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Venn Diagrams

FoundationHigherAQAEdexcel

Understand Venn Diagrams for GCSE Statistics with this free worksheet and full mark scheme — Foundation and Higher exam-style questions with worked answers for AQA and Edexcel. Venn diagrams organise outcomes into sets to work out probabilities involving 'and', 'or' and 'not'.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA and Edexcel specifications. Every worksheet comes with a full mark scheme.

Foundation

WorksheetFoundation · PDFMark SchemeFoundation · PDF

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

A Venn diagram represents sets as overlapping regions, making combined probabilities easy to read.

The overlap contains outcomes belonging to both sets. The region outside both circles contains outcomes belonging to neither, and forgetting it is the most common error — the numbers in all regions must sum to the total.

The notation matters. \(A \cap B\) means A and B, the intersection, read from the overlap. \(A \cup B\) means A or B, the union, read from everything inside either circle. \(A'\) means not A, read from everything outside circle A including the region outside both.

Revision notes

Reading the regions

The overlap contains outcomes in both sets. Each crescent contains outcomes in one set only.

The region outside both circles contains outcomes in neither. All the regions together must sum to the total number of items.

The notation

\(A \cap B\): the intersection, meaning A AND B — the overlap.

\(A \cup B\): the union, meaning A OR B — everything inside either circle. \(A'\): the complement, meaning NOT A — everything outside circle A.

Filling in a Venn diagram

Start with the overlap, because it is counted in both sets.

Then subtract it from each set total to find the crescents. Finally subtract everything from the grand total to find the region outside both.

Key points

  • Venn diagrams show sets as overlapping regions.
  • The overlap contains outcomes in both sets.
  • The outside region contains outcomes in neither.
  • \(A \cap B\) means A and B.
  • \(A \cup B\) means A or B.
  • \(A'\) means not A.

Worked examples

Example 1

A Venn diagram has 8 in A only, 5 in the overlap and 9 in B only, from a total of 30. Work out how many are in neither set. [2 marks]

Working

\(8 + 5 + 9 = 22\) are in at least one setadd the three inner regions
\(30 - 22 = 8\) are in neither setsubtract from the total

Example 2

Set A has 20 members and the overlap with B has 7. Work out how many are in A only. [2 marks]

Working

The overlap is included in the total for A, so subtract it: \(20 - 7\)subtract the overlap from the set total
\(= 13\) in A onlywork out the answer

Example 3

Explain what \(A \cup B\) represents. [2 marks]

Working

It is the union of A and B, meaning A or B or bothstate the meaning
read from everything inside either circle, including the overlapexplain how it is read

Common mistakes

  • Forgetting the region outside both circles.

    It must be included so the regions sum to the total.

  • Not subtracting the overlap from the set totals.

    The overlap is counted within each set's total.

  • Confusing the union and intersection symbols.

    \(\cap\) is 'and'; \(\cup\) is 'or'.

  • Excluding the overlap from the union.

    The union includes the overlap.

Exam tips

  • Fill in the overlap first.
  • Subtract the overlap from each set total.
  • Check all regions sum to the grand total.
  • Learn the three notation symbols precisely.

Key terms

Venn diagram
A diagram showing sets as overlapping regions.
Intersection
\(A \cap B\), outcomes in both sets.
Union
\(A \cup B\), outcomes in either set.
Complement
\(A'\), outcomes not in A.

Related topics

Written and reviewed against the current AQA and Edexcel specifications. Spotted an error? Let us know.