2.4 Mutually Exclusive Events

Defining Mutually Exclusive Events

Two events are mutually exclusive (or disjoint) if they cannot occur at the same time — their intersection is empty, so P(A and B) = 0. For example, when rolling a die, getting a 2 and getting a 5 on the same roll are mutually exclusive since only one number shows.

The Addition Rule

For any two events, P(A or B) = P(A) + P(B) - P(A and B). When events are mutually exclusive, the overlap term disappears, simplifying to P(A or B) = P(A) + P(B). Using this simplified rule when events actually overlap will overcount probability and give an incorrect answer.

Mutually Exclusive vs. Independent

These are different concepts! Mutually exclusive events can't both happen (P(A and B) = 0). Independent events don't affect each other's probability (P(A|B) = P(A)). In fact, two mutually exclusive events with nonzero probabilities are always dependent, since knowing one occurred tells you the other definitely did not.

Key terms

Mutually exclusive events

Events that cannot occur at the same time; P(A and B) = 0.

Addition Rule (general)

P(A or B) = P(A) + P(B) - P(A and B).

Addition Rule (mutually exclusive)

P(A or B) = P(A) + P(B), since P(A and B) = 0.

Mutually exclusive vs. independent

Mutually exclusive = can't both happen; Independent = one doesn't affect the other's probability. They are different ideas.

Relationship between mutually exclusive events

If P(A) > 0 and P(B) > 0 and they're mutually exclusive, they must be dependent, since one occurring rules out the other.

Try an AP-style question

A and B are mutually exclusive, P(A) = 0.3, and P(B) = 0.5. What is P(A or B)?

1. 0.8
2. 0.65
3. 0.2
4. 0.15