated values of P(A|B) and P(A and B) are quite different..
I'd really appreciate any input here as I'm trying to get my basic concepts in probability right.