GCSE Maths Practice: venn-diagrams

Question 3 of 10

GCSE Maths (Higher): Use a Venn diagram idea to find the probability of liking football or basketball when some students like both.

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\( \begin{array}{l}\textbf{In a class of 150 students, 90 like football,}\\\textbf{80 like basketball, and 60 like both.}\\\textbf{What is the probability that a randomly chosen student}\\\textbf{likes football or basketball?}\end{array} \)

Diagram

Choose one option:

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

GCSE Maths: Probability with Venn Diagrams

This question involves two overlapping groups:

  • F = students who like football
  • B = students who like basketball

The word either (or or) means we include:

  • Students who like football only
  • Students who like basketball only
  • Students who like both sports

Only students who like neither sport are excluded.

Why subtraction is needed

When we add the number who like football and the number who like basketball, students who like both sports are counted twice:

  • Once in the football total
  • Once in the basketball total

To correct this, we subtract the overlap once. This gives the inclusion–exclusion formula:

n(F ∪ B) = n(F) + n(B) − n(F ∩ B)

Step-by-step method

  1. Total students: 150
  2. Add the two groups: 90 + 80 = 170
  3. Subtract the overlap: 170 − 60 = 120
  4. Convert to probability: \(\frac{120}{150}\)
  5. Simplify: \(\frac{120}{150} = \frac{4}{5}\)

Common GCSE mistakes

  • Forgetting the overlap and using \(\frac{170}{150}\), which is impossible.
  • Subtracting 60 twice, which removes valid students.
  • Dividing by the wrong total instead of 150.
  • Confusing “or” with “and”: “and” would mean only the 60 students.

Exam tip

If a question uses the word or and mentions students in both groups, always ask yourself:
“Who gets counted twice when I add the totals?”

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