# Probability on finite sample spaces

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

Assign nonnegative weights summing to one on a nonempty finite sample space. An event probability is the sum of the weights of its outcomes.

mko-probability · definition · v0.10.0



## Mathematical statement

Let S be a nonempty finite sample space and F=𝒫(S) its event family, where 𝒫 denotes the full powerset and P denotes the probability law. Assign rational weights w_s≥0 to all s∈S with Σ_(s∈S)w_s=1. For E⊆S, P(E)=Σ_(s∈E)w_s. The formula P(E)=|E|/|S| applies only when all outcomes are explicitly equiprobable.





## Representative formula (exact TeX)

P(E)





## Assumptions


- S is finite and nonempty, its elementary outcomes are distinct, and the event family is the full powerset.
- Weights are exact nonnegative rational numbers summing exactly to one, and E must be a subset of S.





## Symbol scope


- P — finite probability law
- E — event subset of the sample space





## Intuition, definition, and misconceptions

First validate every weight and their total, then sum the weights of the event outcomes. Event cardinality is not a probability in general; elementary outcomes may have different weights.

P(∅)=0 and P(S)=1. The complement is taken relative to the same S, with P(S∖E)=1−P(E). A nonempty event can have probability zero when all its outcomes have zero weight.

The exercises use explicit integer outcome labels and rational weights with a common denominator for exact calculation and model-validity checks. They do not infer real-world experimental probabilities, establish independence or conditional probability, or extend to a general measure-theoretic definition.

Common errors include dividing favorable counts by total counts without equiprobability, or inferring that E is empty from P(E)=0.





## Adopted derivation guide

[Probability on finite sample spaces: validating a weight model, computing event probabilities and complements](https://ocme.evemisslab.com/en/derivations/derivation-mko-probability-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/) — Fix the sample space and event subsets, keeping outcomes distinct from events.
- [Ratio and proportion](https://ocme.evemisslab.com/en/mko/mko-ratio-proportion/) — Equiprobable counts and rational-weight arithmetic require valid ratios with nonzero denominators.





## Verification boundary

Explicit finite rational weights can be recomputed completely; formal Evidence for general probability measures and countable additivity has not been established.

not_formalized

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





## Next obligation

Bind finite PMFs, rational-weight normalization, event sums, and relative complements; use a separate semantic bridge for the measure layer.





## 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/probability.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-probability/index.md)