Event

An event is a subset of the sample space Ω — a collection of outcomes to which a probability can be assigned. “Rolling an even number” is the event {2, 4, 6} in the sample space {1, 2, 3, 4, 5, 6}. The empty set ∅ is the impossible event (probability 0); the full sample space Ω is the certain event (probability 1).

Events combine with set operations: the union A ∪ B is “A or B occurs,” the intersection A ∩ B is “both A and B occur,” and the complement Aᶜ is “A does not occur.” The probability axioms require countable additivity: if events A₁, A₂, … are mutually exclusive (pairwise disjoint), then P(A₁ ∪ A₂ ∪ …) = P(A₁) + P(A₂) + ….

Formally, events are members of a σ-algebra F on Ω — a collection of subsets closed under complement and countable union. The σ-algebra determines which questions about outcomes are “askable” within the probability space. Two events are independent when the occurrence of one does not affect the probability of the other: P(A ∩ B) = P(A) · P(B).