Innumeracy

In one study, Gigerenzer and his colleagues asked doctors in Germany and the United States to estimate the probability that a woman with a positive mammogram actually has breast cancer, even though she’s in a low-risk group: 40 to 50 years old, with no symptoms or family history of breast cancer.  To make the question specific, the doctors were told to assume the following statistics — couched in terms of percentages and probabilities — about the prevalence of breast cancer among women in this cohort, and also about the mammogram’s sensitivity and rate of false positives:

The probability that one of these women has breast cancer is 0.8 percent.  If a woman has breast cancer, the probability is 90 percent that she will have a positive mammogram.  If a woman does not have breast cancer, the probability is 7 percent that she will still have a positive mammogram.  Imagine a woman who has a positive mammogram.  What is the probability that she actually has breast cancer?

Gigerenzer describes the reaction of the first doctor he tested, a department chief at a university teaching hospital with more than 30 years of professional experience:

“[He] was visibly nervous while trying to figure out what he would tell the woman.  After mulling the numbers over, he finally estimated the woman’s probability of having breast cancer, given that she has a positive mammogram, to be 90 percent.  Nervously, he added, ‘Oh, what nonsense.  I can’t do this.  You should test my daughter; she is studying medicine.’  He knew that his estimate was wrong, but he did not know how to reason better.  Despite the fact that he had spent 10 minutes wringing his mind for an answer, he could not figure out how to draw a sound inference from the probabilities.”

When Gigerenzer asked 24 other German doctors the same question, their estimates whipsawed from 1 percent to 90 percent.   Eight of them thought the chances were 10 percent or less, 8 more said 90 percent, and the remaining 8 guessed somewhere between 50 and 80 percent.  Imagine how upsetting it would be as a patient to hear such divergent opinions.

As for the American doctors, 95 out of 100 estimated the woman’s probability of having breast cancer to be somewhere around 75 percent.

The right answer is 9 percent.

More:  Chances Are by Steven Strogatz (New York Times)

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2 thoughts on “Innumeracy

  1. The really important thing Gigerenzer showed, though (as Steve goes on to note), is that people get the right answer (+-) when the problem is posed in frequency terms, rather than individual-case probabilities. (Original paper.) One can even tell a plausible evolutionary story about why this should be so. (That second paper includes more experiments to the same general conclusion, not just paleonathropological speculations.)

    • epanechnikov says:

      Thanks very much for your comment Cosma and for the interesting articles you were kind to mention. Posing a problem in frequency terms can indeed attenuate the base-rate fallacy and I agree with the point you are highlighting. However I don’t believe that the natural frequency hypothesis generally explains why probability judgements succeed or fail. Under the right circumstances people have been shown to be capable of reasoning with explicit probabilities and percentages (see for example Bar-Hillel 1980). Furthermore one must avoid the fallacious application of the reductio ad absordum. Surely, in certain experimental situations some cognitive illusions disappear. But in other common real world situations the illusions are present and hence they can not be denied. Finally (please do correct me if I am wrong) Gigerenzer implies that single-event statistical probabilities (which can not be stated in frequency terms) are meaningless. More specifically, he seems to adopting the Von Mises view of frequency probability. But there are certainly alternative ways of defining statistical probability and hence nothing prevents on from defining probability for single events (an interesting relevant discussion can be found here).

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