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Canonical static concept record

Conditional probability

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Summary

For events $A$ and $B$ with $\mathbb P(B)>0$ , the normalized probability $\mathbb P(A\mid B)=\mathbb P(A\cap B)/\mathbb P(B)$ .

Record metadata

Canonical name: Conditional probability
Concept ID: conditional_probability
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Probability
Domains: Probability, Measure
Alternate terminology: No alternate terminology is currently authored for this record.

Carrier(s)

probability space $(\Omega,\mathcal F,\mathbb P)$
events $A,B\in\mathcal F$

Data

conditioned probability $\mathbb P(A\mid B)$

Axioms / constraints

$\mathbb P(B)>0$

Canonically induces

a probability measure $A\mapsto\mathbb P(A\mid B)$ for fixed $B$

Notes

Conditioning on sigma-algebras or null events requires conditional expectation or regular conditional probabilities; the elementary event formula does not cover those cases.

Concept sources

Wikipedia — Conditional probability — Conditional probability · encyclopedic reference · source ID wp-conditional-probability
nLab — probability theory — probability theory · mathematical reference · source ID nlab-probability-theory
Durrett — Probability: Theory and Examples — Probability: Theory and Examples · textbook · source ID durrett-probability

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Probability space Conditional probability

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: probability_space_to_conditional_probability
Relation type: Canonical construction canonical-construction
Direction: source → target
Endpoint roles: source: Canonically constructs; target: Canonically constructed from
Authored annotation: normalize on a positive-probability event

Authored explanation

For an event $B$ with $\mathbb P(B)>0$ , restricting $\mathbb P$ to intersections with $B$ and dividing by $\mathbb P(B)$ gives the conditional probability measure.

How to interpret this relation type

Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.

Relation sources

Wikipedia — Conditional probability — Conditional probability · encyclopedic reference · source ID wp-conditional-probability
nLab — probability theory — probability theory · mathematical reference · source ID nlab-probability-theory
Durrett — Probability: Theory and Examples — Probability: Theory and Examples · textbook · source ID durrett-probability

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.

No direct relations are authored in this direction.

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