The Mammogram Puzzle

A woman gets a mammogram. The test catches 90% of real cancers and has a 9% false positive rate. Breast cancer prevalence is about 1%. She tests positive. What is the chance she actually has cancer?

Most doctors guess 80 to 90%. The real answer is about 9%. Here is why: out of 1,000 women, 10 have cancer, and the test catches 9 of them. Of the 990 healthy women, about 89 get false positives. So only 9 of about 98 positive results are true, which is 9.2%.

Studies of doctors reading these numbers, popularized by Gerd Gigerenzer, found that most physicians get this badly wrong.