Probability models and techniques permeate many important areas of modern life. A variety of types of random processes, reliability models and techniques, and statistical considerations in experimental work play a significant role in engineering and the physical sciences. The solutions of management decision problems use as aids decision analysis, waiting line theory, inventory theory, time series, cost analysis under uncertainty — all rooted in applied probability theory. Methods of statistical analysis employ probability analysis as an underlying discipline.

Modern probability developments are increasingly sophisticated mathematically. To utilize these, the practitioner needs a sound conceptual basis which, fortunately, can be attained at a moderate level of mathematical sophistication. There is need to develop a feel for the structure of the underlying mathematical model, for the role of various types of assumptions, and for the principal strategies of problem formulation and solution.

Probability has roots that extend far back into antiquity. The notion of “chance” played a central role in the ubiquitous practice of gambling. But chance acts were often related to magic or religion. For example, there are numerous instances in the Hebrew Bible in which decisions were made “by lot” or some other chance mechanism, with the understanding that the outcome was determined by the will of God. In the New Testament, the book of Acts describes the selection of a successor to Judas Iscariot as one of “the Twelve.” Two names, Joseph Barsabbas and Matthias, were put forward. The group prayed, then drew lots, which fell on Matthias.

Early developments of probability as a mathematical discipline, freeing
it from its religious and magical overtones, came as a response to questions
about games of chance played repeatedly. The mathematical formulation
owes much to the work of Pierre de Fermat and Blaise Pascal in the
seventeenth century. The game is described in terms of a well defined trial
(a play); the result of any trial is one of a specific set of distinguishable
outcomes. Although the result of any play is not predictable,
certain “statistical regularities” of results are observed. The possible
results are described in ways that make each result seem equally likely.
If there are *N* such possible “equally likely” results, each is assigned a
probability

The developers of mathematical probability also took cues from early work on the analysis of statistical data. The pioneering work of John Graunt in the seventeenth century was directed to the study of “vital statistics,” such as records of births, deaths, and various diseases. Graunt determined the fractions of people in London who died from various diseases during a period in the early seventeenth century. Some thirty years later, in 1693, Edmond Halley (for whom the comet is named) published the first life insurance tables. To apply these results, one considers the selection of a member of the population on a chance basis. One then assigns the probability that such a person will have a given disease. The trial here is the selection of a person, but the interest is in certain characteristics. We may speak of the event that the person selected will die of a certain disease– say “consumption.” Although it is a person who is selected, it is death from consumption which is of interest. Out of this statistical formulation came an interest not only in probabilities as fractions or relative frequencies but also in averages or expectatons. These averages play an essential role in modern probability.

We do not attempt to trace this history, which was long and halting, though marked
by flashes of brilliance. Certain concepts and patterns which emerged
from experience and intuition called for clarification. We move
rather directly to the mathematical formulation (the “mathematical model”) which
has most successfully captured these essential ideas. This is the model, rooted in the mathematical
system known as measure theory, is called the *Kolmogorov model*, after the brilliant Russian
mathematician A.N. Kolmogorov (1903-1987). Kolmogorov succeeded in bringing together various
developments begun at the turn of the century, principally in the work of E. Borel and H. Lebesgue on
measure theory. Kolmogorov published his epochal work in German in 1933. It was translated
into English and published in 1956 by Chelsea Publishing Company.