GCSE Maths Practice: venn-diagrams

Question 4 of 10

GCSE Maths (Higher): Use a Venn-diagram method to calculate the probability of studying Spanish or French.

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\( \begin{array}{l}\textbf{In a group of 1200 students, 750 study Spanish,}\\\textbf{800 study French, and 600 study both subjects.}\\\textbf{What is the probability that a randomly chosen student}\\\textbf{studies Spanish or French?}\end{array} \)

Diagram

Choose one option:

Add the two totals, subtract the overlap once, then divide by the total number of students.

GCSE Maths (Higher): Venn Diagrams with Two Subjects

When dealing with two subjects that overlap, Venn diagrams help organise information clearly and avoid common counting errors.

Understanding “or” in probability

The word or means we include everyone who is:

  • In the first group only
  • In the second group only
  • In both groups

The only students excluded are those who are in neither group.

The inclusion–exclusion rule

For two overlapping sets A and B:

n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

This formula prevents students who belong to both sets from being counted twice.

Worked example (different data)

In a college of 400 students:

  • 220 study Geography
  • 180 study History
  • 90 study both subjects

Step 1: Add the two subject totals:

\(220 + 180 = 400\)

Step 2: Subtract the overlap:

\(400 - 90 = 310\)

Step 3: Divide by the total number of students:

\(\frac{310}{400} = \frac{31}{40}\)

Why this is Higher tier

  • Larger numbers increase arithmetic complexity
  • Students must correctly interpret overlapping data
  • This structure is often extended to neither or conditional probability questions

Common mistakes to avoid

  • Adding both totals without subtracting the overlap
  • Subtracting the overlap more than once
  • Dividing by the wrong total

Study tip

If you see the word or, always check whether an overlap is mentioned. If it is, inclusion–exclusion is required.

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