What does 'mutually exclusive' mean?
Think: no overlap.
Mutually exclusive events cannot happen at the same time.
Mutually Exclusive Events
Mutually exclusive events cannot happen at the same time. This builds on basic probability and helps when calculating combined probabilities, especially in conditional probability.
Overview
Two events are mutually exclusive if they cannot happen at the same time.
If one event occurs, the other cannot occur in the same trial.
If events A and B are mutually exclusive, then \( P(A \text{ or } B) = P(A) + P(B) \)
The key step is deciding whether events can happen together before choosing the correct probability rule.
What you should understand after this topic
- Understand what mutually exclusive means
- Recognise when events cannot happen together
- Add probabilities for mutually exclusive events
- Avoid adding probabilities when events overlap
- Apply this rule in exam-style questions
Key Definitions
Event
A possible result or outcome in probability.
Mutually Exclusive
Two events that cannot happen at the same time.
Overlap
A situation where two events can happen together.
\(A \text{ or } B\)
The probability that event A happens, or event B happens, or both.
Addition Rule
For mutually exclusive events, add the probabilities.
Single Trial
One experiment, such as one coin toss or one dice roll.
Key Rules
Cannot happen together
Then the events are mutually exclusive.
Can happen together
Then they are not mutually exclusive.
Add the probabilities
\( P(A \text{ or } B) = P(A) + P(B) \)
Check the context
Ask whether both events could happen in the same trial.
Quick Comparison
Mutually exclusive
Rolling a 2 and rolling a 5 on one dice roll.
Not mutually exclusive
Choosing a card that is red and also a king.
Mutually exclusive
Getting heads or tails on one coin toss.
Not mutually exclusive
A student liking maths and liking science.
How to Solve
Step 1: Understand mutually exclusive events
Two events are mutually exclusive if they cannot happen at the same time. In a single trial, only one of the events can occur.
Exam tip: If one event happens, the other cannot happen.
Key question: Can both events happen together? If NO → mutually exclusive.
Step 2: Recognise examples
Dice
Rolling a 2 and a 5 at the same time is impossible.
Coin
Getting heads and tails together is impossible.
Cards (not mutually exclusive)
A card can be red AND a king.
Step 3: Use the addition rule
If events are mutually exclusive, add their probabilities.
\( P(A \text{ or } B) = P(A) + P(B) \)
Why this works: There is no overlap to remove.
This builds on basic probability.
Step 4: When events are not mutually exclusive
If events can happen together, you must subtract the overlap.
\( P(A \text{ or } B) = P(A) + P(B) - P(A \cap B) \)
Exam thinking: Add both, then subtract the overlap.
This connects to conditional probability, where overlap plays a key role.
Step 5: Decide in exam questions
- Read both events carefully.
- Ask if they can happen together.
- If NO → add probabilities.
- If YES → subtract overlap.
Step 6: Visual idea
Mutually exclusive events do not overlap, while other events can overlap.
Example Questions
Edexcel
Exam-style questions inspired by Edexcel GCSE Mathematics, focusing on identifying mutually exclusive events and adding probabilities.
Edexcel
A fair dice is rolled once.
Find the probability of rolling a 2 or rolling a 5.
Edexcel
A fair spinner has 8 equal sections numbered 1 to 8.
Find the probability that the spinner lands on 1 or 3 or 7.
Edexcel
A card is chosen at random from a standard pack of 52 cards.
Explain why choosing a heart and choosing a club are mutually exclusive events.
AQA
Exam-style questions based on the AQA GCSE Mathematics specification, focusing on deciding whether events can happen at the same time.
AQA
A fair dice is rolled once. A student says, "Rolling an even number and rolling a 4 are mutually exclusive events."
Tick one box. True ☐ False ☐
Give a reason for your answer.
AQA
A fair dice is rolled once. Find the probability of rolling a number less than 3 or a number greater than 4.
AQA
A counter is chosen at random from a bag containing 5 red counters, 3 blue counters and 2 green counters.
Find the probability that the counter is blue or green.
OCR
Exam-style questions aligned with OCR GCSE Mathematics, emphasising reasoning with mutually exclusive and non-mutually exclusive events.
OCR
A card is chosen at random from a standard pack of 52 cards.
Are the events "choosing a red card" and "choosing a king" mutually exclusive? Give a reason for your answer.
OCR
Events A and B are mutually exclusive.
\( P(A)=0.35 \quad \text{and} \quad P(B)=0.28 \)
Find \( P(A \text{ or } B) \).
OCR
Events C and D are mutually exclusive.
\( P(C)=\frac{2}{5} \quad \text{and} \quad P(C \text{ or } D)=\frac{7}{10} \)
Find \( P(D) \).
Exam Checklist
Common Mistakes
These are common mistakes students make when working with mutually exclusive events in GCSE Maths.
Assuming “different” means mutually exclusive
Incorrect
A student thinks two different events must be mutually exclusive.
Correct
Events are only mutually exclusive if they cannot happen at the same time. Different events can still occur together.
Adding probabilities when events overlap
Incorrect
A student adds probabilities of events that can happen together.
Correct
You can only add probabilities directly if the events are mutually exclusive. If they overlap, you must account for the intersection.
Ignoring the number of trials
Incorrect
A student assumes events are mutually exclusive without checking the context.
Correct
Check whether the situation involves one trial or multiple trials. For example, getting heads and tails is mutually exclusive in one toss, but not across multiple tosses.
Confusing “or” with “and”
Incorrect
A student mixes up addition and multiplication rules.
Correct
“Or” usually means add probabilities (for mutually exclusive events). “And” means multiply probabilities.
Not recognising simple mutually exclusive events
Incorrect
A student fails to identify obvious cases.
Correct
Events like heads or tails in a single coin toss are mutually exclusive because only one outcome can occur at a time.
Try It Yourself
Practise calculating probabilities of mutually exclusive events.
Foundation
Foundation Practice
Understand mutually exclusive events and calculate probabilities.
A die is rolled. What is the probability of getting a 2 or a 5?
Add probabilities of 2 and 5.
P(2) = 1/6, P(5) = 1/6 → total = 2/6 = 1/3.
A card is picked from a standard deck. What is the probability of getting a king or a queen?
There are 4 kings and 4 queens.
8 out of 52 → 8/52 = 2/13.
A spinner has 4 equal sections: A, B, C, D. What is the probability of landing on A or B?
Add probabilities.
1/4 + 1/4 = 1/2.
Which pair of events is mutually exclusive?
They cannot happen together.
You cannot get 3 and 5 at the same time.
A die is rolled. What is the probability of getting a 1 or a 6?
Add probabilities.
1/6 + 1/6 = 1/3.
If events are mutually exclusive, what happens to their probabilities when finding 'A or B'?
No overlap to worry about.
Mutually exclusive → add probabilities.
A bag has 3 red, 2 blue, 5 green balls. Find P(red or blue).
Add favourable outcomes.
3 + 2 = 5 out of 10 → 1/2.
A student multiplies probabilities for mutually exclusive events. What is wrong?
Mutually exclusive uses addition.
Multiplication is for independent events, not mutually exclusive.
A spinner has 5 equal sections. What is the probability of landing on section 1 or 2?
Add probabilities.
1/5 + 1/5 = 2/5.
Higher
Higher Practice
Use addition rules and identify when events are not mutually exclusive.
Which formula applies to mutually exclusive events?
No overlap.
Mutually exclusive events → add probabilities.
A die is rolled. Find P(getting a number less than 3 or greater than 5).
List outcomes.
Less than 3: 1,2. Greater than 5: 6 → 3/6 = 1/2.
Which pair is NOT mutually exclusive?
Can they happen together?
A person can be tall and play basketball → not mutually exclusive.
A card is chosen. Find P(king or heart).
Not mutually exclusive — subtract overlap.
Kings = 4, hearts = 13, overlap = 1 → (4+13−1)/52 = 16/52 = 4/13.
Why must overlap be subtracted when events are not mutually exclusive?
Double counting issue.
Overlap is counted twice if not removed.
A bag has 5 red, 3 blue, 2 striped (both red and blue counted once). Find P(red or blue).
Subtract overlap if counted twice.
5 + 3 − 0 overlap → 8/10.
A student assumes all events are mutually exclusive. What is the danger?
Think overlap.
Overlap can be counted twice incorrectly.
A die is rolled. Find P(odd or even).
All outcomes are included.
Odd or even covers all possibilities → probability = 1.
Which statement is true for mutually exclusive events?
They cannot happen together.
Mutually exclusive → no overlap → P(A ∩ B) = 0.
A spinner has 8 equal sections. Find P(section 1 or 2 or 3).
Add probabilities.
1/8 + 1/8 + 1/8 = 3/8.
Games
Practise this topic with interactive games.
Games coming soon.
Frequently Asked Questions
What does mutually exclusive mean?
Events that cannot happen together.
How do probabilities combine?
Add them.
Example?
Getting heads or tails in one coin toss.