# Event independence

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

Two events are independent when their intersection probability equals the product of their marginal probabilities; this definition does not rely on conditioning on a zero-probability event.

mko-event-independence · definition · v0.10.0



## 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

[Event independence: deciding by the product rule, with the conditional form, complements and a zero-probability event](https://ocme.evemisslab.com/en/derivations/derivation-mko-event-independence-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Event](https://ocme.evemisslab.com/en/mko/mko-event/) — A, B and their intersection belong to the same event space.
- [Probability on finite sample spaces](https://ocme.evemisslab.com/en/mko/mko-probability/) — The product definition uses the same probability measure.
- [Conditional probability in finite models](https://ocme.evemisslab.com/en/mko/mko-conditional-probability/) — 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.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/event-independence.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-event-independence/index.md)