TheoremStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

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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.

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Probability is additive on every finite pairwise-disjoint family of events

Statement

Let $(A_{i})_{i \in I}$ be a finite pairwise-disjoint family of events in a finite probability space. Then $P ⁣ (⋃_{i \in I} A_{i}) = \sum_{i \in I} P (A_{i}) .$ This includes the empty and one-member families.

Facts & Assumptions

Given: A finite probability space $(Ω, w)$ and a finite pairwise-disjoint family $(A_{i})_{i \in I}$ .

[L1]

Event probability is the finite sum of outcome weights (Finite probability spaces, outcome weights, events, and event probabilities).

[L2]

A finite sum may be reindexed by a bijection, split over a disjoint union, and evaluated in either order over a finite product (Finite commutative-monoid sums are invariant under bijective reindexing, split over disjoint unions, and satisfy the finite Fubini rule).

Proof

technique · direct

1.1

Pairwise disjointness makes $(i, ω) \mapsto ω$ a bijection from ${(i, ω) : i \in I, ω \in A_{i}}$ onto $⋃_{i \in I} A_{i}$ .

given

2.1

Reindexing by this bijection and summing first over each fibre gives $P (⋃_{i} A_{i}) = \sum_{i \in I} \sum_{ω \in A_{i}} w (ω) = \sum_{i \in I} P (A_{i})$ .

step 1.1L1L2

3.1

If $I = \emptyset$ , both sides are the empty sum $0$ ; if $I$ has one member, step 2.1 is the identity $P (A) = P (A)$ .

step 2.1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 55 results over 14 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

C. M. Grinstead and J. L. Snell, Introduction to Probability, 2nd ed., Theorem 1.2 (standard reference, not scraped)
H. Pishro-Nik, Introduction to Probability, Statistics, and Random Processes, Section 1.3.2 (standard reference, not scraped)