GCSE Maths Practice: venn-diagrams

Question 8 of 10

GCSE Maths (Higher): Find the probability that a student studies at least one of two subjects.

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\( \begin{array}{l}\textbf{In a set of 150 students, 90 study Economics,}\\\textbf{100 study Business, and 70 study both subjects.}\\\textbf{What is the probability that a randomly chosen student}\\\textbf{studies Economics or Business?}\end{array} \)

Diagram

Choose one option:

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

GCSE Maths (Higher): Probability of "At Least One"

When a question asks whether a student studies at least one subject, this means:

  • The first subject only
  • The second subject only
  • Both subjects

This is mathematically the same as an OR probability.

Using inclusion–exclusion

If two subjects overlap, simply adding their totals will count some students twice. To avoid this, we use:

n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

Worked example (different data)

In a sixth form of 200 students:

  • 120 study Psychology
  • 90 study Sociology
  • 50 study both subjects

Step 1: Add the two subject totals:

\(120 + 90 = 210\)

Step 2: Subtract the overlap:

\(210 - 50 = 160\)

Step 3: Divide by the total number of students:

\(\frac{160}{200} = \frac{4}{5}\)

Why this is Higher tier

  • Requires careful interpretation of wording
  • Students must identify and subtract the overlap correctly
  • Often extended to "neither" or conditional probability questions

Common mistakes

  • Forgetting to subtract the overlap
  • Subtracting the overlap more than once
  • Dividing by the wrong total

Study tip

If you see at least one, think: total minus neither, or use inclusion–exclusion directly.

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