TheoremStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge 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.

The law of total probability for a finite partition

Statement

Let $B_{0}, \dots, B_{n - 1}$ be a finite partition of $Ω$ with $P (B_{i}) > 0$ for every $i < n$ . Then every event $A$ satisfies $P (A) = \sum_{i < n} P (A ∣ B_{i}) P (B_{i}) .$ Partition cells of probability zero may instead be omitted, since their intersections with $A$ also have probability zero.

Facts & Assumptions

Given: An event $A$ and a finite partition $(B_{i})_{i < n}$ as in the Statement.

[L1]

Probability is additive on a finite pairwise-disjoint family (Probability is additive on every finite pairwise-disjoint family of events).

[L2]

If $P (B) > 0$ , then $P (A \cap B) = P (A ∣ B) P (B)$ (The multiplication rule and finite chain rule for conditional probability).

[L3]

Probability is monotone under event inclusion (Normalization, nonnegativity, monotonicity, complements, and differences in a finite probability space).

Proof

technique · direct

1.1

The events $A \cap B_{i}$ are pairwise disjoint and have union $A$ , so $P (A) = \sum_{i < n} P (A \cap B_{i})$ .

1.2

Each positive-probability cell satisfies $P (A \cap B_{i}) = P (A ∣ B_{i}) P (B_{i})$ .

2.1

Substitution gives the displayed formula. If $P (B_{i}) = 0$ , [L3] gives $0 \leq P (A \cap B_{i}) \leq P (B_{i}) = 0$ , so deleting that cell changes neither side of the unconditioned decomposition.

step 1.1step 1.2L3algebra∎

Depends on

Used by

Bayes' theorem over a finite partition Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 31 results over 11 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 1.3 and equation (4.3) (standard reference, not scraped)
H. Pishro-Nik, Introduction to Probability, Statistics, and Random Processes, Section 1.4.2 (standard reference, not scraped)