Suppose that you have two events A and B that have joint probabilities. From Bayes' theorem, the probability that A occurs given that B has occurred, Prob(A|B) is
where Prob(B|A) is the probability that B occurs given A, Prob(A) is the unconditiional probability of A occurring, and Prob(B) is the unconditional probability of B occurring.
Let's apply this to cancer screening. According to the American Cancer Society, the unconditional probability that a woman who is age 40 develops breast cancer by the time she is 50 is 1.44 percent.
Suppose that we had a screening test for breast cancer that gave a positive result 100 percent of the time that breast cancer is present. That is, let's assume that the test never generates a false negative. However, let's also assume that there is a small chance, say 5 percent, of false positives.
From these figures, the probability of a randomly screened 40-year-old woman receiving a "positive" result is
= 1.0*.0144 + .05*(1-.0144) = .0637
And the probability that a woman with a positive test has cancer is
= 1.0 * .0144 / .0637 = .226
In this case, a positive test means that the woman has only a one-in-four chance of actually having cancer. She is much less likely to have cancer than to not have it.
The test provides information. Without the test, the woman would have only known about a 1-in-70 chance.
However, because of the very low underlying rate of cancer, the test yields many more false positives than true positives (three times as many). Many, many more women without cancer are tested and subject to false positives.
Suppose that the same test is used for a 50 year old woman. According to the American Cancer Society, the woman would have a 2.39 percent chance of developing cancer over the next ten years. The probability that she has cancer given a positive result from our hypothetical test is 33 percent. So, the rate of false positives drops from one in four to one in three.
To be clear, these examples are not arguments for doing away with testing. However, they do show some of the limitations associated with even a very good test and some of the excess costs associated with routine testing.
