Why a Positive Mammogram Is Mostly a False Alarm

Suppose 1% of women aged 40 in routine screening have breast cancer. The mammogram detects 80% of actual cancers. It also gives a positive result for 9.6% of women without cancer. A woman in this age group tests positive. What is the probability she has cancer?

Most people, including physicians, answer 70 to 80 percent. The correct answer, from Bayes' theorem, is about 7.8 percent.

The arithmetic. Out of 1,000 women: 10 have cancer; about 8 of them test positive. Of the 990 without cancer, about 9.6%, or 95, also test positive. Of the 103 positive results, only 8 are actual cancers. 8/103 ≈ 0.078.

This is the base-rate fallacy: people anchor on the test's accuracy and skip past the prior probability of having the disease in the first place. When the prior is small — 1 in 100 here — even a fairly accurate test produces more false positives than true positives, because there are so many more healthy people to test.

Ward Casscells, Arno Schoenberger, and Thomas Grayboys ran the canonical study at Harvard Medical School in 1978. They asked 60 students and staff a version of this problem with a 1-in-1,000 base rate. Only 11 of 60 got the right answer of about 2%. The modal answer was 95% — the test's specificity, treated as if it were the predictive value.

The fix is not new math, just remembering to apply old math. P(disease | positive) = P(positive | disease) · P(disease) / P(positive). The denominator is what people forget. Gerd Gigerenzer has shown that the same problem stated in natural frequencies — 8 out of every 10 women with cancer test positive; of every 990 women without cancer, 95 test positive — is solved correctly by most people without any training. The arithmetic is identical. The framing does the work.