Event independence

Mathematical statement

In one probability space, A and B are independent iff P(A∩B)=P(A)P(B). If P(B)>0, this is equivalent to P(A|B)=P(A). For P(B)=0, an undefined conditional probability must not replace the product definition.

Representative formula (exact TeX)

I(A,B)

Assumptions

• The sample space, exact rational masses and event subsets are completely explicit.
• Only two events are handled; pairwise independence does not stand in for mutual independence of three events.

Symbol scope

• A — first event
• B — second event
• P — common probability measure

Intuition, definition, and misconceptions

The primary qualification always uses the intersection-product equation.

With a positive denominator the conditional equivalence can be checked.

Independence involving an event's complement follows from the same product equation.

Disjointness is not independence; do not calculate the elementary conditional ratio when P(B)=0 or infer mutual independence from pairwise independence.

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

• Event — A, B and their intersection belong to the same event space.
• Probability on finite sample spaces — The product definition uses the same probability measure.
• Conditional probability in finite models — The conditional equivalence is used only with positive denominator.

Verification boundary

Only two events in the same complete finite probability model are checked; no mutual-independence conclusion about an event family follows.

not_formalized

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

Next obligation

Add independence of random variables.

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.