Venn Diagrams
A Venn diagram shows how different sets of outcomes or data overlap, using circles inside a rectangle that represents everything possible. In GCSE Statistics, Venn diagrams record how many items belong to one event, another event, both events, or neither, which makes it straightforward to calculate probabilities involving 'and', 'or' and 'not'.
Before you start
Make sure you're comfortable with these topics first:
Method
- Draw a rectangle to represent all possible outcomes and label it with the total, often written as the sample size.
- Draw one circle for each event, making sure circles that can happen together overlap and circles for mutually exclusive events do not.
- Start filling in the diagram from the overlap (both events happening) first, since this number is usually easiest to place uniquely.
- Work outwards, subtracting the overlap from each event's total to find the 'only this event' regions.
- Place any remaining outcomes, that belong to neither event, inside the rectangle but outside both circles.
- Check every region adds up to the correct total, then use the diagram to read off frequencies or calculate probabilities by dividing a region by the total.
Worked example
In a class of 30 students, 18 study French, 14 study Spanish, and 7 study both French and Spanish. A student is picked at random. Find the probability that the student studies French or Spanish (or both).
- Draw a rectangle labelled 30 (the whole class), with overlapping circles for French and Spanish.
- Place 7 in the overlap, since 7 students study both.
- Only French = 18 - 7 = 11, so write 11 in the French circle outside the overlap.
- Only Spanish = 14 - 7 = 7, so write 7 in the Spanish circle outside the overlap.
- Add every region inside the circles: 11 + 7 + 7 = 25 students study French or Spanish (or both).
- Divide by the total: P(French or Spanish) = 25/30 = 5/6, the final answer.
Practice questions
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Q1In a Venn diagram, what does the region where two circles overlap represent?Show answer
Answer: Outcomes that belong to both events at the same time.
Q2A rectangle labelled 20 contains two non-overlapping circles for events A and B with 8 and 5 inside them. How many outcomes are outside both circles?Show answer
Answer: 7 (20 - 8 - 5 = 7).
Q3Out of 25 pupils, 12 like football, 9 like tennis, and 4 like both. How many like neither sport?Show answer
Answer: 8 (25 - (12 + 9 - 4) = 8).
Q4Using the pupils in the previous question, find P(likes football only).Show answer
Answer: 8/25 (12 - 4 = 8 like football only, out of 25).
Q5A Venn diagram shows 40 people: 22 own a car, 15 own a bike, 6 own both. Find the probability a random person owns neither.Show answer
Answer: 9/40 (40 - (22 + 15 - 6) = 9, so 9/40).
Q6Events A and B are mutually exclusive with P(A) = 0.3 and P(B) = 0.25. Find P(A or B).Show answer
Answer: 0.55 (mutually exclusive, so add: 0.3 + 0.25 = 0.55, no overlap to subtract).
Exam-style questions
Written in the style of a GCSE Statistics exam paper, with a full mark scheme.
In a survey of 50 customers, 28 buy coffee, 19 buy tea, and 9 buy both coffee and tea. Draw a Venn diagram to show this information and find how many customers buy neither drink.
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Of 60 members at a sports club, 35 play badminton, 27 play squash, and 14 play both sports. A member is chosen at random. Find the probability the member plays badminton only, and the probability the member plays neither sport.
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Explain why P(A or B) is not always equal to P(A) + P(B).
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See real GCSE Statistics past-paper questions, with official mark schemes →
Free printable worksheet
Want more practice on paper? Download the venn diagrams worksheet pack - 21 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 20 of GCSE Statistics Foundation Workbook 2 and chapter 26 of GCSE Statistics Higher Workbook 2, the whole course as one free printable PDF.
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