Mutually Exclusive

Mutually Exclusive
Disjoint Events

Events that have no outcomes in common.

Example: Rolling a six-sided die once. Event A = rolling 1 or 2; Event B = rolling 5 or 6. These are disjoint (mutually...

See also

Addition rule, independent events

Key Formula

P(A \cup B) = P(A) + P(B)

Where:

$A, B$ = Two mutually exclusive events
$P(A \cup B)$ = Probability that event A or event B (or both) occurs
$P(A)$ = Probability that event A occurs
$P(B)$ = Probability that event B occurs

Worked Example

Problem: A single six-sided die is rolled. Let event A = rolling a 2 and event B = rolling a 5. Find P(A or B).

Step 1: Determine whether the events are mutually exclusive. A single die shows exactly one number per roll, so you cannot roll a 2 and a 5 at the same time. The events share no outcomes, so they are mutually exclusive.

A \cap B = \emptyset

Step 2: Find the probability of each event individually.

P(A) = \frac{1}{6}, \quad P(B) = \frac{1}{6}

Step 3: Because the events are mutually exclusive, the probability of A and B occurring together is zero.

P(A \cap B) = 0

Step 4: Apply the addition rule for mutually exclusive events: simply add the individual probabilities.

P(A \cup B) = P(A) + P(B) = \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3}

Answer: The probability of rolling a 2 or a 5 is 1/3, or approximately 0.333.

Another Example

This example uses a card deck instead of a die, showing that mutual exclusivity depends on whether two categories can overlap — a card can be a King or a Queen, but never both.

Problem: A card is drawn at random from a standard 52-card deck. Let event A = drawing a King and event B = drawing a Queen. Are A and B mutually exclusive? Find P(A or B).

Step 1: Check for mutual exclusivity. A single card cannot be both a King and a Queen at the same time, so these events are mutually exclusive.

A \cap B = \emptyset

Step 2: Find each probability. There are 4 Kings and 4 Queens in a 52-card deck.

P(A) = \frac{4}{52}, \quad P(B) = \frac{4}{52}

Step 3: Apply the mutually exclusive addition rule.

P(A \cup B) = \frac{4}{52} + \frac{4}{52} = \frac{8}{52} = \frac{2}{13}

Answer: The probability of drawing a King or a Queen is 2/13, or approximately 0.154.

Frequently Asked Questions

What is the difference between mutually exclusive and independent events?

Mutually exclusive events cannot happen at the same time: if one occurs, the other cannot. Independent events, on the other hand, can happen together — knowing one occurred does not change the probability of the other. In fact, if two events both have nonzero probability, they cannot be both mutually exclusive and independent. Mutually exclusive events are actually dependent because the occurrence of one forces the probability of the other to zero.

Can more than two events be mutually exclusive?

Yes. Three or more events are mutually exclusive if no two of them can occur at the same time. For example, when rolling a single die, the events {1}, {3}, and {6} are all mutually exclusive. The addition rule extends naturally: P(A ∪ B ∪ C) = P(A) + P(B) + P(C) when all three are pairwise mutually exclusive.

When do you use the simplified addition rule for mutually exclusive events?

You use P(A ∪ B) = P(A) + P(B) only when A and B share no outcomes. If the events can overlap, you must use the general addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). Forgetting to check for overlap is a common source of errors in probability problems.

Mutually Exclusive Events vs. Independent Events

	Mutually Exclusive Events	Independent Events
Definition	Events that cannot occur at the same time; P(A ∩ B) = 0	Events where one occurring does not affect the other; P(A ∩ B) = P(A)·P(B)
Key formula	P(A ∪ B) = P(A) + P(B)	P(A ∩ B) = P(A) · P(B)
Can both occur?	No — that is the defining property	Yes — both can happen together
Relationship	They are dependent (occurrence of one forces the other to not occur)	They are not mutually exclusive (unless one has probability 0)
Example	Rolling a 2 and rolling a 5 on one die	Rolling a 2 on die 1 and rolling a 5 on die 2

Why It Matters

Mutually exclusive events appear throughout introductory probability and statistics courses, from basic dice and card problems to real-world scenarios like classifying survey responses into non-overlapping categories. Recognizing whether events are mutually exclusive determines which version of the addition rule you use — the simplified form or the general form. Getting this wrong leads to over-counting or under-counting probabilities, which is one of the most frequent mistakes on probability exams.

Common Mistakes

Mistake: Confusing mutually exclusive with independent. Students often assume that because two events 'have nothing to do with each other,' they must be both independent and mutually exclusive.

Correction: These are opposite ideas in practice. If two events with nonzero probabilities are mutually exclusive, they are dependent — the occurrence of one guarantees the other does not occur. Independent events can (and typically do) both occur.

Mistake: Using the simplified addition rule P(A ∪ B) = P(A) + P(B) when events are NOT mutually exclusive.

Correction: Always check whether the events share any outcomes. If they can overlap, you must subtract the overlap: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). For example, drawing a King and drawing a Heart are not mutually exclusive because the King of Hearts belongs to both events.

Related Terms

Event — A set of outcomes; mutually exclusive describes events
Outcome — Individual results that events share or don't share
Addition Rule — Simplifies to direct addition for mutually exclusive events
Independent Events — Often confused with mutually exclusive; distinct concept
Sample Space — The universal set from which mutually exclusive events draw
Complement — An event and its complement are always mutually exclusive
Probability — Mutually exclusive events simplify probability calculations