The Book of Why: The New Science of Cause and Effect (Penguin Science)

by Judea Pearl

Why a weighted average? Because there are many more healthy women (~D) than women with cancer (D). In fact, only 1 in 700 women has cancer, and the other 699 do not, so the probability of a positive test for a randomly chosen woman should be much more strongly influenced by the 699 women who don’t have cancer than by the one woman who does. Mathematically, we compute the weighted average as follows: P(T) = (1/700) × (73 percent) + (699/700) × (12 percent) ≈ 12.1 percent. The weights come about because only 1 in 700 women has a 73 percent chance of a positive test, and the other 699 have a 12 percent chance. Just as you might expect, P(T) came out very close to the false positive rate. Now that we know P(T), we finally can compute the updated probability—the woman’s chances of having breast cancer after the test comes back positive. The likelihood ratio is 73 percent/12.1 percent ≈ 6. As I said before, this is the factor by which we augment her prior probability to compute her updated probability of having cancer. Since her prior probability was one in seven hundred, her updated probability is 6 × 1/700 ≈ 1/116. In other words, she still has less than a 1 percent chance of having cancer. The conclusion is startling. I think that most forty-year-old women who have a positive mammogram would be astounded to learn that they still have less than a 1 percent chance of having breast cancer.