What is a Sample Space?

Examples and why they’re useful.

Sample space notation capturing all the possible outcomes of a single dice role. (Image created by the author with the help of Midjourney).

A “Sample Space” is a way of defining all the possible outcomes of a Statistical Experiment. For example, if you have a six-sided die and you decide to roll it and take the top-facing number, there are six possible outcomes:

Outcomes = {1, 2, 3, 4, 5, 6}

These numbers are the sample space of your dice roll. No matter how many times you roll, one of these numbers is the outcome.

The braces, { … }, are used in sample space notation to wrap up all the possibilities. The idea is more important than the notation, but skip to Learn More if you want to know why.

What if you roll two dice?

Doing this broadens your sample space, but the same idea applies. You list every single possible combination of the two dice:

Outcomes = {(1, 1), (1, 2), (1, 3), … , (6, 4), (6, 5), (6, 6)}

The ellipsis in our sample space definition is an invitation to fill in the gaps. It assumes it’s clear the pattern shown is enough to see the outcomes are all…