Complements and unions

Use what does not happen and subtract overlap when “or” events meet.

9 minutes · intermediate · Original Poker4Quant lesson

What you will learn

Compute unions and complements without assuming disjointness or independence.

Every trial either belongs to an event or its complement

The complement of A is every outcome outside A. Because together they cover the entire sample space with no overlap, P(not A)=1−P(A). “At least one” is often easier to calculate as one minus “none.”

Be precise about the complement. The complement of “exactly one heart in two cards” is zero hearts or two hearts, not just zero hearts. Writing the event in words first prevents an otherwise tidy calculation of the wrong probability.

Or includes the overlap once

For two events, P(A or B)=P(A)+P(B)−P(A and B). If they cannot both happen, the overlap is zero. Independence is a different property: independent events can overlap.

A single card being a heart and a spade are mutually exclusive. A card being an ace and a heart can both happen. For a uniform card, rank and suit are independent, even though “ace” and “heart” have a nonempty intersection.

Worked example

A card is drawn uniformly. What is P(red or king)?

There are 26 red cards and 4 kings.
Two red kings are in both counts.
The union contains 26+4−2=28 cards.

P(red or king)=28/52=7/13. Its complement has 24 cards: black non-kings.

Common mistakes

Adding event probabilities without checking overlap.
Confusing mutually exclusive with independent.
Replacing the complement of exactly one with zero only.

Keep these ideas

A complement includes all and only the excluded outcomes.
Union counts overlap once.
Independence and mutual exclusivity answer different questions.