Sheet № 71 · Foundation + Higher · AQA · Edexcel · OCR
Venn Diagrams –
Venn diagrams are a visual way to organise data and solve probability problems in GCSE Maths. They appear on both Foundation and Higher papers across AQA, Edexcel and OCR, with Higher-tier questions demanding fluency in set notation and three-circle diagrams. Once you can fill in a Venn diagram accurately, you unlock straightforward marks
§Key definitions
Question:
In a class of 30 pupils, 18 like football, 12 like tennis, and 5 like both. (a) Draw a Venn diagram. (b) How many like neither sport? (c) Find the probability that a randomly chosen pupil likes football but not tennis.
(a)
Intersection (both) = 5. Football only = 18 − 5 = 13. Tennis only = 12 − 5 = 7. Neither = 30 − 13 − 5 − 7 = 5.
(b)
5 pupils like neither sport.
(c)
Football but not tennis = 13. P(football but not tennis) = 13/30.
(d)
A' ∩ B = elements in B but not in A = {2, 4, 8, 10} — that is 4 elements. P(A' ∩ B) = 4/12 = 1/3.
§Formulas to memorise
P(event) = Number in the region ÷ Total number in the universal set
Universal set (ξ) — everything being considered.
A ∩ B (intersection) — elements in both A and B (the overlap).
A ∪ B (union) — elements in A or B or both.
A' (complement) — elements not in A.
n(A) — the number of elements in set A.
(A ∩ B)' — elements not in the intersection of A and B.
(A ∪ B)' — elements not in A or B — i.e. those outside both circles.
Start with the intersection — fill in the number or elements that belong to both sets.
The probability of an event equals the number in the relevant region divided by the total:
Worked example
In a class of 30 pupils, 18 like football, 12 like tennis, and 5 like both. (a) Draw a Venn diagram. (b) How many like neither sport? (c) Find the probability that a randomly chosen pupil likes football but not tennis.
Working:
⚠ Common mistakes
- ✗Filling in set totals instead of the "only" region — if 18 like football and 5 like both, write 13 in the "football only" section, not 18.
- ✗Forgetting the rectangle — the rectangle represents the universal set. Without it, you cannot account for items outside all circles.
- ✗Misreading set notation — (A ∪ B)' means everything not in the union, whereas A' ∪ B' means everything not in A or not in B. Use a diagram to check.
- ✗Not starting with the intersection — always fill in the middle first and work outward.
- ✗Double-counting in three-circle diagrams — subtract overlaps carefully at each step.
✦ Exam tips
- →Always start with the intersection — this is the golden rule for filling in Venn diagrams efficiently.
- →Check the total — all regions must sum to n(ξ). If they do not, you have made an arithmetic error.
- →Set notation fluency — at Higher level, expect questions using ∩, ∪ and ' notation. Practise translating words into symbols and vice versa.
- →Three-circle diagrams — work from the centre outward. The triple overlap first, then pairs, then singles, then the outside.
- →Probability from Venn diagrams — divide the number in the relevant region by n(ξ). Make sure you read the correct region for the notation given.