Mutually Exclusive Events in Math: Definition & Examples

Definitionnoun phrase

What are mutually exclusive events?

Mutually exclusive events are events that cannot happen at the same time, so P(A and B) = 0. Rolling a 2 and rolling a 5 on one roll of a number cube are mutually exclusive. For such events the addition rule is simple: P(A or B) = P(A) + P(B).

In Spanish: eventos mutuamente excluyentes
Also called: disjoint events, mutually exclusive, non-overlapping events
Grade level: Statistics

Grade 7Algebra 2StatisticsCCSS 7.SP.C.8CCSS HSS-CP.A.1CCSS HSS-CP.B.7SATACTASVABGEDTEASAP

Mutually Exclusive Events in a picture

When the circles do not overlap, just add the probabilities. When they do overlap, the shared outcomes get counted twice, so you must subtract them once.

Formal definition and notation

Two events A and B in the same sample space are mutually exclusive, or disjoint, when they share no outcomes: A ∩ B = ∅, so P(A and B) = 0. Then P(A or B) = P(A) + P(B). For events that can overlap, the general addition rule subtracts the overlap: P(A or B) = P(A) + P(B) − P(A and B). Several events are mutually exclusive when every pair of them is. An event and its complement are mutually exclusive and together fill the sample space, so P(A) + P(not A) = 1.

Symbol	Read it as	Meaning
P(A or B)	the probability of A or B	the probability that at least one of the events happens; also written P(A ∪ B)
A ∩ B = ∅	A intersect B is the empty set	the events share no outcomes; they are mutually exclusive

Key facts

In a Venn diagram, mutually exclusive events are circles that do not overlap.
Drawing a heart and drawing a king are not mutually exclusive, because the king of hearts is both.
An event and its complement are always mutually exclusive: rolling an even number and rolling an odd number, with 12 + 12 = 1.
For or, add (and subtract any overlap). For and with independent events, multiply. Mixing these up is the most common error.
Mutually exclusive is not the same as independent. If both events have nonzero probability, mutually exclusive events are always dependent: when one happens, the other’s chance drops to 0.

Addition rule for mutually exclusive events

P(A or B) = P(A) + P(B)

P(A), P(B): the probabilities of two events that cannot happen together
P(A or B): the probability that one or the other happens

If the events can overlap, use the general rule P(A or B) = P(A) + P(B) − P(A and B). For mutually exclusive events the last term is 0, so both rules agree.

Try it: one spin, one color

Interactive

Each spin lands in exactly one sector, so the four colors are mutually exclusive. Spin 1,000 times, then add the results for red and blue: the total should be close to 1/8 + 2/8 = 3/8.

This interactive needs JavaScript. The worked examples below show the same ideas step by step.

Mutually Exclusive Events examples, step by step

Example 1: Two faces of a number cube

Warm-up

You roll a number cube once. What is the probability of rolling a 2 or a 5?

One roll cannot show both 2 and 5, so the events are mutually exclusive.
P(2) = 16 and P(5) = 16.
Add: 16 + 16 = 26 = 13.

Answer: P(2 or 5) = 13.

Example 2: Events that overlap

Core

One card is drawn from a standard 52-card deck. What is the probability that it is a heart or a king?

The king of hearts is both, so these events are not mutually exclusive.
P(heart) = 1352 and P(king) = 452, and P(both) = 152.
General addition rule: 1352 + 452 − 152 = 1652 = 413.

Answer: P(heart or king) = 413.

Example 3: One choice per person

Test-style

At a school, each student gets to school in exactly one way. The probability that a student walks is 0.35, and the probability that a student takes the bus is 0.45. What is the probability that a student walks or takes the bus? What is the probability that a student does neither?

Each student uses one way only, so walking and taking the bus are mutually exclusive.
P(walk or bus) = 0.35 + 0.45 = 0.8.
Neither is the complement: 1 − 0.8 = 0.2.

Answer: P(walk or bus) = 0.8, and P(neither) = 0.2.

Common mistakes

✗ MistakeP(heart or king) = 1352 + 452 = 1752.

✓ InsteadSubtract the king of hearts once: 1652 = 413.

Hearts and kings overlap, so plain adding counts the king of hearts twice. Only mutually exclusive events can be added directly.

✗ MistakeFinding P(2 or 5) on one roll as 16 × 16 = 136.

✓ InsteadFor or with mutually exclusive events, add: 16 + 16 = 13.

Multiplying finds the chance that two things both happen, such as a 2 on one roll and a 5 on the next. Or asks for either one.

✗ MistakeCalling two coin flips mutually exclusive because they do not affect each other.

✓ InsteadTwo flips are independent, and both can be heads, so they are not mutually exclusive.

Mutually exclusive means cannot happen together. Independent means one does not change the other’s chance. These are different ideas.

Not to be confused with

Mutually Exclusive Eventsvs.Independent events

Independent events can happen together and do not affect each other: heads on a coin and a 6 on a cube, with P = 12 × 16. Mutually exclusive events cannot happen together at all: P(A and B) = 0.

Mutually Exclusive Eventsvs.Dependent events

Dependent events change each other’s probabilities, like two draws without replacement. Mutually exclusive events (with nonzero chances) are one extreme kind of dependent events: one happening makes the other impossible.

Mutually Exclusive Eventsvs.Complementary events

Complementary events are mutually exclusive and cover every outcome, so their probabilities add to 1, like even and odd on a number cube. Rolling a 2 and rolling a 5 are mutually exclusive but not complementary: 13 ≠ 1.

Quick check: 3 questions

Question 1 of 3

On one roll of a number cube, which pair of events is mutually exclusive?

Show answer and explanation

Answer: C. 3 is odd, so no roll is both a 3 and even. The other pairs share outcomes: 4 and 6 are even and greater than 3, 1 is odd, and 3 and 4 are both less than 5 and more than 2.

Question 2 of 3

Events A and B are mutually exclusive, with P(A) = 0.25 and P(B) = 0.4. What is P(A or B)?

Show answer and explanation

Answer: B. Add: 0.25 + 0.4 = 0.65. 0.1 multiplies them, which is the and rule for independent events; 0.15 subtracts; 0.35 is 1 − 0.65, the chance of neither.

Question 3 of 3

If A and B are mutually exclusive, what is P(A and B)?

Show answer and explanation

Answer: A. They cannot happen together, so the probability that both happen is 0. P(A) × P(B) is the rule for independent events, and P(A) + P(B) is P(A or B).

Mutually Exclusive Events: frequently asked questions

What is the difference between mutually exclusive and independent events?

Mutually exclusive events cannot both happen: one roll cannot be a 2 and a 5. Independent events can both happen, and one does not affect the other: a coin can land heads while a cube shows 5. In fact, two events with nonzero chances cannot be both mutually exclusive and independent, because one happening rules out the other.

What is the formula for mutually exclusive events?

P(A or B) = P(A) + P(B), and P(A and B) = 0. It extends to more events: for one roll, P(1 or 2 or 3) = 16 + 16 + 16 = 12. If you are not sure the events are mutually exclusive, use the general rule and subtract P(A and B).

Are complementary events mutually exclusive?

Yes. An event and its complement, such as rain and no rain tomorrow, can never both happen, so they are mutually exclusive. They also cover every possible outcome, so their probabilities add to exactly 1. That is why P(not A) = 1 − P(A). Not every mutually exclusive pair is complementary, though.

Practice and tools

Probability practicePractice on Effortless Math Probability CalculatorCompute P(A or B) and P(A and B)Set CalculatorCheck whether two sets of outcomes overlap Dice RollerRoll dice to see mutually exclusive outcomes in action

Sources

Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 7.SP.C.8, CCSS HSS-CP.A.1, CCSS HSS-CP.B.7

Cite this entry

Effortless Math Team. “Mutually Exclusive Events.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/mutually-exclusive-events/

Flashcard

TermMutually Exclusive EventsP(A and B) = 0

DefinitionEvents that cannot happen together. One roll: P(2 or 5) = 16 + 16 = 13.

Study the full deck

Browse all terms in the Math Dictionary · More probability terms · Random term

EventAn event is a set of one or more outcomes of a chance experiment, such as “rolling an even number” on a number cube. It is part of the sample space. When outcomes are equally likely, the probability of an event is the number of outcomes in it divided by the number of outcomes in the sample space.Probability · Grade 7, Grade 8, StatisticsSample SpaceThe sample space of a chance experiment is the set of all its possible outcomes, usually written S. For one coin flip, S = {H, T}; for one number cube, S = {1, 2, 3, 4, 5, 6}. Organized lists, tables and tree diagrams help you write it out, and the probabilities of all its outcomes add to 1.Probability · Grade 7, Grade 8, StatisticsOutcomeAn outcome is one possible result of a chance experiment, such as rolling a 4 on a number cube or getting tails on a coin flip. The set of all outcomes is the sample space, and a group of outcomes is an event. When outcomes are equally likely, each one has probability 1 divided by the number of outcomes.Probability · Grade 7, Grade 8, StatisticsAdditionAddition is the operation that puts amounts together to find the total. The numbers you add are called addends, the answer is the sum, and the sign is the plus sign, +. For example, 7 + 5 = 12. On a number line, adding a positive number means moving to the right.Operations · Kindergarten, Grade 1, Grade 2Venn DiagramA Venn diagram is a picture that uses overlapping circles inside a rectangle to show how groups, or sets, are related. Each circle is a set, the overlap holds items in both sets, and the rectangle holds everything being considered. Venn diagrams make it easy to count “and”, “or” and “not” cases and to find probabilities.Probability · Grade 7, Algebra 2, StatisticsCircleA circle is the set of all points in a flat plane that are the same distance from one fixed point, the center. That distance is the radius. A segment through the center from edge to edge is the diameter, twice the radius. The distance around is the circumference, 2π r, and the area inside is π r².Geometry · Kindergarten, Grade 1, Grade 7Even NumberAn even number is an integer that can be divided by 2 with no remainder, so it can be split into two equal groups or into pairs. Even numbers end in 0, 2, 4, 6 or 8, such as 14, 236 and −10. Zero is even. Every even number can be written as 2n for some integer n.Number System · Grade 2, Grade 3, Grade 4Odd NumberAn odd number is a whole number that cannot be split into two equal groups: dividing it by 2 always leaves a remainder of 1. Odd numbers end in 1, 3, 5, 7 or 9, such as 7, 23 and 409. Every odd number can be written as 2k + 1, one more than a multiple of 2.Number System · Grade 2, Grade 3, Grade 4Independent EventsIndependent events are two events where the outcome of one does not change the probability of the other. Flipping heads on a coin does not affect rolling a 5 on a die. For independent events, multiply: P(A and B) = P(A) × P(B), so P(heads and 5) = 12 × 16 = 112.Probability · Grade 7, Algebra 1, Algebra 2ProbabilityProbability measures how likely an event is, from 0 for impossible to 1 for certain. For equally likely outcomes it is the number of favorable outcomes divided by the total number of outcomes: for a fair coin, the probability of heads is 12. Probabilities can be written as fractions, decimals or percents.Probability · Grade 6, Grade 7, Grade 8NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2Dependent EventsDependent events are two events where the outcome of the first changes the probability of the second. Drawing two cards without putting the first one back is the classic example. For dependent events, P(A and B) = P(A) × P(B ∣ A), where P(B ∣ A) is the probability of B once A has happened.Probability · Grade 7, Algebra 2, StatisticsFractionA fraction is a number that names equal parts of a whole, written ab. The bottom number, the denominator, tells how many equal parts make one whole; the top number, the numerator, tells how many of those parts you have. So 34 means 3 of 4 equal parts. Every fraction is also a point on the number line.Number System · Grade 3, Grade 4, Grade 5Two-Way TableA two-way table shows counts for two categorical variables collected from the same group, with one variable in the rows and the other in the columns. Each inside cell is a joint frequency, and the totals in the margins are marginal frequencies. Dividing a cell by a row or column total gives relative frequencies that reveal associations.Statistics · Grade 8, Algebra 1Joint FrequencyJoint frequency is the number of data values that fit two categories at the same time, such as students who are in Grade 7 and play an instrument. In a two-way table, a joint frequency is one of the inside cells, not the totals. Dividing it by the grand total gives the joint relative frequency.Statistics · Grade 8, Algebra 1Relative FrequencyRelative frequency is how often a value or event occurs compared with the total number of observations: frequency ÷ total. It can be a fraction, a decimal or a percent. If 12 of 40 students choose apples, the relative frequency of apples is 1240 = 0.3, or 30%. All the relative frequencies in a table add to 1.Statistics · Grade 7, Grade 8, StatisticsConditional ProbabilityConditional probability is the probability that an event happens given that another event has already happened. It is written P(A ∣ B), read “the probability of A given B”, and found by P(A ∣ B) = P(A and B)P(B). Knowing B shrinks the possible outcomes to only those in B.Probability · Algebra 2, Statistics

Mutually Exclusive Events

What are mutually exclusive events?

Mutually Exclusive Events in a picture

Formal definition and notation

Key facts

Addition rule for mutually exclusive events

Try it: one spin, one color

Mutually Exclusive Events examples, step by step

Example 1: Two faces of a number cube

Example 2: Events that overlap

Example 3: One choice per person

Common mistakes

Not to be confused with

Quick check: 3 questions

Mutually Exclusive Events: frequently asked questions

What is the difference between mutually exclusive and independent events?

What is the formula for mutually exclusive events?

Are complementary events mutually exclusive?

Related terms

Practice and tools

Sources

Flashcard

Login

Login and use all of our services.

Register

Register Fast!

Choose your next math review topic