Consider a woman who has just received a positive result from a mammogram and asks her doctor: Do I have breast cancer for sure, or what are the chances that I have the disease? In a 2007 continuing education course for gynecologists, Gigerenzer asked 160 of these practitioners to answer that question given the following information about women in the region:

- The probability that a woman has breast cancer (prevalence) is 1 percent.
- If a woman has breast cancer, the probability that she tests positive (sensitivity) is 90 percent.
- If a woman does not have breast cancer, the probability that she nonetheless tests positive (false-positive rate) is 9 percent.

What is the best answer to the patient's query?

A. The probability that she has breast cancer is about 81 percent.

B. Out of 10 women with a positive mammogram, about nine have breast cancer.

C. Out of 10 women with a positive mammogram, about one has breast cancer.

D. The probability that she has breast cancer is about 1 percent.

Gynecologists could derive the answer from the statistics above, or they could simply recall what they should have known anyhow. In either case, the best answer is C; only about one out of every 10 women who test positive in screening actually has breast cancer. The other nine are falsely alarmed. Prior to training, most (60 percent) of the gynecologists answered 90 percent or 81 percent, thus grossly overestimating the probability of cancer. Only 21 percent of physicians picked the best answer — one out of 10.

Many physicians do not know the probabilities that a person has any disease given a positive screening test — that is, the positive predictive value of that test. Nor can they estimate it from conditional probabilities such as the test's sensitivity (the probability of testing positive in the presence of the disease) and the false-positive rate. Such innumeracy causes undue fear. Months after receiving a false-positive mammogram, one in two women reported considerable anxiety about mammograms and breast cancer, and one in four reported that this anxiety affected her daily mood and functioning.

Doctors would more easily be able to derive the correct probabilities if the statistics surrounding the test were presented as natural frequencies. For example:

- Ten out of every 1,000 women have breast cancer.
- Of these 10 women with breast cancer, nine test positive.
- Of the 990 women without cancer, about 89 nonetheless test positive.

Thus, 98 women test positive, but only nine of those actually have the disease. After learning to translate conditional probabilities into natural frequencies, 87 percent of the gynecologists understood that one in 10 is the best answer. Similarly, psychologist Ros Bramwell of the University of Liverpool in England and his colleagues reported in 2006 that only one out of 21 obstetricians could correctly estimate the probability of an unborn child actually having Down syndrome given a positive test. When they were given the relevant natural frequencies, 13 out of 20 obstetricians arrived at the correct answer.