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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