Event

Given a sample space $\Omega$, an event $A$ is a subset $A\subseteq \Omega$, a collection of outcomes to which the probability measure $P$ assigns a value $P(A)\in [0,1]$. Events can be combined using set operations, union ('or'), intersection ('and'), and complement ('not'), and the collection of all events under consideration must form a sigma-algebra for $P$ to be well-defined on it.

• Every event is a subset of the sample space, but not every subset of the sample space is necessarily a valid event in a rigorous measure-theoretic treatment, only those subsets belonging to the chosen sigma-algebra are assigned a probability, a subtlety that matters for uncountable sample spaces.
• Events can be combined with the standard set operations, the union of two events represents 'at least one occurs', the intersection represents 'both occur', and the complement represents 'the event does not occur', giving probability theory its close relationship to set theory and Venn diagrams.
• An elementary event (a single outcome) and a compound event (a set of two or more outcomes) are both technically 'events' in the formal sense, the distinction is purely about how many underlying outcomes the event's subset contains.