These calculations show that a positive test result for a given test means a lot more when the base rate is high than when it is low. Thus, if doctors use the specified test as a screening test in the general population, and if the rate of colon cancer in that general population is only 0.003, then a positive test result by itself does not show that the patient has cancer. In contrast, if doctors in stead use the specified test as a diagnostic test only for people with certain symptoms, and if the rate of colon cancer among people with those symptoms is 0.3, then a positive test result does show that the patient probably has can- cer, though the test still might be mistaken. Bayes’s theorem, thus, reveals the right ways and the wrong ways to use and interpret such tests.
Notice also what happens to the probabilities when additional tests are performed. In our original example, one positive test result raises the probability of cancer from the base rate of 0.003 to our solution of 0.083. Now suppose that the doctor orders an additional independent test, and the result is again posi- tive. To apply Bayes’s theorem at this point, we can take the probability after the original positive test result (0.083) as the prior probability or base rate in calculating the probability after the second positive test result. This method makes sense because we are now interested not in the general population but only in the subpopulation that already tested positive on the first test. The so- lution after two tests [Pr(h|e)], where “e” is now two independent positive test results in a row, is 0.731. Next, if the doctor orders a third independent test, and if the result is positive yet again, then Pr(h|e) increases to 0.988. Bayes’s theorem, thus, reveals the technical rationale behind the commonsense prac- tice of ordering additional tests. Problems arise only when doctors put too much faith in a single positive test result without doing any additional tests.