# Chapter 8. Concepts of Probability Theory

In this chapter, we introduce the general concepts of probability theory. Probability theory serves as the quantification of risk in finance. To estimate probabilistic models, we have to gather and process empirical data. In this context, we need the tools provided by statistics. We will see that many concepts from Part One can be extended to the realm of probability theory.

We begin by introducing a few preliminaries such as formal set operations, right-continuity, and nondecreasing functions. We then explain probability, randomness, and random variables, providing both their formal definitions and the notation used in this field.

# HISTORICAL Development of ALTERNATIVE Approaches to PROBABILITY

Before we introduce the formal definitions, we provide a brief outline of the historical development of probability theory and the alternative approaches since probability is, by no means, unique in its interpretation. We will describe the two most common approaches: relative frequencies and axiomatic system.

## Probability as Relative Frequencies

The relative frequencies approach to probability was conceived by Richard von Mises in 1928 and as the name suggests formulates probability as the relative frequencies *f*(*x*_{i}) introduced in Chapter 2. This initial idea was extended by Hans Reichenbach. Given large samples, it was understood that *f*(*x*_{i}) was equal to the true probability of value *x*_{i}. For example, if *f*(*x*_{i}) is small, then the true probability of value ...