Curriculum fit: GCSE Maths Probability and sets (Foundation and Higher; set notation is mainly Higher). US: High School Statistics and Probability (sets and Venn diagrams in probability).
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What Is It?
A set is a collection of items, written inside curly brackets, such as A = {2, 4, 6}. A Venn diagram uses overlapping circles inside a rectangle to show how sets overlap. The rectangle represents the universal set, written ξ.
Venn diagrams are useful for sorting data and for working out probabilities.
Key Formulas and Ideas
- A ∩ B (intersection): items in both A and B.
- A ∪ B (union): items in A or B or both.
- A′ (complement): items not in A.
- n(A ∪ B) = n(A) + n(B) − n(A ∩ B).
- P(A) = n(A) ÷ n(ξ).
Worked Examples
Set Operations
A = {2, 4, 6, 8, 10} and B = {4, 8, 12}. Find A ∩ B and A ∪ B.
- Items in both sets: 4 and 8.
- All items in either set: 2, 4, 6, 8, 10, 12.
✓ Answer A ∩ B = {4, 8}; A ∪ B = {2, 4, 6, 8, 10, 12}
Counting With a Venn Diagram
In a class of 30, 18 play football, 12 play tennis and 5 play both. How many play neither?
- Both = 5. Football only = 18 − 5 = 13. Tennis only = 12 − 5 = 7.
- At least one sport: 13 + 5 + 7 = 25.
- Neither: 30 − 25 = 5.
✓ Answer 5 students
Probability
Using the class above, find the probability that a randomly chosen student plays tennis only.
- Tennis only = 7 students.
- P = 7 ÷ 30.
✓ Answer 7/30
Common Mistakes to Avoid
- Putting the 'both' number in the football circle and the tennis circle again.
- Forgetting to subtract the overlap when counting the union.
- Forgetting the region outside the circles.
- Mixing up ∩ and ∪.
Practice Questions
Frequently Asked Questions
What does n(A) mean?
The number of elements in set A.
How do I start filling in a Venn diagram?
Begin with the overlap (the intersection), then work outwards, and finally find the number outside both circles.
Keep Going With These Guides
Probability Basics: How to Calculate the Chance of an Event
Learn how to calculate probability, use 'not' events and find the probability of two events together, with worked examples for KS3 and GCSE.
KS3GCSEMiddle SchoolHigh SchoolTree Diagrams and Conditional Probability Explained
Draw tree diagrams for independent and dependent events, including picks without replacement, with worked examples and common mistakes.
GCSEHigh SchoolHCF and LCM Using Prime Factorisation (GCF and LCM)
Write numbers as products of primes, then find the highest common factor (greatest common factor in the US) and the lowest common multiple.
GCSEMiddle SchoolHigh SchoolSources and Review
- DfE: GCSE mathematics subject content and assessment objectives
- Common Core State Standards: Mathematics (high school)
Reviewed by StudyMorrow editorial review on 5 Nov 2026.