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In probability theory, the conventional mathematical model of randomness is a probability space. It formalizes three interrelated ideas by three mathematical notions. First, a sample point (called also elementary event), — something to be chosen at random (outcome of experiment, state of nature, possibility etc.) Second, an event, — something that will occur or not, depending on the chosen elementary event. Third, the probability of an event.

Alternative models of randomness (finitely additive probability, non-additive probability) are sometimes advocated in connection to various probability interpretations.

Introduction

The notion "probability space" provides a basis of the formal structure of probability theory. It may puzzle a non-mathematician for several reasons:

• it is called "space" but is far from geometry;

• it is said to provide a basis, but many people applying probability theory in practice neither understand nor need this quite technical notion.

These puzzling facts are explained below. First, a mathematical definition is given; it is quite technical, but the reader may skip it. Second, an elementary case (finite probability space) is presented. Third, the puzzling facts are explained. Next topics are countably infinite probability spaces, and general probability spaces.

Definition

A probability space is a measure space such that the measure of the whole space is equal to 1.

In other words: a probability space is a triple $\textstyle (\Omega ,{\mathcal {F}},P)$ consisting of a set $\textstyle \Omega$ (called the sample space), a σ-algebra (called also σ-field) $\textstyle {\mathcal {F}}$ of subsets of $\textstyle \Omega$ (these subsets are called events), and a measure $\textstyle P$ on $\textstyle (\Omega ,{\mathcal {F}})$ such that $\textstyle P(\Omega )=1$ (called the probability measure).

Elementary level: finite probability space

On the elementary level, a probability space consists of a finite number $n$ of sample points $\omega _{1},\dots ,\omega _{n}$ and their probabilities $p_{1},\dots ,p_{n}$ — positive numbers satisfying $p_{1}+\dots +p_{n}=1.$ The set $\Omega =\{\omega _{1},\dots ,\omega _{n}\}$ of all sample points is called the sample space. Every subset $A\subset \Omega$ of the sample space is called an event; its probability is the sum of probabilities of its sample points. For example, if $A=\{\omega _{1},\omega _{8},\omega _{9}\}$ then $\mathbb {P} (A)=p_{1}+p_{8}+p_{9}$.

A random variable $X$ is described by real numbers $x_{1},\dots ,x_{n}$ (not necessarily different) corresponding to the sample points $\omega _{1},\dots ,\omega _{n}.$ Its expectation is $\mathbb {E} (X)=x_{1}p_{1}+\dots +x_{n}p_{n}.$

The puzzling facts explained

Why "space"?

Fact: it is called "space" but is far from geometry.

Explanation: see Space (mathematics).

What is it good for?

Fact: it is said to provide a basis, but many people applying probability theory in practice do not need this notion. For them, formulas (such as the addition rule, the multiplication rule, the inclusion-exclusion rule, the law of total probability, Bayes' rule etc.) are instrumental; probability spaces are not, they reign but do not rule.

Explanation 1. Likewise, one may say that points are of no use in geometry. Formulas connecting lengths and angles (such as Pythagorean theorem, law of sines etc.) are instrumental; points are not.

However, these useful formulas follow from the axioms of geometry formulated in terms of points (and some other notions). It would be very cumbersome and unnatural, if at all possible, to reformulate geometry avoiding points.

Similarly, the formulas of probability follow from the axioms of probability formulated in terms of probability spaces. It would be very cumbersome and unnatural, if at all possible, to reformulate probability theory avoiding probability spaces.