Thinking and Deciding

124 NORMATIVE THEORY OF PROBABILITY
of probability as a proportion of cases, then we want to know what proportion of the
cases with D also have H. This is just the number of cases with D and H divided
by the number with D.
Formula 1 expresses an important implication of the multiplication rule: our
judgment of the conditional probability of A given B — our judged probability of A
if we learned that B were true — should be equal to the ratio of our judgment of the
probability of A and B to our judgment of the probability of B.
Formula 1 does not help us much, because we don’t know p(H & D). But we do
know p(D|H) and p(H), and we know from the multiplication rule that p(D & H) =
p(D|H) · p(H). Of course p(D & H) is the same as p(H & D), so we can replace
p(H & D) in formula 1 to get:
p(D|H) · p(H)
p(H|D) = (5.2)
p(D)
Formula 2 is useful because it refers directly to the information we have. In the
example, p(D—H) is .90, and p(H) is .10, except for p(D). But we can calculate that
too. There are two ways for D to occur; it can occur with H or without H (that is,
with ∼H. These are mutually exclusive, so we can apply the additivity rule to get:
p(D) = p(D & H)+p(D & ∼H)
= p(D|H) · p(H)+p(D|∼H) · p(∼H)
This leads (by substitution into formula 2) to formula 3:
p(D|H) · p(H)
p(H|D) =
(5.3)
p(D|H) · p(H)+p(D|∼H) · p(∼H)
Formulas 2 and 3 are called Bayes’s theorem. They are named after Rev. Thomas
Bayes, who first recognized their importance in a theory of personal probability
(Bayes 1764/1958). In formula 3, p(H|D) is usually called the posterior probability
of H, meaning the probability after D is known, and p(H) is called the prior probability,
meaning the probability before D is known. p(D|H) is sometimes called the
likelihood of D.
Formula 3 is illustrated in Figure 5.1. The whole square represents all of the
possible patients. The square is divided, by a vertical line, into those patients with
cancer, on the right (indicated with p(H) =.1), and those without cancer (indicated
with p(∼ H) =.9). The height of the shaded region in each part represents the
conditional probability of a positive test result for each group, p(D|H) =.9 and
p(D| ∼H) =.2, respectively. The darker shaded region represents the possible
patients with cancer and positive results. Its area (.09) represents the probability
of being such a patient, which is the product of the prior probability (.1) and the
conditional probability of a positive result (.9). Likewise, the area of the lighter
shaded region (.18) represents the probability of being a patient with a positive result
and no cancer. Possible patients with positive results are in the two shaded regions.