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Edexcel IGCSE Maths (4MA1) · Unit 6: Statistics & Probability

Edexcel IGCSE 4MA1 Sets and Venn Diagrams: exam questions and answers, notes and worked examples

Updated October 2026 · By Pete Bromfield

A set is a collection of elements in curly brackets. $A \cup B$ (the union) is everything in A or B or both, $A \cap B$ (the intersection) is everything in both, and $A^{'}$ is everything in the universal set $E$ that is not in A. In a Venn diagram, fill the overlap first, then the rest of each circle, then the outside, and check that the regions add up to the total: $n (A \cup B) = n (A) + n (B) - n (A \cap B)$ .

Unit 6: Statistics & Probability
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Notes
Formulas
Worked examples
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FAQ

Sets and Venn Diagrams notes

Set notation

$x \in A$ means $x$ is an element of A; $n (A)$ is the number of elements in A; $\emptyset$ is the empty set; $B \subset A$ means every element of B is also in A. With $E = {1, 2, \dots, 10}$ and $A = {multiples of 3}$ , $A = {3, 6, 9}$ and $n (A^{'}) = 7$ .

Union, intersection and complement

$A \cup B$ is "A or B", $A \cap B$ is "A and B". For $A = {2, 4, 6, 8}$ and $B = {1, 2, 3, 4}$ : $A \cap B = {2, 4}$ and $A \cup B = {1, 2, 3, 4, 6, 8}$ . $A^{'} \cap B$ is the part of B outside A, here ${1, 3}$ ; $(A \cup B)^{'}$ is the region outside both circles.

Counting with a Venn diagram

Fill the regions from the inside out. Of 40 students, 25 play football, 18 play tennis and 8 play both. Football only is 17 and tennis only is 10, so 35 play at least one and 5 play neither. The formula gives the same: $25 + 18 - 8 = 35$ .

Three sets

Start with the centre (in all three), then the three "exactly two" regions (subtract the centre from each pair), then each set on its own, and finally the outside. Check that all eight regions add up to the total.

Key formulas

Sets Learn it: $n (A \cup B) = n (A) + n (B) - n (A \cap B)$

Learn it = not printed on the 4MA1 Higher formulae sheet, so learn it by heart. The whole 4MA1 formula sheet (free PDF) → · 4MA1 formulas to learn

Sets and Venn Diagrams worked examples

Worked example 1: Union and intersection

$A = {1, 2, 3, 4, 5}$ , $B = {4, 5, 6, 7}$ . Find $A \cup B$ and $A \cap B$ .

Solution

Union: everything in either set, ${1, 2, 3, 4, 5, 6, 7}$ .
Intersection: in both, ${4, 5}$ .

3 steps

Tip: ∪ is 'or', ∩ is 'and'.

Worked example 2: Counting with a Venn diagram

$n (A) = 15$ , $n (B) = 12$ , $n (A \cup B) = 20$ . Find $n (A \cap B)$ .

Solution

$n (A \cup B) = n (A) + n (B) - n (A \cap B)$ .
$20 = 27 - n (A \cap B)$ , so $n (A \cap B) = 7$ .

3 steps

Tip: The intersection is counted twice in n(A) + n(B).

Worked example 3: Venn problem solving

In a class of 30, 18 study French, 15 study Spanish and 4 study neither. How many study both?

Solution

Studying at least one: $30 - 4 = 26$ .
Both $= 18 + 15 - 26 = 7$ .

3 steps · 1 to fill in

Tip: Draw the diagram and check all regions total 30.

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Sets and Venn Diagrams exam-style questions

Try each question before you open the answer (one has its full worked solution open). The mark schemes use Edexcel-style marks: M for method, A for accuracy (after the method mark), B for an independent result.

Question 1 · easy · 2 marks

$ξ = {whole numbers from 1 to 15}$ , A = {even numbers} and B = {multiples of 3}. (a) List the members of $A \cap B$ . (b) Find $n (A \cup B)$ .

Show the answer and mark scheme

B1 6, 12
B1 10

Answer: (a) 6, 12 (b) 10

Question 2 · easy · 2 marks

A sports centre asked 50 of its members which facilities they use. 27 use the gym, 21 use the pool and 6 use both the gym and the pool. How many of these members use neither the gym nor the pool?

Show the answer and mark scheme

M1 27 + 21 − 6 = 42 (or 21 + 6 + 15 in the Venn diagram)
A1 8

Answer: 8

Question 3 · medium · 3 marks

$ξ = {whole numbers from 1 to 12}$ , P = {prime numbers} and O = {odd numbers}. (a) List the members of $P^{'} \cap O$ . (b) List the members of $(P \cup O)^{'}$ . (c) Find $n (P \cap O^{'})$ .

Full worked solution Free, no sign-in

B1 1, 9
B1 4, 6, 8, 10, 12
B1 1

Answer: (a) 1, 9 (b) 4, 6, 8, 10, 12 (c) 1

Every question here has a full worked solution. See them all →

Question 4 · medium · 3 marks

$n (ξ) = 90$ , $n (A) = 48$ , $n (B) = 35$ and $n (A^{'} \cap B^{'}) = 22$ . Find $n (A \cap B)$ .

Show the answer and mark scheme

M1 n(A ∪ B) = 90 − 22 = 68
M1 48 + 35 − 68
A1 15

Answer: 15

Question 5 · hard · 4 marks

120 guests were asked which drinks they had: tea (T), coffee (C) and juice (J). 55 had tea, 62 had coffee and 40 had juice. 30 had both tea and coffee, 20 had both coffee and juice, and 18 had both tea and juice; these numbers include the 12 guests who had all three. How many guests had none of the three drinks?

Show the answer and mark scheme

M1 12 in the centre and 18, 8, 6 in the 'exactly two' regions
M1 T only 19, C only 24, J only 14
M1 total inside the circles 101 (or 55 + 62 + 40 − 30 − 20 − 18 + 12)
A1 19

Answer: 19

Question 6 · hard · 5 marks

There are 60 students in a club. 28 play hockey (H) and 20 play netball (N). $x$ students play both, and $2 x$ students play neither. (a) Find the value of $x$ . (b) One student is chosen at random. Find the probability that this student plays exactly one of the two sports.

Show the answer and mark scheme

M1 H only 28 − x and N only 20 − x
M1 (28 − x) + x + (20 − x) + 2x = 60
A1 x = 12
M1 (28 − 12) + (20 − 12) = 24 students play exactly one
A1 24/60 = 2/5 oe

Answer: (a) $x = 12$ (b) $\frac{2}{5}$

Question 7 · hard · 2 marks

$n (ξ) = 40$ , $n (A) = 25$ and $n (B) = 22$ . Find the least possible value of $n (A \cap B)$ .

Show the answer and mark scheme

M1 n(A ∪ B) ≤ 40, so n(A ∩ B) ≥ 25 + 22 − 40
A1 7

Answer: 7

Question 8 · very hard · 5 marks

Each of 64 students studies at least one of Art (A), Biology (B) and Chemistry (C). 7 study all three subjects. 12 study Art and Biology only, 9 study Biology and Chemistry only, and 5 study Art and Chemistry only. Twice as many students study only Biology as study only Chemistry, and the number who study only Art is 3 more than the number who study only Chemistry. (a) Find the number of students who study only Chemistry. (b) A student who studies Biology is chosen at random. Find the probability that this student also studies Chemistry.

Show the answer and mark scheme

M1 only C = x, only B = 2x, only A = x + 3
M1 7 + 12 + 9 + 5 + x + 2x + x + 3 = 64 oe
A1 x = 7
M1 n(B) = 14 + 12 + 9 + 7 = 42 and n(B ∩ C) = 9 + 7 = 16
A1 16/42 = 8/21 oe

Answer: (a) 7 (b) $\frac{8}{21}$

Common sets and Venn diagrams mistakes

Counting the overlap twice. $n (A)$ includes the overlap. Subtract it from each circle before you fill in the "only" regions.
Forgetting the outside region. Elements of the universal set that are in neither set go outside the circles but inside the rectangle.
Mixing up ∪ and ∩. ∪ (union) is everything in either set; ∩ (intersection) is only what is in both. The union is never smaller than the intersection.
Listing an element twice. Each element appears once in a set, and in exactly one region of the Venn diagram.

Where students lose marks across 4MA1, from the examiner reports →

Sets and Venn Diagrams FAQ

What does A′ mean?

A′, the complement of A, is every element of the universal set that is not in A. So n(A) + n(A′) = n(ℰ).

How do I solve a Venn diagram word problem?

Draw the diagram, put the number in the overlap first, then work out each 'only' region and the outside region. Check the four regions add up to the total.

Can I use a calculator for sets and Venn diagrams in 4MA1?

Yes. A calculator may be used in all four 4MA1 papers, so both Higher papers (1H and 2H) allow one. Most marks are for the method you write down, so show your working.

How long are the 4MA1 Higher papers?

Papers 1H and 2H are 2 hours each and 100 marks each, worth 50% each. Each tier has its own formulae sheet in the exam.

Next steps

Statistics and Probability revision notes · 4MA1 formula sheet · Taking Cambridge 0580 instead? Cambridge 0580 sets and venn diagrams questions

Every question and worked example here is original, written for IGCSE Math Revision rather than copied from Pearson papers, and every result was re-solved independently before publishing.

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Edexcel IGCSE 4MA1 Sets and Venn Diagrams: exam questions and answers, notes and worked examples

Try it first

Sets and Venn Diagrams notes

Set notation

Union, intersection and complement

Counting with a Venn diagram

Three sets

Key formulas

Sets and Venn Diagrams worked examples

Worked example 1: Union and intersection

Solution

Worked example 2: Counting with a Venn diagram

Solution

Worked example 3: Venn problem solving

Solution

Sets and Venn Diagrams exam-style questions

Question 1 · easy · 2 marks

Question 2 · easy · 2 marks

Question 3 · medium · 3 marks

Full worked solution Free, no sign-in

Question 4 · medium · 3 marks

Question 5 · hard · 4 marks

Question 6 · hard · 5 marks

Question 7 · hard · 2 marks

Question 8 · very hard · 5 marks

Common sets and Venn diagrams mistakes

Sets and Venn Diagrams FAQ

Next steps

Practise sets and Venn diagrams with your progress saved

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