To describe of the outcome of an uncertain event, we often speak of the probability of its occurrence. Weatherpersons tell the probability of rain, engineers predict the reliability (or probability of success) of a system, quality control managers measure the probability of a defect, gamblers estimate their chance (or probability) of winning, doctors tell their patients the risk (or probability of failure) of a medical procedure and legislators and economists try to guess the probability of an economic downturn. For many, risk and uncertainty are unpleasant aspects of daily life; while for others, they are the essence of adventure. To the analyst they are the inescapable result of almost every activity.

For mathematical programming models, with a few exceptions, we neglected the effects of uncertainty and assumed that the results of our decisions were predictable and deterministic. This abstraction of reality allows large and complex decision problems to be modeled and solved using the powerful methods of mathematical programming. Stochastic models explicitly include uncertainty as part of the model usually with the help of probability models. Decision making is much more difficult because the models are primarily descriptive rather than prescriptive.

In this chapter we present the language of probability that is used to model situations whose outcomes cannot be predicted with certainty. In important instances, reasonable assumptions allow single random variables to be described using one of the named discrete or continuous random variables. The chapter summarizes the formulas for obtaining probabilities and moments for these distributions. When combinations of several random variables affect the system output, the method of simulation is often used to learn about a system. Here we provide an introduction to the subject.