Mutually Exclusive Events

This lesson explains mutually exclusive events and the addition rule for probability, as set out in the AQA GCSE Mathematics specification. You will learn what it means for events to be mutually exclusive, how to use the addition rule P(A or B) = P(A) + P(B), and how exhaustive events relate to the complement rule.

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What Are Mutually Exclusive Events?

Two events are mutually exclusive if they cannot happen at the same time. When one occurs, the other is impossible.

Examples

Events	Mutually Exclusive?	Reason
Rolling a 2 and rolling a 5 on one die	Yes	A single die shows only one number
Picking a red card and picking a heart from a pack of cards	No	Hearts are red — both can happen together
Getting heads and getting tails on one coin flip	Yes	A coin can only show one side
Choosing a boy and choosing someone who wears glasses	No	A boy could wear glasses

Exam Tip: To check if events are mutually exclusive, ask: "Can both events happen at the same time?" If the answer is no, they are mutually exclusive.

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The Addition Rule (OR Rule)

For mutually exclusive events A and B:

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

This extends to any number of mutually exclusive events:

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

This works because there is no overlap between mutually exclusive events — you cannot "double count".

Worked Example 1

A fair die is rolled. Find the probability of rolling a 2 or a 5.

Rolling a 2 and rolling a 5 are mutually exclusive (you cannot roll both at the same time).

P(2 or 5) = P(2) + P(5) = 1/6 + 1/6 = 2/6 = 1/3

Worked Example 2

A bag contains 4 red, 6 blue, and 5 yellow counters. A counter is picked at random. Find the probability of picking a red or yellow counter.

• Total counters = 4 + 6 + 5 = 15
• Picking red and picking yellow are mutually exclusive.

P(red or yellow) = P(red) + P(yellow) = 4/15 + 5/15 = 9/15 = 3/5

Exam Tip: The addition rule P(A or B) = P(A) + P(B) only works when A and B are mutually exclusive. If the events can happen together, you must subtract the overlap: P(A or B) = P(A) + P(B) - P(A and B). This extended formula is mainly tested at Higher tier.

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Exhaustive Events

A set of events is exhaustive if the events cover all possible outcomes. At least one of the events must happen.

For example, when rolling a die, the events {1, 2, 3, 4, 5, 6} are exhaustive because every possible result is included.

If events are both mutually exclusive and exhaustive, their probabilities must add up to 1.

graph TD
    A[All Possible Outcomes] --> B[Event A]
    A --> C[Event B]
    A --> D[Event C]
    B ---|Mutually exclusive| C
    C ---|Mutually exclusive| D
    B ---|P A + P B + P C = 1| D

Worked Example 3

A spinner can land on red, blue, or green. These are the only options.

Colour	Probability
Red	0.35
Blue	0.25
Green	?

Since the events are mutually exclusive and exhaustive:

P(red) + P(blue) + P(green) = 1

0.35 + 0.25 + P(green) = 1

P(green) = 1 - 0.35 - 0.25 = 0.4

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The Complement Rule

The complement of an event A (written A') is "event A does not happen".

A and A' are always mutually exclusive and exhaustive, so:

P(A) + P(A') = 1

Rearranging:

P(A') = 1 - P(A)

Worked Example 4

The probability that a student passes an exam is 0.85. What is the probability that the student does not pass?

P(not pass) = 1 - 0.85 = 0.15

Worked Example 5

A card is picked at random from a standard pack of 52 playing cards. Find the probability that the card is not a king.

• P(king) = 4/52 = 1/13

P(not king) = 1 - 1/13 = 12/13

Exam Tip: Use the complement rule whenever it is easier to calculate the probability of something NOT happening. This is especially useful for "at least one" problems — P(at least one) = 1 - P(none).

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Using the Addition Rule with the Complement

Worked Example 6

A biased die has the following probabilities:

Score	1	2	3	4	5	6
Probability	0.1	0.15	0.15	0.2	0.15	0.25

Find P(score is greater than 4).

Scores greater than 4 are 5 and 6. These are mutually exclusive:

P(5 or 6) = P(5) + P(6) = 0.15 + 0.25 = 0.4

Alternatively, use the complement: P(greater than 4) = 1 - P(4 or less)

P(4 or less) = 0.1 + 0.15 + 0.15 + 0.2 = 0.6

P(greater than 4) = 1 - 0.6 = 0.4 (same answer).

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Common Mistakes

Mistake	Correction
Adding probabilities of events that are not mutually exclusive	Check whether the events can happen at the same time first
Probabilities not adding up to 1 for exhaustive events	All outcomes must be accounted for
Forgetting to check if events are mutually exclusive before adding	Always verify before applying the addition rule
Confusing "or" with "and"	"Or" uses addition (for mutually exclusive); "and" uses multiplication

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Summary

• Mutually exclusive events cannot happen at the same time.
• For mutually exclusive events: P(A or B) = P(A) + P(B).
• Exhaustive events cover all possible outcomes; their probabilities add up to 1.
• The complement rule: P(not A) = 1 - P(A).
• Use the complement rule for "at least one" problems.
• Always check that events are mutually exclusive before adding their probabilities.