---
title: "Probability Spaces: Sample Spaces, Events and Axioms"
description: "Set probability on its axioms: a sample space, events as subsets, and a measure that is nonnegative, totals one and adds over disjoint events. Every rule met earlier follows from those three."
canonical: https://lightmysky.com/learn/mathematics/probability-spaces-sample-spaces-events-and-axioms-mt_4gsBeqJtIQ
source: https://lightmysky.com/learn/mathematics/probability-spaces-sample-spaces-events-and-axioms-mt_4gsBeqJtIQ.md
retrieved: 2026-09-12
---

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# Probability Spaces: Sample Spaces, Events and Axioms

Set probability on its axioms: a sample space, events as subsets, and a measure that is nonnegative, totals one and adds over disjoint events. Every rule met earlier follows from those three.

Subject: Mathematics · Area: Probability · Ages 18 to 19
Page: https://lightmysky.com/learn/mathematics/probability-spaces-sample-spaces-events-and-axioms-mt_4gsBeqJtIQ

## Ready when they can

- Write the sample space and an event for an experiment
- Derive the complement and general addition rules from the axioms
- Give an assignment of probabilities that breaks the axioms and say which one

## Lesson: Sample spaces, events and the three axioms

The dice stall gave every total a probability without ever naming what those totals were chosen from. Now name it: list every possible result of an experiment, and that list is the sample space. An event is just a collection of results from it. Toss a coin and roll a die: 2 faces times 6 faces gives 12 pairs, so that sample space holds 12 outcomes.

Three axioms govern every chance: each chance sits between 0 and 1, the whole space has chance exactly 1, and disjoint events add. The complement rule follows: P(not A) equals 1 minus P(A). Roll a die: the even event {2, 4, 6} leaves the complement {1, 3, 5}.

**Example.** Add with care when events overlap: P(A or B) equals P(A) plus P(B) minus P(A and B). The overlap was counted twice, so subtract it once. With P(A) 0.5, P(B) 0.4 and P(both) 0.1, the union is 0.5 plus 0.4 minus 0.1, which is 0.8.

Check an assignment against each axiom before you trust it. A union of 1.5 breaks the 0 to 1 bound, and P(S) of 1.2 breaks the whole-space total. A Venn diagram shows why: the rectangle is the whole space and must total exactly 1.

**Recap.** Name the space, name the event, then let the three axioms check every chance.

## Practice

13 questions on this page, each with its working shown.

## Needs first

- [The Addition Rule and Mutually Exclusive Events](https://lightmysky.com/learn/mathematics/the-addition-rule-and-mutually-exclusive-events-mt_QsIusFefeE)
- [Discrete Random Variables and Probability Distributions](https://lightmysky.com/learn/mathematics/discrete-random-variables-and-probability-distributions-mt_yKztNkvvq3)

## Opens up

- [Joint, Marginal and Conditional Probability](https://lightmysky.com/learn/computing/joint-marginal-and-conditional-probability-mt__Rj-hOSMEw)
- [Randomised Algorithms and the Probabilistic Method](https://lightmysky.com/learn/computing/randomised-algorithms-and-the-probabilistic-method-mt_4ShN_WlT0-)
- [Conditional Probability, Independence and Bayes' Theorem](https://lightmysky.com/learn/mathematics/conditional-probability-independence-and-bayes-theorem-mt_wIi_XRzwwK)