Innumeracy and the False Positive

I did not read this article in the same way as some of those who have cited it, possibly because I'm infinitely fascinated by mathematics and mathematical reading (I'd refer that to aleph null inifintiely).

from "Mammogram Math"
John Allen Paulos

A little vignette with made-up numbers may shed some light. Assume there is a screening test for a certain cancer that is 95 percent accurate; that is, if someone has the cancer, the test will be positive 95 percent of the time. Let’s also assume that if someone doesn’t have the cancer, the test will be positive just 1 percent of the time. Assume further that 0.5 percent — one out of 200 people — actually have this type of cancer. Now imagine that you’ve taken the test and that your doctor somberly intones that you’ve tested positive. Does this mean you’re likely to have the cancer? Surprisingly, the answer is no.

To see why, let’s suppose 100,000 screenings for this cancer are conducted. Of these, how many are positive? On average, 500 of these 100,000 people (0.5 percent of 100,000) will have cancer, and so, since 95 percent of these 500 people will test positive, we will have, on average, 475 positive tests (.95 x 500). Of the 99,500 people without cancer, 1 percent will test positive for a total of 995 false-positive tests (.01 x 99,500 = 995). Thus of the total of 1,470 positive tests (995 + 475 = 1,470), most of them (995) will be false positives, and so the probability of having this cancer given that you tested positive for it is only 475/1,470, or about 32 percent! This is to be contrasted with the probability that you will test positive given that you have the cancer, which by assumption is 95 percent.

I love to learn how wrong my thinking can be on any matter--such a chain of thought had never occurred to me.