Important Bayes' Theorem question

Question

A blood test for a certain disease tests positive 90% of the time, when the disease is present. The same test shows up positive 2.5% of the time, when the disease is not present. 2% of the population has the disease. If a patient tests positive for the disease, what is the probability that he/she actually has the disease?

a. 90%

b. 57.6%

c. 42.4%

d. 1.8%

Answer 1

The total probability of the test returning positive is 0.9(0.02) + 0.25(0.98), of which only the first term is applicable if patient has the disease. So the probability is 0.9(0.02)/ [0.9(0.02) + 0.025(0.98)] = 0.018/(0.018+0.0245).

By inspection, this is slightly less than 50%, so the only possible answer among the ones given (and, in fact, the actual answer) is 42.4%, which is answer C.