Thinking and Deciding

112 NORMATIVE THEORY OF PROBABILITY
Constructing probability judgments
According to the frequency theory and the logical theory, probability statements are
reflections of objective facts in the world. According to the personal view, probability
statements are judgments, and probability judgments need to be constructed by
appropriate methods.
Probability as willingness to bet
What should we be trying to do when we make a probability judgment? Many theorists
have argued that we should think about our personal willingness to bet that the
proposition in question is true. Surely, the probability of a proposition should be one
of the determinants of our willingness to act as though the proposition were true —
to “bet on it” (see Chapter 10). In addition, the idea that probability can be defined
in terms of a tendency to take action ties the concept of probability closely to observable
behavior. However, the idea that probability judgments can be constructed by
thinking about our willingness to bet has drawbacks.
Ramsey (1931) argued that if we believe that the probability of an event is .5,
this amounts to saying that we are equally willing to bet on the event’s happening
as on it not happening. To take a more complicated case, suppose I value $3 three
times as much as I value $1. Suppose you ask me the probability that it will rain
tomorrow, and I say that I would be willing to bet, at three-to-one odds, that it will.
This means that I would be willing to accept $1 if it does rain, on condition that I
would pay $3 if it does not. Moreover, I would be just as willing to do the opposite,
to pay $1 if it rains or accept $3 if it does not. According to the betting interpretation
of probability, this means that my subjective probability of rain is .75. I would regard
it as three times as likely to rain (.75) as not to rain (.25).
Over a long series of making such bets, based on this same degree of belief,
I would come out even, winning and losing the same amounts, if my judgments
corresponded to the relative frequencies of the events. If I collect $1 every time
the .75-probability event happens and pay $3 every time the .25-probability event
happens, then, out of 100 events, I will collect about $75 for the more frequent event
and pay out about $75 (25 times $3) for the less frequent event. This fact may help
to make clear the use of the concept of betting in the measurement of probability.
In sum, by this view, the way to construct personal probability judgments is to ask
ourselves what odds we would consider fair. (We shall return to the connection
between the personal view and relative frequencies.)
It is not clear that thinking about bets is the best way to elicit probability judgments
(Shafer, 1981). We think about probabilities, and, more generally, the strengths
of our beliefs, because our probability judgments help us to make decisions. We often
do not know what decision to make until we think about the strengths of our
beliefs about the consequences of the various options before us. We cannot consult
our decisions to determine what we believe, because we need to know what we
believe in order to make our decisions.