GCSE Maths Practice: venn-diagrams

Question 7 of 10

GCSE Maths (Higher): Use a Venn-diagram method to calculate the probability of liking Art or Music.

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\( \begin{array}{l}\textbf{In a set of 1000 students, 650 like Art,}\\\textbf{700 like Music, and 500 like both.}\\\textbf{What is the probability that a student likes}\\\textbf{either Art or Music?}\end{array} \)

Diagram

Choose one option:

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

GCSE Maths (Higher): Using Venn Diagrams for “OR” Probability

When a question asks for the probability of A or B, it means:

  • In A only
  • In B only
  • In both A and B

The key challenge is avoiding double-counting elements that belong to both sets.

The inclusion–exclusion rule

For two overlapping sets A and B:

n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

This rule works in all GCSE Venn-diagram problems involving two sets.

Worked example (different data)

In a year group of 300 students:

  • 180 study Biology
  • 140 study Chemistry
  • 60 study both subjects

Step 1: Add the two subject totals:

\(180 + 140 = 320\)

Step 2: Subtract the overlap to avoid double-counting:

\(320 - 60 = 260\)

Step 3: Divide by the total number of students:

\(\frac{260}{300} = \frac{13}{15}\)

Why this is Higher tier

  • Students must identify the overlap correctly
  • Large numbers increase the risk of arithmetic errors
  • Questions may later extend to neither or conditional probability

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 the question uses the word or, always ask yourself:
“Who would be counted twice if I just added the totals?”

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