Probability

Venn diagrams and sets

Q: What does the upside-down U mean in a Venn diagram?

A \cap B means the intersection: things in both A and B. The U shape, A \cup B, means the union: things in A or B or both.

12 past-paper questions, easiest first. Try each one, then reveal the answer.

Foundation & Higher10 of 12 from Foundation papers

Practise this topic online

Revision guide by The Full Marks

Venn diagrams and sets: what you need to know

Venn diagrams sort things into overlapping circles so you can see which belong to one set, both sets or neither. GCSE questions ask you to fill one in and then find probabilities from it.

Both tiers need two- and three-set Venn diagrams and set notation; 'given that' (conditional) probabilities from a Venn diagram are Higher only.

Key facts

$\xi$ is the universal set: everything in the question, including items outside the circles.
$A \cap B$ (intersection) means in A and B: the overlap.
$A \cup B$ (union) means in A or B or both: everything inside either circle.
$A'$ (complement) means not in A.
Always fill in the overlap first, then the rest of each circle, then the outside.

Completing a Venn diagram from totals

Put the number in both sets in the overlap (work it out if needed).
Circle A only = total in A − overlap. Do the same for B.
Outside = overall total − everything inside the circles.
Check all regions add up to the overall total.

Worked examples

Example 1 Foundation · grade 4

$\xi = \{1, 2, 3, \ldots, 12\}$ , A = multiples of 3, B = even numbers. A number is chosen at random. Find $P(A \cap B)$ and $P(A \cup B)$ .

Show the solution

A = {3, 6, 9, 12}, B = {2, 4, 6, 8, 10, 12}
1, 5, 7 and 11 are in neither set, so they go outside both circles.
$A \cap B = \{6, 12\}$ , so $P(A \cap B) = \frac{2}{12} = \frac{1}{6}$
$A \cup B = \{2, 3, 4, 6, 8, 9, 10, 12\}$ , 8 numbers
$P(A \cup B) = \frac{8}{12} = \frac{2}{3}$

Example 2 Higher · grade 6

In a group of 60 students, 35 study French, 28 study Spanish and 9 study neither. (a) How many study both? (b) Find the probability a student studies French only. (c) A student who studies Spanish is chosen. Find the probability they also study French.

Show the solution

(a) $35 + 28 + 9 = 72$ , but there are only 60 students. The extra $72 - 60 = 12$ were counted twice, so 12 study both.
(b) French only $= 35 - 12 = 23$ , so $P = \frac{23}{60}$
(c) Only look at the 28 Spanish students; 12 of them study French.
$P = \frac{12}{28} = \frac{3}{7}$

Common mistakes

Putting the total for A in the circle and forgetting to subtract the overlap.
Forgetting the items that belong to neither set: they go inside the rectangle, outside the circles.
In 'given that' questions, dividing by the overall total instead of the number in the given set.
Missing edge cases when sorting numbers: 1 is a square number, an odd number and a factor of every number.

Exam tips

Check your diagram: all the regions must add up to the total in the question.
Three-circle Venn diagrams can appear on both tiers. Start with the centre, where all three overlap, and work outwards.

Questions people ask

What does the upside-down U mean in a Venn diagram?

$A \cap B$ means the intersection: things in both A and B. The U shape, $A \cup B$ , means the union: things in A or B or both.

How do you find probability from a Venn diagram?

Add the numbers in the regions you want and divide by the total of all the numbers, including those outside the circles.

Exam questions: Venn diagrams and sets (12, easiest first)

Venn diagrams and sets3 marksGrade 3No calculatorOn both tiers

November 2021 · Paper 1 Foundation · Q21

Place even numbers into a two-set Venn diagram from set lists.

Show answer

Answer	Overlap: 6, 18; A only: 12; B only: 2, 14; outside: 4, 8, 10, 16	3 marks

Worked solution Checked against the mark scheme

$\mathcal{E} = \{2, 4, 6, 8, 10, 12, 14, 16, 18\}$

In both A and B: 6, 18

A only: 12. B only: 2, 14

In neither: 4, 8, 10, 16

Watch out: Leaving out the numbers outside both circles, or including 0 or 20.

Venn diagrams and sets3 marksGrade 4No calculatorOn both tiers

June 2024 · Paper 1 Foundation · Q23

Read a set complement from a Venn diagram and find the probability of the union of two sets.

Show answer

(a)	$10, 11, 13, 14, 16, 17$	1 mark
(b)	$\frac{5}{9}$	2 marks

Worked solution Checked against the mark scheme

(a) $P'$ is everything not in $P$ : $10, 11, 13, 14, 16, 17$

(b) $P \cup Q = \{10, 12, 14, 15, 18\}$ , which is 5 of the 9 numbers.

Probability $= \frac{5}{9}$

Watch out: Leaving out the numbers outside both circles in (a), or only counting the overlap in (b).

Venn diagrams and sets2 marksGrade 4CalculatorOn both tiers

November 2024 · Paper 2 Foundation · Q23

Spot the two errors in a Venn diagram drawn from given sets.

Show answer

Answer	Remove the 17 from 'A only' so it is only in the overlap; put 1 in the region outside both circles	2 marks

Worked solution Checked against the mark scheme

The universal set is 1, 3, 5, ..., 19.

17 is in both A and B, so it belongs only in the overlap; Tom also wrote it in A only.

1 is odd but in neither A nor B, so it must go inside the rectangle, outside both circles.

Watch out: Forgetting that 1 is an odd number between 0 and 20.

Venn diagrams and sets5 marksGrade 4No calculatorOn both tiers

June 2023 · Paper 1 Foundation · Q24

Fill in a Venn diagram for odd and square numbers up to 10, then find the probability of not being square.

Show answer

(a)	$A$ only: 3, 5, 7; both: 1, 9; $B$ only: 4; outside: 2, 6, 8, 10	3 marks
(b)	$\frac{7}{10}$	2 marks

Worked solution Checked against the mark scheme

(a) Odd: 1, 3, 5, 7, 9. Square: 1, 4, 9.

In both: 1, 9. Odd only: 3, 5, 7. Square only: 4. Neither: 2, 6, 8, 10.

(b) $B'$ means not square: $10 - 3 = 7$ numbers, so $P = \frac{7}{10}$

Watch out: Forgetting that 1 is a square number.

Venn diagrams and sets5 marksGrade 4CalculatorOn both tiers

November 2020 · Paper 2 Foundation · Q22

Sort the numbers 1 to 10 into a Venn diagram of even numbers and factors of 10, then find a probability for the intersection.

Show answer

(a)	$A$ only: 4, 6, 8; $A \cap B$ : 2, 10; $B$ only: 1, 5; outside: 3, 7, 9	3 marks
(b)	$\frac{2}{10}$	2 marks

Worked solution Checked against the mark scheme

(a) Even: 2, 4, 6, 8, 10. Factors of 10: 1, 2, 5, 10.

Both: 2, 10. Only $A$ : 4, 6, 8. Only $B$ : 1, 5. Neither: 3, 7, 9.

(b) $A \cap B$ has 2 of the 10 numbers, so the probability is $\frac{2}{10} = \frac{1}{5}$ .

Watch out: Forgetting that 1 and 10 are factors of 10.

Venn diagrams and sets4 marksGrade 4CalculatorFoundation

November 2019 · Paper 3 Foundation · Q20

Place the numbers 1 to 13 in a two-set Venn diagram for multiples of 3 and even numbers.

Show answer

Answer	Both: 6, 12. Multiples of 3 only: 3, 9. Even only: 2, 4, 8, 10. Outside: 1, 5, 7, 11, 13	4 marks

Worked solution Checked against the mark scheme

Multiples of 3: 3, 6, 9, 12. Even: 2, 4, 6, 8, 10, 12.

In both (overlap): 6, 12

Multiples of 3 only: 3, 9. Even only: 2, 4, 8, 10

Outside both circles: 1, 5, 7, 11, 13

Watch out: Putting 6 and 12 in both circles separately rather than in the overlap.

Venn diagrams and sets5 marksGrade 4CalculatorOn both tiers

June 2019 · Paper 3 Foundation · Q24

Fill in a Venn diagram from set lists, then find the probability a random number is in both sets.

Show answer

(a)	Intersection: 6, 9; A only: 1, 5, 8; B only: 2; outside: 3, 4, 7	3 marks
(b)	$\frac{2}{9}$	2 marks

Worked solution Checked against the mark scheme

(a) In both A and B: 6, 9

A only: 1, 5, 8. B only: 2. Neither: 3, 4, 7.

(b) $A \cap B = \{6, 9\}$ , 2 numbers out of 9.

$P = \frac{2}{9}$

Watch out: Forgetting to place 3, 4 and 7 outside both circles.

Venn diagrams and sets3 marksGrade 4No calculatorFoundation

June 2018 · Paper 1 Foundation · Q18

List the union of two sets of numbers and describe their intersection.

Show answer

(a)	$15, 17, 19, 20, 21, 23, 25$	2 marks
(b)	Odd multiples of 5 (15 and 25)	1 mark

Worked solution Checked against the mark scheme

$A = \{15, 20, 25\}$ , $B = \{15, 17, 19, 21, 23, 25\}$

(a) Union is everything in either: 15, 17, 19, 20, 21, 23, 25

(b) Intersection is in both: 15 and 25, the odd multiples of 5.

Watch out: Mixing up union and intersection.

Venn diagrams and sets6 marksGrade 4CalculatorOn both tiers

November 2018 · Paper 2 Foundation · Q20

Place set elements in a three-set Venn diagram, then find a probability for an intersection.

Show answer

(a)	A only: 2, 10, 14; A∩B∩C: 8; B only: 6; B∩C only: 20; C only: 18, 22; outside: 4, 12, 16, 24	4 marks
(b)	$\frac{1}{12}$	2 marks

Worked solution Checked against the mark scheme

(a) $\mathcal{E} = \{2, 4, 6, ..., 24\}$ , 12 numbers.

8 is in all three sets. 20 is in B and C only. 6 is B only; 2, 10, 14 are A only; 18, 22 are C only.

4, 12, 16, 24 go outside the circles.

(b) $A \cap B = \{8\}$ , so P $= \frac{1}{12}$

Watch out: Forgetting to place the numbers that are in no set.

Venn diagrams and sets4 marksGrade 5CalculatorFoundation

Specimen · Paper 3 Foundation · Q20

Draw a Venn diagram from set notation for a universal set, set A, their intersection and union.

Show answer

Answer	A only: 4, 8, 10; A ∩ B: 2, 6; B only: 1, 3, 9; outside both: 5, 7	4 marks

Worked solution Checked against the mark scheme

A (even numbers) = {2, 4, 6, 8, 10}.

Overlap: 2 and 6. So A only: 4, 8, 10.

$A \cup B$ adds 1, 3, 9, which must be in B only.

Numbers in neither set: 5 and 7, written outside the ovals but inside the rectangle.

Watch out: Leaving 5 and 7 off the diagram, or putting 2 and 6 in A only as well.

Venn diagrams and sets6 marksGrade 6CalculatorHigher

June 2022 · Paper 2 Higher · Q16

Fill in a three-set Venn diagram from given facts, then find a conditional probability from it.

Show answer

(a)	A only 44, A∩B only 8, A∩C only 6, all three 4, B only 12, B∩C only 10, C only 0, outside 16	4 marks
(b)	$\frac{12}{62}$	2 marks

Worked solution Checked against the mark scheme

(a) All three: 4. A and B only: 8. B and C total 14, so B and C only $= 14 - 4 = 10$ .

Nobody uses C alone, so A and C only $= 20 - 4 - 10 - 0 = 6$ .

A only $= 62 - 8 - 4 - 6 = 44$ . Outside: 16.

B only $= 100 - 16 - 44 - 8 - 4 - 6 - 10 = 12$

(b) Among the 62 A users, $8 + 4 = 12$ also use B: $P = \frac{12}{62} = \frac{6}{31}$

Watch out: In (b), dividing by 100 instead of the 62 people who use A.

Venn diagrams and sets5 marksGrade 8CalculatorHigher

June 2018 · Paper 3 Higher · Q20

Complete a three-set Venn diagram from language information, then find the probability that two people chosen without replacement both speak only Spanish.

Show answer

Answer	$\frac{6}{490}$	5 marks

Worked solution Checked against the mark scheme

Fill the Venn diagram from the centre out:

All three: 2. French and Spanish only: 4. German and Spanish only: $7 - 2 = 5$ .

German only: 0 (all German speakers speak another). French and German only: $10 - 2 - 5 = 3$ . French only: $31 - 2 - 4 - 3 = 22$ .

Spanish only: $50 - 8 - 22 - 3 - 2 - 4 - 5 = 6$

P(both) $= \frac{6}{50} \times \frac{5}{49} = \frac{30}{2450} = \frac{6}{490}$

Watch out: Treating the choices as with replacement ( $\frac{6}{50} \times \frac{6}{50}$ ).

Also tested in these questions

Their main topic is different, but part of each one tests this topic.

June 2024 · Paper 3 Foundation · Q27: Complete a probability tree for two biased coins and find the probability both land heads.
November 2023 · Paper 2 Higher · Q21: Use a conditional probability from a Venn diagram to form an equation and find the total number of items.
November 2022 · Paper 3 Foundation · Q23: Use given totals to work out how many boys chose a particular language.
June 2022 · Paper 2 Foundation · Q20: Complete a probability tree for two train journeys and find the chance both are late.
November 2018 · Paper 1 Foundation · Q17: Complete a frequency tree about men, women and coffee, then find a probability from it.
November 2018 · Paper 2 Foundation · Q14: Judge two probability claims about a fair dice and list the outcomes of a dice and a coin.
June 2018 · Paper 2 Higher · Q8: Use given facts about gender and holiday preference to find the probability that a chosen female prefers Spain.