2 CHAPTER 1 INTRODUCTIO N
Most historians consider the work of Fermat (1601-1665) and Pascal (1623-1662)
on games of chance to be the first significant contribution to the study of probability;
however, many of Fermat's and Pascal's ideas can be traced to earlier works of Cardan,
Kepler, and Galileo. There is also some evidence that the Romans, many centuries before,
used mortality tables1 to predict human lifespans. Since Fermat and Pascal's time, nearly
every great mathematician has made some contribution to probability. Among the more
famous contributors are the Bernoullis, Laplace, DeMoivre, Poisson, DeMorgan, Venn,
Bayes, Markov, and Kolmogorov. A complete and readable account of the history of
the subject from the early 17th to the mid-19th century is given in the classic book by
Todhunter listed at the end of this chapter. Subsequent developments up to the early
20th century are discussed in the scholarly book of the famous economist John Maynard
Keynes, which is also listed at the end of the chapter.
While many scholars have studied probability purely for its intellectual and philo-
sophical appeal, a good deal of the motivation for the subject has come, and continues
to come, from practical problems outside of mathematics. Indeed, the development of
probability since Fermat's time has been heavily influenced by investigations in gaming,
demography, insurance, genetics, and quantum physics, to name just a few. Moreover,
the subject itself has had profound implications on everything from economics to engi-
neering and, it could be argued, has played a significant role in the history of the world
over the last 200 years. To give a simple example, consider marine insurance, whose
issuance can be justified by the well-known law of averages: The availability of marine
insurance enabled commercial shipping to develop on a large scale (because it freed mar-
itime shippers from the worry of financial ruin due to a catastrophe at sea), which in turn
contributed to the economic and political ascendancy of Britain in the 19th century and
to international commerce as we know it.2
Today, probability is used in a wide range of fields including engineering, finance,
medicine, meteorology, and management. We will encounter numerous applications of
probability to these and other fields throughout this book.
1.2 How Is Uncertainty Quantified?
If we agree that probability, from a scientific perspective, is the study of mathematical
techniques for making quantitative inferences about uncertainty, then for the subject to
have any meaningful content, we must have some precise way of quantifying uncertainty
and making inferences about that quantification. That there is considerable controversy
over how to precisely formulate such a quantification of uncertainty is an understatement,
to say the least. Indeed, some philosophers have gone so far as to argue that the very
notion of uncertainty cannot be precisely quantified since to do so would, in effect, make
One approach to quantifying uncertainty is to use the concept of relative frequency.
To describe this concept, consider an experiment with several possible outcomes which
mortality table lists the number of deaths each year for a hypothetical group of individuals assumed to be
born at the same time.
We will have more to say about the connection between insurance and probability in § 1.4.