DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Independent events, pairwise independence, and mutual independence of a finite family

Definition

Two events $A$ and $B$ are independent when $P (A \cap B) = P (A) P (B) .$

A finite family $(A_{i})_{i \in I}$ is pairwise independent when every two distinct members are independent. It is mutually independent when for every nonempty $J \subseteq I$ , $P ⁣ (⋂_{j \in J} A_{j}) = \prod_{j \in J} P (A_{j}) .$ The empty-subfamily identity is also valid under the conventions $P (Ω) = 1$ and $\prod_{j \in \emptyset} P (A_{j}) = 1$ .

Depends on

Finite probability spaces, outcome weights, events, and event probabilities

Used by

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. Matousek and J. Vondrak, The Probabilistic Method, Definition 1.1.4 (standard reference, not scraped)
M. Bucic, Probabilistic Method, Definitions A.14-A.15 (standard reference, not scraped)