Thinking and Deciding

108 NORMATIVE THEORY OF PROBABILITY
weather-forecasting example. Most of us intuitively understand what the forecaster
is trying to do. A theory of probability that disallows this activity is a strange theory
indeed, and once again we are compelled to look for alternatives. 2
The logical theory
The logical theory of probability is the theory usually assumed in most introductory
treatments of probability, especially with respect to card games and dice. The
theory is useful when we can define a set of logically equivalent, or exchangeable,
propositions. The evidence for each of these propositions is the same, and any two
of them can be exchanged without affecting our beliefs in their truth, so their probabilities
must be the same. Typically, when we make calculations in card games,
we regard every card remaining in a deck as equally likely to be drawn, because the
only evidence we have for any particular card is simply that it is somewhere in the
deck and we have exactly this evidence for each of the cards. The cards are logically
equivalent.
Suppose I flip three coins. We generally think of the three flips as exchangeable,
because we think that the order does not matter, and we think that the names we give
to the coins do not matter either. There are eight compound events (three-flip units)
that can be made up of these basic, exchangeable propositions: HHH (heads, heads,
heads), THH (tails, heads, heads), HTH, HHT, TTH, THT, HTT, and TTT. In half of
these compound events, the first coin comes up heads; hence, the probability of this
event is 1/2. In1/4 of these worlds, the first two coins come up heads; hence the
probability of both of these events is 1/4, and so on. Of course, each of these events
corresponds to an infinite set of atomic possible worlds.
Like the frequency view, the logical view holds that probability is objective —
that is, it is something that can be known. When two individuals give different probability
judgments, at most one can be correct. As in logic, once the premises are
accepted, the conclusion follows. The assumptions here concern exchangeability.
An advantage of the logical view is that it solves the problem raised by the coin.
It explains quite well why we believe that the probability of a coin’s coming up heads
is 1/2.
A disadvantage is that it is usually impossible to find exchangeable events. I
cannot sensibly enumerate equally likely possibilities in order to calculate the probability
of rain tomorrow, in the way that I would enumerate the possible poker hands
in order to calculate the probability of a flush. Therefore, like the frequency theory,
the logical theory renders many everyday uses of probability nonsensical. When I
want to evaluate my physician’s judgment of the probability that I have strep throat,
it does not help to ask about the proportion of exchangeable events in which I would
2 Another, more subtle, argument (from Hacking, 1965) against this view is that some sequences may
have no limit. Consider the sequence HTTHHHTTTTTT .... Inthissequence,Hisfollowedbyenough
Ts so that there are 2/3 Ts, then T is followed by enough Hs so that there are 2/3 Hs, and so on,
alternating. If you were dropped into the middle of the sequence somewhere, it would be reasonable to
think that your chance of landing on a T would be .5, yet this sequence has no limit.