*In a hospital, the probability that a patient has Disease M is $0.003$. A test is carried out for Disease M. This test has a probability p of giving a positive result if patient has Disease M and the same probability p of giving a negative result if patient does not have Disease M. A patient is given the test. Taking p to be $0.897$, find the probability that the result of the test is positive.
This problem can be very easily solved with the aid of a tree diagram. What I have a problem with is this:
To find the p(test positive), I simply take $0.003p + 0.997(1-p)$, and of course I'll then use $p = 0.897$. I know this in an extremely mechanistic manner and I feel like I don't truly understand it intuitively. What I'm finding now is p(patient has disease and test positive) + p(patient does not have disease and test positive). Why does this automatically give me p(test positive)? Why are they equivalent? Can somebody explain in a way which would allow me to understand this intuitively? Ridiculous as it may sound, why can't I just say, the probability of the test being positive is simply p + (1-p), i.e. disregarding whether the patient actually has the disease or not? I mean, the question asks for the probability of the test being positive - it didn't ask about the status of the patient!