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.