Thinking and Deciding

WELL-JUSTIFIED PROBABILITY JUDGMENTS 113
Comparison with a chance setup
The idea of betting, however, can be useful in the task of explaining to judges (for
example, experts) what their task is when we ask them to make probability judgments.
We need not use anything as difficult to think about as betting odds. Instead,
we can ask judges to compare the situation to some sort of device with known probabilities,
what Hacking (1965) calls a chance setup, such as an honest roulette wheel
or a game with a “spinner” (a pointer that spins). For example, suppose I want to
assess my probability judgment that a Republican would win the U.S. presidency in
the next election. Instead of thinking about what sort of odds I would be willing to
take in a bet, I might simply imagine a spinner, like the one shown here:
I might ask myself which is more likely: that a Republican will win or that the point
of the spinner will land in the shaded part of the circle. I adjust the size of the shaded
part until I am unable to say which is more likely. My probability judgment, when
I have finished this adjustment, is the size of the shaded part (as a proportion of the
whole). In practice, it would be a good idea to try this several times, sometimes
starting with a small area and adjusting upward, sometimes starting with a large area
and adjusting downward. Such techniques as this are used routinely in the formal
analysis of decisions (Chapter 10).
The chance-setup approach to probability assessment is consistent with the gambling
approach. If I thought that the point of the spinner was more likely to land on
the shaded part, I would be more willing to bet on that outcome. If I thought that
it was less likely to land there, I would rather bet that a Republican would become
president than that the point of the spinner would land on the shaded part. Although
this approach helps judges understand the significance of probability judgments for
action, it is important that they also understand the fact that they can construct their
probability judgments by thinking about the evidence for their beliefs rather than by
thinking about what choice they are inclined to make before they think about the
evidence at all.
Well-justified probability judgments
Are there rules or constraints that will help in constructing probability judgments? If
there are, these rules may serve as reasons we can give for our judgments, when we
are asked to justify them, as well as methods for constructing them.