What is important to know about Venn diagrams?

The universal set ξ contains all elements under consideration.

What is important to know about Venn diagrams?

A ∩ B (intersection) = elements in both A and B (overlap region).

What is important to know about Venn diagrams?

A ∪ B (union) = elements in A or B or both.

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Statistics & Probability · Higher

Venn diagrams

Venn diagrams show the relationships between sets using overlapping circles. They are used to organise data and calculate probabilities, including conditional probabilities.

Revising venn diagrams the night before an exam? These are condensed revision notes — the facts, formulas and mistakes examiners actually mark, with no long explanations. Prefer a full walkthrough with step-by-step reasoning instead? Search “Venn diagrams GCSE” on the site for the complete guide.

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Key facts to remember

1The universal set ξ contains all elements under consideration.
2A ∩ B (intersection) = elements in both A and B (overlap region).
3A ∪ B (union) = elements in A or B or both.
4A' (complement of A) = elements not in A.
5The sum of all frequencies in a Venn diagram = total frequency.
6P(A) = frequency in A ÷ total.

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Worked examples

Example 1

40 students: 18 like maths (M), 22 like English (E), 7 like both. Fill in a Venn diagram and find P(M only).

Working

M only = 18 − 7 = 11
E only = 22 − 7 = 15
Neither = 40 − 11 − 7 − 15 = 7
P(M only) = 11/40

AnswerP(M only) = 11/40

Example 2

From the same diagram, find P(M | E): probability of liking maths given they like English.

Working

Given E: restrict to students who like English = 22
Of those, likes maths too = 7
P(M | E) = 7/22

Answer7/22

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Common mistakes

✗Placing the total frequency of each set in the overlap, rather than calculating the "only" regions.

✗Adding the two set totals to get the union without subtracting the intersection.

✗Forgetting the "neither" region when filling in the Venn diagram.

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Exam tips

✓Always fill in the intersection first, then calculate the "only" regions and finally the "neither" region.

✓Check your diagram adds up to the total before answering probability questions.

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