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.

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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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