--- title: "Probability Spaces: Sample Spaces, Events and Axioms" description: "Probability has a sample space, events as subsets, and a measure governed by three axioms: it is nonnegative, totals one and adds over disjoint events. Every rule met earlier follows from these axioms" 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: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Probability Spaces: Sample Spaces, Events and Axioms Probability has a sample space, events as subsets, and a measure governed by three axioms: it is nonnegative, totals one and adds over disjoint events. Every rule met earlier follows from these axioms. 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 naming the set those totals came from. The set of every possible result of an experiment is its sample space. An event is a collection of results from it. Flip two coins: each flip has 2 results, so the sample space {HH, HT, TH, TT} holds 2 times 2 = 4 outcomes. Three axioms govern every chance. A chance is never negative. The whole sample space has chance exactly 1. Disjoint events, which cannot both happen, add: P(A or B) = P(A) + P(B). The complement of an event is every result outside it, and the axioms give its chance: P(not A) = 1 - P(A). Roll a die: the event 'at least 5' is {5, 6}, so its complement is {1, 2, 3, 4}. *(drawing: Disjoint events add and the whole space has chance 1, so P(not A) = 1 - P(A). That is never above 1, because P(A) is never negative.)* **Example.** Add with care when events overlap. Split 'A or B' into three pieces that cannot overlap: only A, both, and only B. By the third axiom, P(A or B) is those three chances added. P(A) already holds only A and both, and P(B) holds only B and both, so P(A) + P(B) counts both twice. Subtract it once: P(A or B) = P(A) + P(B) - P(A and B). With P(A) 0.7, P(B) 0.5 and P(both) 0.3, the union is 0.7 + 0.5 - 0.3 = 0.9. Check an assignment against each axiom before you trust it. A negative chance, such as -0.1, breaks the first axiom. A whole space with chance 0.9 breaks the second. A chance cannot pass 1 either, because P(A) = 1 - P(not A) and P(not A) is never negative. So P(A) = 1.3 is not allowed. **Recap.** Name the space, name the event, then let the three axioms check every chance. *Original lesson and question revisions authored for this dataset.* ## 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)