utfallsrommet

Utfallsrommet, in probability theory, is the set of all possible results of a random experiment. It is often denoted by Ω (or sometimes S) and, together with a σ-algebra F of events, forms a measurable space (Ω, F). The probability measure P assigns probabilities to events in F, with P(Ω) = 1. The choice of utfallsrommet reflects the nature of the experiment and the level of precision of the outcomes.

In a finite example, rolling a fair six-sided die has Ω = {1, 2, 3, 4, 5, 6}, with each individual outcome having probability 1/6. In continuous settings, the utfallsrommet might be an interval such as [0, 1] or the real line, and events are subsets of Ω that are measurable (often Borel sets).

Events are subsets of the utfallsrommet. If A ⊆ Ω is an event, its probability P(A) describes the chance that the result lies in A. Random variables are mappings X: Ω → R (or to another measurable space) that translate outcomes into numerical values and induce probability distributions via P(X ∈ B) = P({ω ∈ Ω : X(ω) ∈ B}).

Discrete models commonly take F as the power set of Ω, while continuous models use a richer σ-algebra such as the Borel σ-algebra. Understanding the utfallsrommet is foundational for defining independence, conditional probability, and limit laws, and it underpins statistical inference and modeling.

In Norwegian and other Scandinavian texts, the term utfallsrommet corresponds to the English “sample space” or “outcome space.”