# Sample space

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

A sample space lists all possible elementary outcomes for one specified random experiment.

mko-sample-space · concept · v0.10.0



## Mathematical statement

If a random experiment has sample space S, every possible elementary outcome x satisfies x∈S. Membership in the space does not itself assume that outcomes are equally likely.





## Representative formula (exact TeX)

x\in S





## Assumptions


- The random experiment and observation scale are explicitly specified.
- S contains all possible elementary outcomes of the experiment under that description.





## Symbol scope


- x — possible outcome
- S — sample space
- ∈ — membership relation





## Intuition, definition, and misconceptions

For one coin toss, S={heads,tails} may be used. If the observation changes to two successive tosses, the sample space contains four ordered outcomes. The sample space depends on the experiment and observation scheme.

A sample space answers which outcomes may occur; an event is a set of those outcomes. Equal likelihood and probability assignment are additional data of a later probability model.

Completeness is relative to the specified experiment. Omitting a possible outcome gives later events and probability calculations the wrong domain.

Common mistakes include assuming that outcomes in one sample space must be equally likely or conflating a single experimental outcome with an event set.





## Adopted derivation guide

[Sample space: two observation scales for two coin tosses, and outcomes that are not equally likely](https://ocme.evemisslab.com/en/derivations/derivation-mko-sample-space-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Set](https://ocme.evemisslab.com/en/mko/mko-set/) — A sample space collects all possible outcomes of an experiment as a set.





## Verification boundary

This entry establishes only the sample-space concept. It assigns no probability, uniformity, or independence and adds no Evidence Object.

not_formalized

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





## Next obligation

Choose an outcome type Ω, distinguish elementary outcomes, event sets, and probability measures, and do not add a uniform distribution without an explicit condition.





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