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Avoiding a Nightmare by Using Math

The answer lies in trigonometry. -Sherlock Holmes

Don't worry if you never learned trigonometry--the answers here lie in arithmetic.

Medical test results often come back positive or negative, as if the result were a certainty. Of course, there is the accuracy, but if the accuracy is 99% or so, what does that really mean? That you should get your affairs in order? Before you call your probate attorney, let's take an example from the book Calculated Risks by Gerd Gigerenzer.

Let's say you're a 40-something year old woman with no symptoms of breast cancer. You have a positive mammogram. What are the odds you have breast cancer?

Using some assumptions about test accuracy and rates of disease based on real data, the odds that you'd have breast cancer are one in eleven according to Gigerenzer. (If you were way off, don't feel bad--most of the physicians Gigerenzer tested were way off, too--and they had the data in front of them. Not that that's comforting in every way.)

A different example: what if you have no risk factors, but have a positive test for HIV? Your odds of actually having HIV are about 50%. But, if the experience of one of Gigerenzer's students who shopped around for good medical advice in Berlin is any indication, you're not likely to hear that from an AIDS counselor. In fact, one counselor--a physician--said, "With certainty, [false positives] do not occur; if there are false results, then only false negatives, occurring when the antibodies have not yet formed." Several of the counselors searched for information on false positives but couldn't find any.

Those with a positive test for a disease have reason to be cautiously optimistic. With a little data and some arithmetic, using a book on probability like Calculated Risks can help you decide how to proceed.
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