Thinking and Deciding

WELL-JUSTIFIED PROBABILITY JUDGMENTS 115
p(A & B) =p(A) · p(B), simply the product of the two probabilities. For
example, p(king & red) =p(king) · p(red) =(1/13) · (1/2) = 1/26.
Such rules put limits on the probability judgments that are justifiable. For example,
it is unjustifiable to believe that the probability of rain is .2, the probability of
snow is .3, and the probability of rain or snow is .8. If we make many different judgments
at one time, or if our past judgments constrain our present judgments, these
constraints can be very strong. As a rule, however, these constraints do not determine
a unique probability for any proposition. Reasonable people with exactly the same
evidence can still disagree.
In practice, the constraints can be useful in checking probability estimates. You
can often estimate probabilities in different ways, and then use these rules to check
them. If you think that the probability of precipitation other than rain and snow is
zero, and if you judge the probabilities of rain as .2, snow as .3, and precipitation as
.8, then you know your judgments disagree, and you can use this fact to adjust them.
One consequence of rules 1 and 2 is that the probability of mutually exclusive and
exhaustive propositions must add up to 1. (“Exhaustive” means that the propositions
considered are the only possible ones.) Psychological research has shown that people
can easily be induced to violate this rule. Robinson and Hastie (1985) asked subjects
to read mystery stories. At several points in the stories, evidence was revealed that
implicated or ruled out one character or another as guilty of the crime. At each
point, subjects were asked to indicate the probability that each of the characters was
guilty. The probabilities assigned to the different subjects added up to about 2 for
most subjects, unless the subjects were explicitly instructed to check the probabilities
to make sure that they added up only to 1. When one character was ruled out, the
probabilities for the others were not usually raised so as to compensate, and when a
clue implicated a particular suspect, the probabilities assigned to the others were not
usually lowered. (Later, we shall see many other violations of the rules.)
The rules of coherence can help us to construct probability judgments by giving
us a way to check them. To return to the mystery, for example, if we regard the
probability that the butler did it as .2, the neighbor as .3, and the parson as .8, we
know, from the fact that the figures add up to more than 1, that something is wrong
and that we have reason to think again about the relation between these numbers and
the evidence we have for the propositions in question. Of course, coherence alone
cannot tell us which numbers we ought to adjust. The rules can also help us to decide
which of two probability judgments is better justified, by examining the consistency
of each of the judgments with other judgments.
Coherence rules and expected utility
Why are these rules normative? You might imagine a less demanding definition of
coherence, in which the only requirement is that stronger beliefs (those more likely
to be true) should be given higher numbers. This would meet the most general goal
of quantifying the strength of belief. In this case, the square of probability p would