## A Concise Introduction to the Theory of Probability Spaces

Have you ever wondered what is meant by a probability space from a concrete or technical view point? This post aims to clarify what is meant by the term *probability space *from the perspective of a mathematician or probability theorist, but in layman’s terms with as little mathematics as possible.

Rather than dive deep into the technical details, the point of this post is to make the concept more concrete with mathematics and build intuition around the mathematics involved. The aim is for you to understand what a probability space is concretely yet intuitively.

## The Sample Space

Suppose you have a random phenomenon. We’ll use the simplest example: the roll of a six sided die. A sample space is the set of all possible outcomes of the random phenomenon. In our case, the sample space is the set \(\Omega\) = {1,2,3,4,5,6} where each number represents the number the die landed on. You’ll notice the upper-case omega \(\Omega\) was used to denote the sample space. This is common convention.

#### Mutually Exclusive Outcomes

One property of the sample space is that the random outcome can only take on one value in the sample space. This is known as *mutual exclusivity.* Because a die must land on one of six sides our sample space is mutually exclusive because only one outcome may occur. The die cannot simultaneously land on 2 and 4.

#### All Outcomes Must be Included

\(\Omega\) must contain all possible outcomes. This is known as the sample space being *exhaustive*. Thus, \(\Omega\) = {2,4,5} isn’t an exhaustive sample space for our die roll experiment. The die could land on 1 which isn’t allowed in this non-exhaustive sample space.

#### The Sample Space Must Not Include Extra Outcomes

\(\Omega\) = {1,2,3,4,5,6,Red} would not make sense in our die roll example. “Red” could represent the color of a randomly chosen vehicle is red, but our experiment involves the roll of a die which takes on values 1, 2, 3, 4, 5, 6 necessarily.

## The Set of Events

In formal probability theory, to define a probability space we must define the set of events we are interested in. To be continued…

Barrett Duna is a professional data scientist versed in all the major machine learning and AI algorithms. He graduated from UCLA with a B.S. in Mathematics/Economics and is currently admitted to George Mason University for the M.S. Data Analytics Engineering program.