The Language of Argument

2 5 9

ba y e s ’ s T h e o r e m

those symptoms is 0.3, then a positive test result does show that the pa-

tient probably has cancer, though the test still might be mistaken. Bayes’s

theorem, thus, reveals the right ways and the wrong ways to use and in-

terpret such tests.

Notice also what happens to the probabilities when additional tests are per-

formed. 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

positive. 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.

Construct tables to confirm the above calculations of probabilities after a

second and third positive test result.

exercise Vii

1. What would Wendy’s chances of having colon cancer be if the other
   probabilities remained the same as in the original example, except that
   the probability that a person in the general population has colon cancer
   only 0.1 percent (or 0.001)?


2. What would Wendy’s chances of having colon cancer be if the other
   probabilities remained the same as in the original example, except that
   the probability that a person in the general population has colon cancer
   1 percent (0.01)?


3. What would Wendy’s chances of having colon cancer be if the other
   probabilities remained the same as in the original example, except that
   the conditional probability that the test is positive, given that the patient
   has colon cancer, is only 50 percent (or 0.5)?


4. What would Wendy’s chances of having colon cancer be if the other
   probabilities remained the same as in the original example, except that
   the conditional probability that the test is positive, given that the patient
   has colon cancer, is 99 percent (or 0.99)?

exercise Viii

(continued)

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