Union and Intersection of Events¶

Definition¶

Let $A$ and $B$ be events in a sample space. The union $A\cup B$ is the event that at least one of $A$ or $B$ occurs, and the intersection $A\cap B$ is the event that both $A$ and $B$ occur.

Interpretation¶

The union combines events by allowing either event to happen. The intersection focuses only on outcomes shared by both events.

Two panels with overlapping discs A and B: the whole shaded area of both discs is the union, and the shaded lens where they overlap is the intersection.

Adding the probabilities of the two discs counts their overlap twice, so its probability must be subtracted once. De Morgan's identities relate these operations to complements: $(A\cup B)^c=A^c\cap B^c$ and $(A\cap B)^c=A^c\cup B^c$ $B^{c}$ .

Union of sets collects outcomes belonging to at least one of the events. Intersection of sets retains only outcomes shared by both; treating those sets as events connects set operations with probabilistic statements.

Key Results¶

$A\cup B$ occurs when at least one of the events occurs.
$A\cap B$ occurs when both events occur.
Inclusion-exclusion connects the probability of a union to the probabilities of the two events and their intersection.

Examples¶

The event 'draw a heart or a king' is a union of two events.
The event 'draw a red king' is an intersection of two events.
The event that a customer buys online and uses a coupon is an intersection.

Prerequisites¶

Sources¶

Probability Theory for EOR · Week 1

Keywords: union of events, intersection of events Symbols: intersection, union

Last updated: 20 October 2026 03:02