During the early development of probability theory, the evolution was based more on intuition rather than mathematical axioms. The axiomatic basis for probability theory was provided by A. N. Kolmogorov in 1933 and his approach conserved the theoretical ideas of all other approaches. This chapter is based on the axiomatic approach and starts with this notion.

In this section we develop the notion of probability measure and present its basic properties.

When an ordinary die is rolled once, the outcome cannot be accurately predicted; we know, however, that the set of all possible outcomes is {1,2,3,4,5,6}. An experiment like this is called a *random experiment.*

**Definition 1.1 (Random Experiment)** *An experiment is said to be random if its result cannot be determined beforehand*.

It is assumed that the set of possible results of a random experiment is known. This set is called a *sample space.*

**Definition 1.2 (Sample Space)** *The set* Ω *of all possible results of a random experiment is called a sample space. An element* Ω *is called an outcome or a sample point.*

EXAMPLE 1.1

Experiment: Flipping a fair coin. The possible results in this case are ...

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