Thinking and Deciding

106 NORMATIVE THEORY OF PROBABILITY
In sum, we can study numerical probabilities for three reasons. First, they can tell
us about the rationality of belief — its conformity to a normative model — as long as
we are careful not to assume that our subjects know how to assign numbers in a way
that conforms to the normative model. Second, we can study people’s understanding
of the normative model itself. Such an understanding can be useful to people who
want to think carefully about the relationships among certain sets of their beliefs,
such as physicians trying to diagnose a disease. Third, because probabilities are used
to communicate belief strengths, we can study the accuracy of communication.
What is probability?
At the outset, one might ask whether probability judgments are really “judgments”
at all. Aren’t they much more objective than that? Crapshooters who know that the
probability of snake eyes is 1/36 are not making a judgment; they are using the result
of a mathematical calculation. Surely the weather forecaster and the surgeon are
doing something like that too — or are they? How do we tell whether a probability
judgment is correct? Is there an objective standard?
The question of what it means to say that a probability statement is “correct”
has been the subject of competing theories, and the disagreement continues today.
I would like to begin this discussion by doing some reflective thinking about three
theories that bear on this issue (see Hacking, 1975; Savage, 1954; von Winterfeldt
and Edwards, 1986).
The question of when probability statements are correct has two subquestions:
(1) How should we make or construct well-justified probability judgments? For
example, when is a surgeon justified in saying that an operation has a .90 probability
of success? (2) How should we evaluate such judgments after we know the truth of
the matter? For example, how would we evaluate the surgeon’s probability judgment
if the operation succeeds, or if it fails?
These are not the same question. I may be well justified in saying that a substance
is water if I have observed that the substance is odorless and colorless, with a boiling
point of 212 degrees Fahrenheit and a freezing point of 32 degrees Fahrenheit. I
would find that I had been wrong in hindsight, however, if the chemical formula of
the substance turned out to be XYZ rather than H2O (Putnam, 1975). In the case of
water, we accept the chemical formula as the ultimate criterion of correctness, the
ultimate evaluation of statements about what is water and what is not. We would still
consider people to be well justified if they judged the substance to be water on the
basis of its properties alone, even if our evaluation later showed that their judgment
had been incorrect.
I shall discuss three theories. The first says that probability statements are about
frequencies, the second says that they are about logical possibilities, and the third
says that they are personal judgments. The frequency theory gives the same answer
to the construction question as to the evaluation question, and so does the logical