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Experiment (Probability Theory)

A mathematical model of a repeatable trial whose possible outcomes form a sample space, measurable events form a sigma-algebra, and probabilities are assigned by a measure.

Version
v1 · 2026-09-28 · History
Domain-specific #
9357
Domain group
Formal Sciences
Origin domain
Mathematics
Subdomain
Probability Theory → Mathematics
Aliases
Random experiment, Probability experiment, Stochastic experiment

Core Idea

In probability theory an experiment specifies what one trial can produce. Mutually exclusive complete outcomes form Ω; measurable events are sets in F; and P assigns probabilities satisfying the axioms. One ω occurs on a trial, while every event containing ω is said to occur.

Repeated trials can be analyzed separately or composed into a larger experiment whose outcomes are sequences. Repeatability is a modeling ideal, not a claim that physical conditions can be reproduced infinitely exactly.

Structural Signature

Sig role-phrases:

  • Trial specification — Defines what is repeated and observed. It is required procedure. Counterfactual: Changing procedure changes the experiment.
  • Sample space Ω — Lists mutually exclusive complete outcomes. It is required carrier. Counterfactual: Omitted possibilities make event claims incomplete.
  • Sigma-algebra F — Selects events assigned probabilities. It is required event structure. Counterfactual: Not every subset is automatically measurable in general.
  • Probability P — Assigns normalized countably additive likelihoods. It is required measure. Counterfactual: Frequencies alone do not define the abstract model.
  • Realized outcome ω — Marks one result of a trial. It is required realization. Counterfactual: An event may contain it, but is not itself necessarily the elementary result.

What It Is Not

  • It is not necessarily a scientific intervention study.
  • An event is not the same as an elementary outcome.
  • A list of possibilities without F and P is not a complete probability model.
  • Empirical frequency estimates P but is not identical to the measure.
  • Closest near-miss. A trial is one execution; a composed experiment can collect many trials into one larger outcome tuple.

Scope of Application

  • Games of chance. Tosses, draws, and rolls give finite spaces.
  • Reliability. System states and failures define outcomes and events.
  • Statistical sampling. Observed data are outcomes under a sampling model.
  • Stochastic processes. Path spaces treat whole trajectories as outcomes.

Clarity

State procedure, granularity, Ω, F, and P. The same physical act can support several experiments depending on what is observed. Independence belongs to a joint probability model, not to repetition alone.

Manages Complexity

A probability space compresses repeatable uncertainty into outcomes, sets, and a measure, enabling many questions without rerunning the procedure. The abstraction hides model misspecification and dependence unless tested.

Abstract Reasoning

  1. Define one execution and observation rule.
  2. Enumerate or characterize mutually exclusive complete outcomes.
  3. Choose a sigma-algebra of meaningful events.
  4. Assign and justify P.
  5. For repetitions, construct the joint space and state dependence.

Knowledge Transfer

The model transfers wherever uncertainty can be represented by a probability space. Calling an uncertain episode an experiment is insufficient until outcomes and measure are supplied.

Examples

Canonical

Two ordered coin tosses have four outcomes, and the event at least one head is the subset containing three of them.

Mapped back: procedure → two tosses; space → HH,HT,TH,TT; event → at least one H; realization → one ordered pair.

Applied / In Practice

One hundred tosses can be modeled as one experiment on length-100 sequences, with individual tosses as trials.

Mapped back: unit → toss; composition → 100-sequence; events → sets of sequences.

Structural Tensions

T1 — Physical Procedure versus Mathematical Model. One procedure admits different outcome granularity and probability assumptions.

Diagnostic: Which distinctions answer the question?

T2 — Outcome versus Event. Exactly one elementary outcome realizes while many sets containing it occur.

Diagnostic: Are those levels being conflated?

Structural–Framed Character

Probability Experiment is strongly structural with model-framed interpretation.

Structural Core vs. Domain Accent

The skeleton is outcome space plus measurable sets and measure. Probability supplies its axioms and trial semantics.

This entry presupposes Probability measure.

  • Approved root. No reviewed parent entails this probability-space interpretation of a trial.

  • Related — probability space, outcome, and event. They are constitutive components.

Relationships to Other Abstractions

Local relationship map for Experiment (Probability Theory)Parents appear above the current abstraction, mutual partners to the right, and children below. Node labels state whether each abstraction is prime or domain-specific; colors identify relation types.Experiment(Probability Theory)DOMAINDomain-specific abstraction: Probability measure — presupposesProbabilitymeasureDOMAIN

Current abstraction Experiment (Probability Theory) Domain-specific

Parents (1) — more general patterns this builds on

  • Experiment (Probability Theory) presupposes Probability measure Domain-specific

    A Probability-Theory Experiment presupposes a Probability Measure because its events receive countably additive probabilities totaling one on the sample space.

Hierarchy paths (2) — routes to 2 parentless roots

Neighborhood in Abstraction Space

Experiment (Probability Theory) sits in a crowded region of the domain-specific corpus (33rd percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Decision & System Modeling Frameworks (30 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With

  • Scientific experiment. Tell: Manipulates or observes systems to test hypotheses.
  • Trial. Tell: One execution, unless usage treats the full repeated design as a trial.
  • Event. Tell: A set of outcomes.
  • Sample. Tell: Observed data generated under the experiment model.

References

  • Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Experiment_(probability_theory) (revision 1351247495).
  • Preserved source candidate: http://www-math.bgsu.edu/~albert/m115/probability/sample_space.html
  • Preserved source candidate: https://web.archive.org/web/20001016182602/http://www-math.bgsu.edu/~albert/m115/probability/sample_space.html
  • Preserved source candidate: http://www.mhhe.com/engcs/electrical/papoulis/
  • Preserved source candidate: http://www.futureaccountant.com/probability/study-notes/trial-result-event-outcome.php

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.