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

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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.
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Bayes' theorem over a finite partition

Statement

Let $B_{0}, \dots, B_{n - 1}$ be a finite partition with $P (B_{i}) > 0$ for every $i < n$ . If $P (A) > 0$ , then for every $j < n$ , $P (B_{j} ∣ A) = \frac{P (A ∣ B_{j}) P (B_{j})}{\sum_{i < n} P (A ∣ B_{i}) P (B_{i})} .$

Facts & Assumptions

Given: A finite partition, an event $A$ of positive probability, and an index $j < n$ as in the Statement.

[L1]

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

[L2]

The law of total probability gives $P (A) = \sum_{i} P (A ∣ B_{i}) P (B_{i})$ (The law of total probability for a finite partition).

Proof

technique · direct

1.1

By conditional probability and symmetry of intersection, $P (B_{j} ∣ A) = P (A \cap B_{j}) / P (A)$ .

2.1

The numerator in step 1.1 is $P (A ∣ B_{j}) P (B_{j})$ , and [L2] is the denominator.

L1L2

3.1

Substitution yields the formula; its denominator is positive because it equals $P (A) > 0$ .

step 1.1step 2.1algebra∎

Depends on

Used by

Bayes' theorem for choosing one of two urns and observing a colour Example

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 9 results over 5 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., Bayes' Formula (standard reference, not scraped)
H. Pishro-Nik, Introduction to Probability, Statistics, and Random Processes, Section 1.4.3 (standard reference, not scraped)