I am reading an essay about common statistical errors, and they give this example to illustrate confusion over conditional probabilities:
Now I understand the concept of the problem and how people made an error in assessing the probability. The stupidity of the average physician aside, how did they arrive at the correct probability being “about 7.5%?” I’m thinking it should be almost 10%. Is this just a case of extreme rounding or am I missing something?
If we have a 0.1 probability of a benign tumor returning a result of malignant, and a 0.8 probability of a malignant tumor returning a result of malignant, and a 0.01 chance that the tumor is actually malignant, then shouldn’t the probability that it is actually malignant in light of the test result be (0.1 * 0.99) - (0.8 * 0.01) = 0.091?
Right, the conditional probability of A given B [aka, P(A | B)] is the same as the probability of (A and B) divided by the probability of B. This tells us that P(actually malignant | test says malignant) = P(actually malignant and test says malignant)/P(test says malignant) = P(actually malignant and test says malignant)/(P(actually malignant and test says malignant) + P(not actually malignant but test says malignant)). Then expanding out these probabilities using the provided data gives you the answer you wrote.
Put another way, which is perhaps more intuitively obvious though of course entirely equivalent, the probability of two events occurring is the probability of the first one * the conditional probability of the second one given the first one