TheoremStatement: AI-adaptedProof: AI-adaptedprecheck passaudited 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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Two positive-probability events are independent exactly when conditioning on either leaves the other's probability unchanged

Statement

Let $A$ and $B$ satisfy $P (A) > 0$ and $P (B) > 0$ . The following are equivalent:

$A$ and $B$ are independent;
$P (A ∣ B) = P (A)$ ;
$P (B ∣ A) = P (B)$ .

Facts & Assumptions

Given: Positive-probability events $A$ and $B$ .

[L1]

Conditional probability is $P (A ∣ B) = P (A \cap B) / P (B)$ for $P (B) > 0$ (Conditional probability $P (A ∣ B)$ for $P (B) > 0$ ).

[L2]

Independence means $P (A \cap B) = P (A) P (B)$ (Independent events, pairwise independence, and mutual independence of a finite family).

Proof

technique · direct

1.1

If $A$ and $B$ are independent, divide the identity in [L2] by $P (B) > 0$ to obtain $P (A ∣ B) = P (A)$ .

L1L2algebra

1.2

The same calculation with $A$ and $B$ interchanged gives $P (B ∣ A) = P (B)$ .

L1L2algebra

1.3

Conversely, multiplying either conditional identity by its positive conditioning probability gives the product identity in [L2].

L1L2algebra

2.1

Thus each of conditions 2 and 3 is equivalent to condition 1, proving all three equivalent.

step 1.1step 1.2step 1.3∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

J. Matousek and J. Vondrak, The Probabilistic Method, Section 1.1 (standard reference, not scraped)
C. M. Grinstead and J. L. Snell, Introduction to Probability, 2nd ed., Section 4.1 (standard reference, not scraped)