TheoremStatement: AI-adaptedProof: AI-adaptedprecheck 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.

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

Dependency tree · two levels

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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)