DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge 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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 31 results over 10 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click 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)