GCSE Maths revision · Probability and statistics

Venn diagrams and sets

Foundation and Higher

The idea

In a Venn diagram, the overlap $A \cap B$ is in both sets; the union $A \cup B$ is in either or both; the complement $A'$ is everything not in $A$ . A sample space lists every outcome.

Fill the overlap first, then the rest of each circle, then the outside.
Check everything adds up to the total.
Probability $=$ the region's count over the total.

Watch out $A \cap B$ is the overlap only; $A \cup B$ includes the overlap once.

See it worked

Worked exampleNon-calculator4 marks

$\xi = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}$ . $A$ is the set of multiples of 3 and $B$ is the set of even numbers.

(a) List the members of $A \cap B$ .

(b) How many members does $A \cup B$ have?

(c) A number is picked at random from $\xi$ . Work out $P(A')$ .

(a) $A = \{3, 6, 9, 12\}$ and $B = \{2, 4, 6, 8, 10, 12\}$ .
$A \cap B$ (intersection) is the numbers in both: 6 and 12.
(b) $A \cup B$ is everything in $A$ or $B$ or both: $\{2, 3, 4, 6, 8, 9, 10, 12\}$ .
(c) $A'$ is the complement: numbers not in $A$ . There are $12 - 4 = 8$ of them.
$P(A') = \frac{8}{12} = \frac{2}{3}$ .

Answer (a) $\{6, 12\}$ ; (b) 8; (c) $\frac{2}{3}$

Your turn

Question 1Non-calculator3 marks

Two fair six-sided dice are rolled and their scores are added.

(a) Work out the probability that the total is 7.

(b) Work out the probability that the total is more than 10.

Question 2Non-calculator2 marks

Four cards are labelled 1, 1, 2 and 3. Two cards are chosen at random without replacement. Work out the probability that their numbers add to 4. Give a fraction in its simplest form.

Question 3Non-calculator5 marks

There are 50 students in a year group. $F$ is the set who play football and $H$ is the set who play hockey. $2x$ play football only, $x$ play both, $3x - 4$ play hockey only and 6 play neither.

(a) A student is picked at random. Work out $P(H)$ .

(b) Work out $P((F \cup H)')$ .

In past papers

124 official questions in 117 papers since 2017. The latest:

All GCSE Maths past papers

GCSE Maths Pro

£119 once

At least 12 months of access

By paying you agree to our Terms and get access straight away, which ends your 14-day cancellation right.

Start free

No VAT is charged.