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Sample Space

Defines Sample Space, sample spaces

The sample space Ω of a probabilistic experiment is the set of all possible outcomes. For a coin flip, Ω = {heads, tails}. For rolling a die, Ω = {1, 2, 3, 4, 5, 6}. For measuring a temperature, Ω might be the real numbers or some interval.

An event is a subset of the sample space — a collection of outcomes that may or may not occur. Not every subset need be an event; in general, events form a σ-algebra: a collection of subsets closed under complement, countable union, and countable intersection, and containing Ω and ∅. A probability measure assigns a number in [0, 1] to each event.

The triple (Ω, F, P) — sample space, σ-algebra of events, probability measure — is a probability space, the formal foundation for probabilistic reasoning. Different sample spaces model different experiments, and the choice of sample space determines what questions can be asked and answered.

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@misc{emsenn2026-sample-space,
  author    = {emsenn},
  title     = {Sample Space},
  year      = {2026},
  url       = {https://emsenn.net/library/math/domains/probability/terms/sample-space/},
  publisher = {emsenn.net},
  license   = {CC BY-SA 4.0}
}