GCSE Maths Practice: venn-diagrams

Question 2 of 10

GCSE Maths (Higher): Use a Venn-diagram idea to find the probability of “Physics or Chemistry” when some students like both.

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\( \begin{array}{l}\textbf{In a class of 120 students, 45 like Physics,}\\\textbf{55 like Chemistry, and 30 like both.}\\\textbf{What is the probability that a randomly chosen student}\\\textbf{likes Physics or Chemistry?}\end{array} \)

Diagram

Choose one option:

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

GCSE Maths: Venn Diagrams and “OR” Probability

In GCSE Maths, Venn diagrams are often used to organise information about two (or more) sets. In this question, the sets are:

  • P = students who like Physics
  • C = students who like Chemistry

The keyword either (or or) means we want everyone who is in Physics, or in Chemistry, or in both. The only students we do not include are those who like neither subject.

Why we subtract the overlap

If we add the numbers who like Physics and Chemistry, students who like both subjects get counted twice:

  • Once in the Physics total
  • Once in the Chemistry total

So we must subtract the overlap once to correct the double-counting. This is the inclusion–exclusion idea:

n(P ∪ C) = n(P) + n(C) − n(P ∩ C)

Step-by-step method (works every time)

  1. Identify totals: total students = 120.
  2. Write the OR formula: OR = Physics + Chemistry − Both.
  3. Calculate the union: 45 + 55 − 30 = 70.
  4. Turn it into a probability: \(\frac{70}{120}\).
  5. Simplify if needed: \(\frac{70}{120} = \frac{7}{12}\) (divide top and bottom by 10).

Worked examples

Example 1 (clubs): Out of 80 students, 28 are in the Drama club, 35 are in the Music club, and 10 are in both. Find the probability a student is in Drama or Music.

  • Union = 28 + 35 − 10 = 53
  • Probability = \(\frac{53}{80}\)

Example 2 (sports): In a year group of 150 students, 60 play football, 50 play basketball, and 15 play both. Probability a student plays football or basketball?

  • Union = 60 + 50 − 15 = 95
  • Probability = \(\frac{95}{150} = \frac{19}{30}\)

Example 3 (languages): In a class of 40, 18 study Spanish, 16 study French, and 6 study both. Probability a student studies Spanish or French?

  • Union = 18 + 16 − 6 = 28
  • Probability = \(\frac{28}{40} = \frac{7}{10}\)

Common mistakes to avoid

  • Forgetting to subtract the overlap and doing 45 + 55 = 100 (this double-counts the 30).
  • Subtracting the overlap twice (only subtract it once).
  • Dividing by the wrong total (always divide by the total number of students: 120).
  • Mixing up “or” and “and”: “and” would mean only the overlap (both).

Real-life applications

This exact structure appears in real decisions, not just in GCSE Maths questions. For example:

  • Marketing: how many customers bought Product A or Product B, when some bought both.
  • Healthcare: how many patients show symptom X or symptom Y, when some show both.
  • School planning: how many students need support in English or Maths, when some need both.

Any time you combine two groups with overlap, inclusion–exclusion prevents double-counting.

FAQ

1) What does “either” mean in GCSE Maths?
It means one or the other or both, unless the question says “either but not both” (which is different).

2) When do I use subtraction?
Use subtraction when two totals overlap and the overlap is included in both totals.

3) Do I have to simplify the fraction?
Sometimes yes (especially at Higher). \(\frac{70}{120}\) is correct, and simplifying to \(\frac{7}{12}\) is also correct.

Study tip

Always say to yourself: “If I add the two groups, who gets counted twice?” That sentence tells you exactly when to subtract the overlap.

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