Mutually Exclusive Events: Formula, Rules and Examples

Mutually Exclusive Events are events that cannot occur together in the same trial. Their intersection is empty, so the probability of their simultaneous occurrence is zero. This page explains the addition formula, pairwise cases, differences from non-mutually exclusive events, and solved probability examples for aptitude questions.

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What Are Mutually Exclusive Events?

Two events are mutually exclusive when they cannot occur at the same time. Mathematically, for events A and B, A ∩ B = ∅ and P(A ∩ B) = 0.

For mutually exclusive events, the probability that at least one event occurs is the sum of their individual probabilities: P(A ∪ B) = P(A) + P(B). For example, when one die is rolled, getting 2 and getting 5 are mutually exclusive because one roll cannot show both numbers.

Mutually Exclusive Events Video Lessons

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Complementary and Mutually Exclusive Events

Understand complementary and mutually exclusive events in probability, including their definitions, differences, and how they relate to possible outcomes in an experiment.

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Mutually Exclusive Events Formula & Tricks

Important Formulas

Intersection of mutually exclusive events
A ∩ B = ∅; P(A ∩ B) = 0

There is no common outcome between mutually exclusive events.

Addition formula
P(A ∪ B) = P(A) + P(B)

Use this when A and B cannot occur together.

General addition rule
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

This applies to any two events. For mutually exclusive events, the intersection term is zero.

Three or more pairwise mutually exclusive events
P(A₁ ∪ A₂ ∪ ... ∪ Aₙ) = P(A₁) + P(A₂) + ... + P(Aₙ)

The probabilities can be added when every pair of events has no common outcome.

Quick Tricks

Check the overlap before adding probabilities

If one outcome cannot satisfy both events, the events are mutually exclusive and their probabilities can be added directly.

Example: On a die, getting an even number and getting an odd number are mutually exclusive. Therefore, P(even or odd) = 1/2 + 1/2 = 1.
Use the complement when all listed outcomes are covered

If mutually exclusive events together contain every possible outcome, their total probability is 1.

Example: For a die, the events of rolling 1, 2, 3, 4, 5, or 6 are pairwise mutually exclusive, so their probabilities add to 6 × 1/6 = 1.

Mutually Exclusive Events Concepts

Identifying mutually exclusive events

Events are mutually exclusive if their common part contains no outcome.

Represent the events as sets of outcomes. If A ∩ B is empty, the events are mutually exclusive. If even one outcome belongs to both sets, they are not mutually exclusive.

Example: For a die, A = {1, 2} and B = {5, 6}. Since A ∩ B = ∅, A and B are mutually exclusive.

Applying the addition theorem

For mutually exclusive events, the probability of either event is the sum of their probabilities.

Use P(A or B) = P(A) + P(B). For a fair die, let A be getting 1 and B be getting 6. Then P(A or B) = 1/6 + 1/6 = 1/3.

Example: If P(A) = 0.35 and P(B) = 0.25 and A, B are mutually exclusive, then P(A ∪ B) = 0.35 + 0.25 = 0.60.

Pairwise mutually exclusive events

A group of events is pairwise mutually exclusive when every two different events in the group are mutually exclusive.

For pairwise mutually exclusive events A, B, and C, all pairwise intersections are empty: A ∩ B = B ∩ C = C ∩ A = ∅. Hence, P(A ∪ B ∪ C) = P(A) + P(B) + P(C).

Example: The outcomes of rolling 1, rolling 3, and rolling 6 on one die are pairwise mutually exclusive. Their combined probability is 1/6 + 1/6 + 1/6 = 1/2.

Mutually exclusive versus independent events

Mutually exclusive events cannot occur together, whereas independent events do not change each other's probabilities.

For mutually exclusive events, P(A ∩ B) = 0. For independent events, P(A ∩ B) = P(A)P(B). Two events with positive probabilities cannot be both mutually exclusive and independent, because P(A)P(B) would be positive while their intersection probability is zero.

Example: Getting a head and getting a tail on one toss are mutually exclusive. Getting a head on the first toss and a head on the second toss are independent events.

Mutually Exclusive Events Visual Guide & Data

Use the table and chart below to understand the data better.

Comparison of event relationships

The intersection determines whether two events are mutually exclusive or independent.

Event relationship Intersection condition Probability rule
Mutually exclusive A ∩ B = ∅ P(A ∪ B) = P(A) + P(B)
Not mutually exclusive A ∩ B ≠ ∅ P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Independent P(A ∩ B) = P(A)P(B) The occurrence of one does not affect the other

Note: Events with positive probabilities cannot be both mutually exclusive and independent.

Disjoint sets for mutually exclusive events

Visual guide: Disjoint sets for mutually exclusive events
Events A and B have no common region, so their intersection is empty.

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Practice Mutually Exclusive Events Questions

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1√1156 + √0.36 + √0.0025 = ?→ 2At what percentage above the cost price of an article is to be marked to gain 20%, after allowing a discount of 25%?→ 3A dealer purchased a washing machine for ₹ 7660. If he allows a discount of 12% on its marked price and still gains 10%, then the marked price of the machine is :→ 4Which fraction is equivalent to 0.245?→ 5The price of an item decreases from 500 to 425. What is the percentage decrease?→ 6Selling price of an article is Rs. 2400. Profit percentage is 20 percent. What is the cost price of the article?→ 7During a sale 50% of the goods are sold at 19% profit, 20% of the remaining goods are sold at 40% profit and the still remaining goods are sold at a loss of 25%. If there is an overall profit of x% , then what is the value of x ?→ 8Anwar’s salary is ₹19,000 per month. He spends ₹4,500 on house rent, ₹1,500 on bills and the rest of the amount is his monthly savings. Find his savings (in ₹) in a year, if in the month of his birthday, he spent his complete monthly savings for his birthday celebration.→ 9Two students appeared for an exam. One student scored 6 marks more than the other, and his score was 60% of their sum. Find the sum of their scores.→ 10The income of Amit in 2019 was ₹25,300. He gets an increment of 10% every year. What was his income (in ₹) in 2021?→

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Quick Revision Notes

Mutually Exclusive Events: Revision Points

Recall the conditions and formulas used to solve mutually exclusive event questions.

  • Mutually exclusive events cannot occur together.
  • A ∩ B = ∅ and P(A ∩ B) = 0.
  • For mutually exclusive events, P(A ∪ B) = P(A) + P(B).
  • Use the general addition rule when events have a common outcome.
  • Pairwise mutually exclusive events allow direct addition of all individual probabilities.
  • Events with positive probabilities cannot be both mutually exclusive and independent.

Related Topic Explainers

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Mutually Exclusive Events FAQs

What is the formula for mutually exclusive events?

For events A and B, P(A ∪ B) = P(A) + P(B), because P(A ∩ B) = 0.

Are getting an even number and getting an odd number on a die mutually exclusive?

Yes. A single die result cannot be both even and odd, so their intersection is empty.

If P(A) = 0.4 and P(B) = 0.3, what is P(A or B) when the events are mutually exclusive?

P(A ∪ B) = 0.4 + 0.3 = 0.7.

Can mutually exclusive events be independent?

Only in the special case where at least one event has probability zero. If both probabilities are positive, they cannot be independent because their intersection is zero.

What is the difference between mutually exclusive and independent events?

Mutually exclusive events cannot occur together, so P(A ∩ B) = 0. Independent events satisfy P(A ∩ B) = P(A)P(B), meaning one event does not affect the other.

How do you solve a question involving three pairwise mutually exclusive events?

Add their probabilities. For example, if the probabilities are 1/5, 1/4, and 1/10, their union probability is 1/5 + 1/4 + 1/10 = 11/20.

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