Thinking and Deciding

WHAT IS PROBABILITY? 107
theory. The personal theory gives different answers. I shall recommend the personal
theory, with some features stolen from the other two.
The frequency theory
According to the frequency theory, probability is a measure of the relative frequency
of particular events. By the simplest, most literal form of this theory, the probability
of a smoker’s getting lung cancer is simply the proportion of smokers who have gotten
lung cancer in the past. According to this theory, unless a probability statement is
based on such a proportion, it is meaningless and should not be made. If a probability
statement differs from the observed relative frequency, it is unjustified and incorrect.
This theory runs into trouble right away. It would certainly make life difficult for
weather forecasters. What could they possibly mean when they say that the probability
of rain is 50%? They might mean “On days like today, it has rained 50% of
the time in the past,” but obviously they do not really mean this. Besides, if they did,
they would have the problem of saying in what way the days they had considered
were “like today.” If these days were like today in being December 17, 1993, then
there is only one such day, and the probability of rain would either be 1 or 0, and we
will not know which it is until today is over or until it rains (whichever comes first).
If those other days were like today in being December 17, regardless of the year, a
simple record of past years would suffice for forecasting, and we could save a lot of
money on satellites and weather stations. If those other days were like today in the
precise configuration of air masses, then, once again, there probably were not any
such days except today. The problem is thus that any event can be classified in many
different ways. The relative frequency of the event, compared to other events in a
certain class, depends on how it is classified.
The true frequency theorist might agree that weather forecasters are being nonsensical
when they give probability judgments. But the problems of the theory do not
end here. Suppose I flip a coin, and it comes up heads seven out of ten times — not
an unusual occurrence. What is the probability of heads on the next flip? The simple
frequency view would have it that the probability is .7, yet most of us would hold
that the probability is .5, because there is no reason to think that the probabilities
of heads and tails are different unless we have some reason to think that the coin is
biased. The simple frequency view flies in the face of strong intuitions. That by itself
is not reason to give it up, but it surely is reason to look at some of the alternatives.
Some have attempted to save the frequency view from this last objection by modifying
it. They would argue that probability is not just the observed frequency, but
rather the limit that the observed frequency would approach if the event were repeated
over and over. This seems to take care of the coin example. We think that
the probability of heads is .5 because we believe that if we continued to flip the coin
over and over, the proportion of heads would come closer and closer to .5.
Why do we believe this, though? We have not actually observed it. This approach
already gives considerable ground to the view that probability is a judgment, not
simply a calculation of a proportion. Moreover, this view does not deal with the