On a finite probability space with and , multiplying by turns the formula into

the inclusion-exclusion principle of combinatorics.

Induction on . The case is . Assume the formula holds for events. Then

Applying the induction hypothesis to both unions of events and collecting terms gives the formula for events.

Indicator algebra reduces the formula to expanding a product. For two events,

More generally, for events ,

Taking expectation and using term by term yields