GCSE Maths Practice: venn-diagrams

Question 5 of 10

GCSE Maths (Higher): Use a Venn diagram approach to find the probability that a student studies neither of two subjects.

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\( \begin{array}{l}\textbf{In a group of 500 students, 350 study History,}\\\textbf{400 study Geography, and 300 study both subjects.}\\\textbf{What is the probability that a randomly chosen student}\\\textbf{studies neither History nor Geography?}\end{array} \)

Diagram

Choose one option:

For “neither”, subtract the OR total from the overall total.

GCSE Maths (Higher): Probability with Venn Diagrams

This question requires two linked steps:

  1. Find how many students are in at least one set (History or Geography).
  2. Use this to find how many are in neither set.

Step 1: Find the union

Students who study History or Geography include:

  • History only
  • Geography only
  • Both subjects

Use inclusion–exclusion to avoid double-counting:

350 + 400 − 300 = 450

Step 2: Find those who study neither

The total number of students is 500. If 450 study at least one subject, the remainder study neither:

500 − 450 = 50

Final probability

Probability that a randomly chosen student studies neither subject:

\(\frac{50}{500} = \frac{1}{10}\)

Why this is Higher tier

  • Requires understanding of complements
  • Two linked calculations
  • Common exam trap if the union step is skipped

Exam tip

If a question asks for neither, always find OR first, then subtract from the total.

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