TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck 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 · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on 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)