One Hundred Interesting Mathematical Calculations, Number 9: False Positives
Suppose that we have a test for a disease that is 98% accurate: if one has the disease, the test comes back "yes" 98% of the time (and "no" 2% of the time), and if one does not have the disease, the test comes back "no" 98% of the time (and "yes" 2% of the time). Suppose further that 0.5% of people--one out of every two hundred--actually has the disease.
Your test comes back "yes." How worried should you be? How likely is it that you have the disease?
Suppose just for ease of calculation that we have a population of 10000, of whom 50--one in every two hundred--have the disease. On average, the fifty who have cancer will contribute 49 "yes" tests and one "no" test. On average, the 9950 who do not have cancer will contribute 9751 "no" tests and 199 "yes" tests.
If you test "no" you can be very happy indeed: there is only one chance in 9752 that you are the unlucky guy who had the disease and yet tested negative.
If you test "yes" you are less happy. But there are 248 "yes" tests, and only 49 of those people have the disease. The chances that you are disease-free are 80.24 percent.
This is the so-called false positive problem: it shows itself wherever you have an imperfect signal of an unlikely event, and it leads to situations in which most of your positive signals are false positives: fake signals, not real indicators of the problem or the event at all.
From John Allen Paulos's Innumeracy.Posted by DeLong at January 13, 2003 05:22 PM | Trackback