Sample space

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

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

Prerequisites

• 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.