Venn Diagrams

This lesson covers the use of Venn diagrams in probability, including set notation, as required by the AQA GCSE Mathematics specification. Venn diagrams are a powerful visual tool for organising data and calculating probabilities, especially when events overlap.

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What Is a Venn Diagram?

A Venn diagram uses overlapping circles inside a rectangle to represent sets (groups) and their relationships. The rectangle represents the universal set (all items being considered).

graph TD
    subgraph Universal Set
        subgraph Set A
            A1[A only]
            AB[A and B]
        end
        subgraph Set B
            AB
            B1[B only]
        end
        N[Neither A nor B]
    end

Each region of the Venn diagram represents a different group:

Region	Meaning
A only (not in B)	Items in set A but not in set B
B only (not in A)	Items in set B but not in set A
A and B (overlap)	Items in both set A and set B
Outside both circles	Items in neither set A nor set B

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Set Notation

The AQA specification requires you to understand and use set notation.

Notation	Name	Meaning
A	Set A	The set of elements in A
A'	Complement of A	Everything not in A
A ∩ B	Intersection	Elements in both A and B
A ∪ B	Union	Elements in A or B or both
n(A)	Number in A	The number of elements in set A

Exam Tip: Remember the symbols — the ∩ (intersection) looks like an "n" for "and" (both), and the ∪ (union) looks like a "u" for "united" (together). These symbols appear frequently in AQA exam questions.

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Drawing a Venn Diagram from Given Information

Worked Example 1

In a class of 30 students:

• 18 study History
• 15 study Geography
• 7 study both

Draw a Venn diagram and find how many study neither subject.

Step 1: Start with the overlap: both History and Geography = 7

Step 2: History only = 18 - 7 = 11

Step 3: Geography only = 15 - 7 = 8

Step 4: Neither = 30 - 11 - 7 - 8 = 4

Region	Number
History only	11
Both	7
Geography only	8
Neither	4
Total	30

Exam Tip: Always start filling in a Venn diagram from the intersection (the overlap in the middle). This prevents errors because the overlap is part of both sets. Then work outwards.

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Reading Probabilities from a Venn Diagram

Once you have completed the Venn diagram, you can calculate probabilities by dividing the relevant count by the total.

Worked Example 2

Using the data from Worked Example 1 (30 students total):

(a) P(History) = n(History) / total = 18/30 = 3/5

(b) P(History and Geography) = P(H ∩ G) = 7/30

(c) P(History or Geography) = P(H ∪ G) = (11 + 7 + 8)/30 = 26/30 = 13/15

(d) P(neither) = 4/30 = 2/15

(e) P(not History) = P(H') = (8 + 4)/30 = 12/30 = 2/5

(f) P(Geography but not History) = 8/30 = 4/15

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The Addition Rule for Non-Mutually Exclusive Events

When events overlap, the addition rule needs adjusting:

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

We subtract the intersection to avoid counting it twice (once in A and once in B).

Worked Example 3

In a bag of 40 sweets, 22 are chocolate, 18 are toffee, and 6 are both chocolate and toffee.

P(chocolate or toffee) = P(C) + P(T) - P(C ∩ T) = 22/40 + 18/40 - 6/40 = 34/40 = 17/20

Exam Tip: If events are mutually exclusive (no overlap), then P(A ∩ B) = 0, and the formula simplifies to P(A ∪ B) = P(A) + P(B). This matches the addition rule from the earlier lesson.

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Three-Set Venn Diagrams

Some questions involve three overlapping sets. The same principles apply, but there are more regions to fill in.

Worked Example 4

60 students were asked about the sports they play. The results were:

• 30 play football (F)
• 25 play tennis (T)
• 20 play cricket (C)
• 10 play football and tennis
• 8 play football and cricket
• 5 play tennis and cricket
• 3 play all three

Fill in the Venn diagram step by step:

Step 1: All three = 3

Step 2: F and T only (not C) = 10 - 3 = 7

Step 3: F and C only (not T) = 8 - 3 = 5

Step 4: T and C only (not F) = 5 - 3 = 2

Step 5: F only = 30 - 7 - 3 - 5 = 15

Step 6: T only = 25 - 7 - 3 - 2 = 13

Step 7: C only = 20 - 5 - 3 - 2 = 10

Step 8: Neither = 60 - 15 - 7 - 5 - 3 - 13 - 2 - 10 = 5

Region	Count
F only	15
T only	13
C only	10
F ∩ T only	7
F ∩ C only	5
T ∩ C only	2
F ∩ T ∩ C	3
Neither	5
Total	60

P(plays exactly one sport) = (15 + 13 + 10)/60 = 38/60 = 19/30

P(plays at least two sports) = (7 + 5 + 2 + 3)/60 = 17/60

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Venn Diagrams with Probabilities (Instead of Frequencies)

Sometimes the diagram is labelled with probabilities rather than counts.

Worked Example 5

Events A and B are such that P(A) = 0.4, P(B) = 0.5, and P(A ∩ B) = 0.15.

Region	Probability
A only	0.4 - 0.15 = 0.25
A ∩ B	0.15
B only	0.5 - 0.15 = 0.35
Neither	1 - 0.25 - 0.15 - 0.35 = 0.25

P(A ∪ B) = 0.25 + 0.15 + 0.35 = 0.75

P(A') = 1 - 0.4 = 0.6

P(A' ∩ B) = 0.35 (B only region)

Exam Tip: Whether a Venn diagram uses counts or probabilities, the method is the same. With probabilities, the regions should add up to 1. With counts, they add up to the total number.

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Summary