512 18. PROBABILITY AND STATISTICSof the Sun, we use a deterministic model (geometric astronomy, in this case) to
study and predict it. If it occurs sometimes under conditions frequently associated
with it, we rely on probabilistic models. Some earlier scientists and philosophers
regarded probability as a measure of our ignorance. Kepler, for example, believed
that the supernova of 1604 in the constellation Serpent may have been caused by
a random collision of particles; but in general he was a determinist who thought
that our uncertainty about a roll of dice was merely a matter of insufficient data
being available. He admitted, however, that he could find no law to explain the
apparently random pattern of eccentricities in the elliptical orbits of the six planets
known to him.
Once the mathematical subject got started, however, it developed a life of its
own, in which theorems could be proved with the same rigor as in any other part
of mathematics. Only the application of those theorems to the physical world
remained and remains clouded by doubt. We use probability informally every day,
as the weather forecast informs us that the chance of rain is 30% or 80% or 100%,^3
or when we are told that one person in 30 will be afflicted with Alzheimer's disease
between the ages of 65 and 74. Much of the public use of such probabilistic notions
is, although not meaningless, at least irrelevant. For example, we are told that the
life expectancy of an average American is now 77 years. Leaving aside the many
questionable assumptions of environmental and political stability used in the model
that produced this fascinating number, we should at least ask one question: Can
the number be related to the life of any person in any meaningful way? What plans
can one base on it, since anyone may die on any given day, yet very few people can
confidently rule out the possibility of living past age 90?^4
The many uncertainties of everyday life, such as the weather and our health,
occur mixed with so many possibly relevant variables that it would be difficult
to distill a theory of probability from those intensely practical matters. What is
needed is a simpler and more abstract model from which principles can be extracted
and gradually made more sophisticated. The most obvious and accessible such
models are games of chance. On them probability can be given a quantitative
and empirical formulation, based on the frequency of wins and losses. At the same
time, the imagination can arrange the possible outcomes symmetrically and in many
cases assign equal probabilities to different events. Finally, since money generally
changes hands at the outcome of a game, the notion of a random variable (payoff
to a given player, in this case) as a quantity assuming different values with different
probabilities can be modeled.
1.1. Cardano. The systematic mathematization of probabiliy began in sixteenth-
century Italy with Cardano. Cardano gambled frequently with dice and attempted
to count the favorable cases for a throw of three dice. His table of values, as reported
by Todhunter (1865, p. 3) is as follows.
(^3) These numbers are generated by computer models of weather patterns for squares in a grid
representing a geographical area. The modeling of their accuracy also uses probabilistic notions
(see Problem 18.1).
(^4) The Russian mathematician Yu. V. Chaikovskii (2001) believes that some of this cloudiness is
about to be removed with the creation of a new science he calls aleatics (from the Latin word
alen, meaning dice-play or gambling)- We must wait and see. A century ago, other Russian
mathematicians confidently predicted a bright future for "arithmology." Prophecy is the riskiest
of all games of chance.