Thinking and Deciding

126 NORMATIVE THEORY OF PROBABILITY
theorem is that it is sometimes easier to assess the diagnostic ratio than to assess the
two likelihoods that make it up.
This version also makes clear the two main determinants of the posterior odds:
the prior odds and the diagnostic ratio. The posterior odds is simply the product of
these two. Substituting the numbers from our example, we have
.09 .90 .10
= ·
.18 .20 .90 .
The important point is that the posterior probability should depend on two things:
what the evidence says (the diagnostic ratio), and what we believed before we got
the evidence (the prior odds). As we shall see, this is not always obvious.
Here is an exercise in which Bayes’s theorem is used to construct a probability
judgment. It is Sunday morning at 7 A.M., and I must decide whether to trek down to
the bottom of my driveway to get the newspaper. On the basis of past experience, I
judge that there is an 80% chance that the paper has been delivered by now. Looking
out of the living room window, I can see exactly half of the bottom of the driveway,
and the paper is not in the half that I can see. (If the paper has been delivered, there is
an equal chance that it will fall in each half of the driveway.) What is the probability
that the paper has been delivered? The footnote has the answer. 7
We should bear in mind the purpose of Bayes’s theorem. It allows us to construct
judgments of the probability of some hypothesis (here, that the paper has been delivered)
given some data that we have observed (the absence of a view of the paper),
on the basis of judgments about the probability of the data given the hypothesis and
about the prior probability of the hypothesis. It is often possible to bypass the use of
Bayes’s theorem and judge the probability of the hypothesis given the data directly.
I could have done this in the case of the newspaper. Even in such cases, however,
Bayes’s theorem can be used to construct a “second opinion,” as a way of checking
judgments constructed in other ways. We shall see in Chapter 6 that the theorem can
provide a particularly important kind of second opinion, because direct judgments
are often insensitive to prior probabilities.
Why frequencies matter
Now let us return to the question of why probability judgments — and the beliefs they
express — should be influenced by relative frequencies: That is, just what is wrong
with believing that a randomly selected smoker’s chance of getting lung cancer is
.01 when 200 out of the last 1,000 smokers have gotten lung cancer? It seems that
there is something wrong, but the personal view, as described so far, has trouble
saying what, as long as a person’s probability judgments are consistent with each
7The prior probability is of course .80. If the paper has been delivered, there is a .50 probability that I
will not see it in the half of the driveway that I can see. Thus, p(D|H) =.50, whereDis not seeing the
paper. If the paper has not been delivered (∼ H), p(D|∼H) =1. So, using formula 3, the probability
.50·.80
of the paper’s having been delivered is
, or .67. If I want the paper badly enough, I should
.50·.80+1·.20
take the chance, even though I do not see it.