## Conditional Probability Calculations

__Excerpts from __

Paulos J.A. (1990) *Innumeracy*, Penguin Books, England. pp 66 – 67

“An interesting elaboration on the concept of conditional probability is known as Bayes’ theorem, first proved by Thomas Bayes in the eighteenth century. It’s the basis for the following rather unexpected result, which has important implications for drug or AIDS testing.

Assume that there is a test for cancer which is 98 percent accurate; i.e., if someone has cancer, the test will be positive 98% of the time, and if one doesn’t have it, the test will be negative 98 percent of the time. Assume further that .5 percent-one out of two hundred people-actually have cancer. Now imagine that you’ve taken the test and that your doctor somberly informs you that you’ve tested positive. The question is: How depressed should you be? The surprising answer is that you should be cautiously optimistic. To find out why, let’s look let’s look at the conditional probability of your having cancer, given that you’ve tested positive.

Imagine that 10,000 tests for cancer are administered. Of these, how many are positive? On the average, 50 of these 10,000 people (.5 percent of 10,000) will have cancer, and so, since 98 percent of them will test positive, we will have 49 positive tests. Of the 9,950 cancerless people, 2 percent of them will test positive, for a total of 199 positive tests (.02 x 9,950 = 199). Thus, of the total of 248 positive tests (199 + 49 = 248), most (199) are false positives, and so the conditional probability of having cancer given that one tests positive is only 49/248, or about 20 percent! (This relatively low percentage is to be contrasted with the conditional probability that one tests positive, given that one has cancer, which, by assumption is 98 percent.)

This unexpected figure for a test that we assumed to be 98 percent accurate should give legislators pause when they contemplate instituting mandatory or widespread testing for drugs or AIDS or whatever. Many tests are less reliable: a recent article in *The Wall Street Journal*, for example, suggests that the well-known Pap tests for cervical cancer is only 75 percent accurate.” (pg 66)

“Lie-detection tests are notoriously inaccurate, and calculations similar to the above demonstrate why truthful people who flunk polygraph tests usually outnumber liars. To subject people who test positive to stigmas, especially when most of them may be false positives, is counterproductive and wrong.” (pg 67)