Fill the four regions of a two-set Venn diagram from the information given, use set notation, and read probabilities from the regions. It is on Foundation and Higher, aimed at grade 5. Free questions with answers are on this page.
A Venn diagram of two sets has four regions: only A, only B, both (intersection A∩B), and neither. The union A∪B is everything in at least one set. Fill the overlap first when a question gives n(A∩B), then the 'only' regions, then outside. The whole rectangle is the universal set, total n.
P(A∣B) at this grade is the overlap over the B circle, not over the whole rectangle. Set-notation questions want the region shaded or a probability, not an essay. Two-set questions that give n(A), n(B) and n(A∪B) use n(A∪B)=n(A)+n(B)−n(A∩B).
Fill from the overlap outwards. n(A∪B)=n(A)+n(B)−n(A∩B). A∩B is both; A∪B is either; A′ is not A. Probability is a region over the total, unless the question restricts to one circle.
Objectives
Find the probability of A or B from the regions.
Find the probability of A and B, and of neither.
Find the probability of A, which includes the overlap.
Key points
Fill in a Venn diagram starting from the overlap and working outwards, including the 'neither' region.
Use set notation: A∩B (and/both), A∪B (or/either), A′ (not A), ξ (the universal set).
List the elements of sets built from numbers (multiples, factors, primes).
Find a probability as a region count over the total.
Worked examples
Example 1
Only A = 2, both = 1, only B = 3, neither = 4. Find P(A or B).
2 marks
Find total in sample space: 2 + 1 + 3 + 4 = 10
Add regions inside A or B: 2 + 1 + 3 = 6
Form probability fraction: 6/10 = 3/5
Answer: 3/5
Common slips
Slip: Amber counted 8 Sum all 4 regions: only A, both, only B, neither.
Slip: Amber excluded the overlap to get 5 A or B includes both crescents and the intersection.
Slip: Ashley wrote 1/2 Divide 6 by 10 and simplify.
Example 2
Only A = 2, both = 1, only B = 3, neither = 4. What is the probability of A?
2 marks
Sum all regions for denominator: 2 + 1 + 3 + 4 = 10
Sum regions belonging to circle A: only A + both = 2 + 1 = 3
Form fraction: 3/10
Answer: 3/10
Common slips
Slip: Belinda used 6 Denominator must be the total of all regions.
Slip: Boris took only A = 2 The set A includes the central intersection.
Slip: Belinda wrote 1/5 3 out of 10 gives 3/10.
Example 3
Only A = 6, both = 2, only B = 4, neither = 8. Find P(A and B).
3 marks
Calculate total: 6 + 2 + 4 + 8 = 20
Identify intersection "A and B": both = 2
Form and simplify fraction: 2/20 = 1/10
Answer: 1/10
Common slips
Slip: Clayton used 12 Include all 4 regions for the total.
Slip: Clayton added only A to get 8 "A and B" is exclusively the shared overlap.
Slip: Clayton wrote 2/5 2 out of 20 simplifies to 1/10.
Questions and answers
GCSE Maths venn diagrams questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
Only A = 6, both = 1, only B = 2, neither = 11. Find P(A or B).
Show answer
Answer: 9/20
Inside circles: 6 + 1 + 2 = 9. Total: 20.
Fraction: 9/20.
M1 — Inside circles: 6 + 1 + 2 = 9. Total: 20.
A1 — Fraction: 9/20.
Question 2 Easy2 marks
Only A = 5, both = 2, only B = 3, neither = 10. Find P(A or B).
Show answer
Answer: 1/2
Inside circles: 5 + 2 + 3 = 10. Total: 20.
Fraction: 10/20 = 1/2.
M1 — Inside circles: 5 + 2 + 3 = 10. Total: 20.
A1 — Fraction: 10/20 = 1/2.
Question 3 Medium2 marks
Only A = 2, both = 1, only B = 3, neither = 4. Find P(A and B).
Show answer
Answer: 1/10
Overlap: 1. Total: 10.
Fraction: 1/10.
M1 — Overlap: 1. Total: 10.
A1 — Fraction: 1/10.
Question 4 Medium2 marks
Only A = 1, both = 3, only B = 2, neither = 4. Find P(A and B).
Show answer
Answer: 3/10
Overlap: 3. Total: 10.
Fraction: 3/10.
M1 — Overlap: 3. Total: 10.
A1 — Fraction: 3/10.
Question 5 Hard2 marks
Only A = 3, both = 3, only B = 6, neither = 3. What is the probability of neither?
Show answer
Answer: 1/5
Neither: 3. Total: 15.
Fraction: 3/15 = 1/5.
M1 — Neither: 3. Total: 15.
A1 — Fraction: 3/15 = 1/5.
Question 6 Hard2 marks
Only A = 6, both = 1, only B = 2, neither = 11. What is the probability of A?
Show answer
Answer: 7/20
Set A: 6 + 1 = 7. Total: 20.
Fraction: 7/20.
M1 — Set A: 6 + 1 = 7. Total: 20.
A1 — Fraction: 7/20.
Question 7 Exam-style3 marks
Only A = 5, both = 3, only B = 2, neither = 10. Find P(A or B).
Show answer
Answer: 1/2
Total in universe is 5 + 3 + 2 + 10 = 20.
Inside A or B is 5 + 3 + 2 = 10.
Probability is 10 / 20 = 1/2.
M1 — Sum all 4 regions to find universal total 20
M1 — Sum regions belonging to A or B: 5 + 3 + 2 = 10
A1 — 10/20 = 1/2
Common mistakes
Most lost marks on this topic are one slip.
Leaving the overlap out of A or B
"A or B" is the union and includes both crescents plus the intersection
Omitting the overlap when finding P(A)
Set A consists of (only A) + (both)
Forgetting the neither region when calculating the total denominator
The sample space includes all 4 regions
Spot the slip
only A = 2, both = 1, only B = 3, neither = 4. Ada forgets the overlap when she counts A or B. What is the probability of A or B?
3/5
1/2
1/10
Show the slip
3/5 A or B is only A, both, and only B: 6 out of 10.
Exam tips
Be ready to do this without a calculator. Paper 1 is the non-calculator paper.
If the question says “show that”, the result is given. The marks are for the steps that reach it.
The method is the same on AQA-style and Edexcel-style papers.
Always check the four regions add to the total and write the check down — it is a free confidence boost and shows method.
Read the question twice for the word 'both' — it tells you whether the overlap is given or has to be found.
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.