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

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Two-event inclusion-exclusion: $P (A \cup B) = P (A) + P (B) - P (A \cap B)$

Statement

For events $A$ and $B$ in a finite probability space, $P (A \cup B) = P (A) + P (B) - P (A \cap B) .$

Facts & Assumptions

Given: Events $A$ and $B$ in a finite probability space.

[L1]

If $C \subseteq D$ , then $P (D ∖ C) = P (D) - P (C)$ (Normalization, nonnegativity, monotonicity, complements, and differences in a finite probability space).

[L2]

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

Proof

technique · direct

1.1

The sets $A$ and $B ∖ A$ are disjoint and have union $A \cup B$ , so $P (A \cup B) = P (A) + P (B ∖ A)$ .

1.2

Since $B ∖ A = B ∖ (A \cap B)$ , one has $P (B ∖ A) = P (B) - P (A \cap B)$ .

2.1

Substitution of step 1.2 into step 1.1 gives the formula.

step 1.1step 1.2algebra∎

Depends on

Used by

The union bound can be strict for overlapping events Example

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 29 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 1.4 (standard reference, not scraped)
H. Pishro-Nik, Introduction to Probability, Statistics, and Random Processes, Section 1.3.3 (standard reference, not scraped)

Two-event inclusion-exclusion: P(A∪B)=P(A)+P(B)−P(A∩B)P(A∪B)=P(A)+P(B)−P(A∩B)