Event

Mathematical statement

The finite elementary model in this entry explicitly uses the full event family F=P(S), so an event E may be any subset of S. In a general model (S,F), whether S is finite or infinite, an event must belong to F. Event E occurs when the observed outcome belongs to E.

Representative formula (exact TeX)

E\subseteq S

Assumptions

• The random experiment and sample space S are explicitly specified.
• Finite elementary exercises explicitly use F=P(S); other models explicitly provide an event family F, and each event must belong to F.

Symbol scope

• E — event set
• S — sample space
• ⊆ — event contained in the sample space

Intuition, definition, and misconceptions

A single outcome is an element of the sample space, while an event is usually a set of outcomes. These roles must not be conflated.

The empty event contains no outcomes and the certain event contains the whole sample space. These names do not mean this entry has assigned probability values to them.

Every subset is an event only when the full powerset F=P(S) is explicitly selected. In a general model, even a finite S still requires checking E∈F.

Common mistakes include treating an event as one outcome, listing events without a sample space or event family, or assuming that finite sample spaces automatically make every subset measurable.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Sample space — An event must be defined relative to an explicit domain of outcomes.
• Subset — An elementary event is represented as a subset of the sample space.

Verification boundary

Finite-set checks verify only concrete inclusion relations; they do not establish general measurability or probability axioms.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Formalize the finite elementary model F=P(S) separately from the general E∈F and MeasurableSet boundary, with version and axiom receipts.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.