Revising Probabilities 549orem expresses the conditional probability needed for the decision in terms of
the reverse conditional probability and the prior probability.^1
Bayes’ theorem is more than a numerical formula. More generally, it expresses
the way new information affects a decision maker’s probability assessments. The
decision maker begins with a prior probability assessment; this is the second brack-
eted term in Equation 13.4. He or she revises this prior assessment in light of new
information. Note that the revised probability, Pr(WƒG), depends directly on the
prior probability, Pr(W). Other things being equal, the larger one’s prior proba-
bility, the larger will be one’s revised probability. (The only exception is in the case
of perfectinformation, where Pr(WƒG) is unity regardless of the prior assessment.)
Of course, the other factor affecting the revised probability is the new infor-
mation itself (the first bracketed term in Equation 13.4). If the factor
[Pr(GƒW)/Pr(G)] is greater than 1—that is, if Pr(GƒW) is greater than Pr(G)—
the information will cause the partners to revise upward their probability of
striking oil. But this is exactly what we would expect. If the frequency of a good
test is greater for wet sites than the overall frequency of good results (for all
sites, wet and dry), this means that a good test is a positive indicator of oil. The
bigger the ratio Pr(GƒW)/Pr(G), the larger the upward revision. Let’s look at
a quick example illustrating Bayes’ theorem.HEALTH RISKS FROM SMOKING About 1 in 12 American adults is a heavy
smoker. One way to assess the health risk of heavy smoking is to study the pop-
ulation of individuals who have lung cancer. Among individuals suffering from
lung cancer, the proportion of heavy smokers is 1 in 3. Based on these facts,
by what factor does the risk of lung cancer increase due to heavy smoking?
Using Bayes’ theorem is the key to answering this question. Analogous to
Equation 13.4, we writewhere LC denotes lung cancer and S a heavy smoker. We know that
[Pr(SƒLC)/Pr(S)] (1/3)/(1/12) 4. Then, from the preceding equation,Pr(LCƒS)cPr(SƒLC)
Pr(S)d Pr(LC),(^1) An expanded version of Bayes’ theorem is obtained by taking the right-hand side of Equation 13.2
and substituting it for the denominator in Equation 13.4:
The partners have available numerical values for all the right-hand-side variables and thus can cal-
culate Pr(WƒG) directly. Note that the numerator is Pr(G&W), and this term is repeated in the
denominator along with Pr(G&D). From this version of Bayes’ theorem, we see that the magnitude
of Pr(WƒG) depends directly on the frequency of the event “good and wet” relative to the frequency
of “good and dry.”
Pr 1 WƒG 2
Pr 1 GƒW 2 Pr 1 W 2
Pr 1 GƒW 2 Pr 1 W 2 Pr 1 GƒD 2 Pr 1 D 2
.
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