GCSE Maths Practice: venn-diagrams

Question 7 of 10

This question uses a Venn diagram to calculate probability with overlapping groups.

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\( \begin{array}{l}\textbf{A Venn diagram shows information about students who} \\ \textbf{like swimming and tennis in a class of 30 students.} \\ \textbf{12 students like swimming, 18 like tennis, and} \\ \textbf{8 like both sports.} \\ \textbf{Find the probability that a randomly chosen student} \\ \textbf{likes swimming or tennis. Give your answer in its} \\ \textbf{simplest form.}\end{array} \)

Diagram

Choose one option:

Always fill the overlap first when completing a Venn diagram.

Understanding Venn Diagrams in Probability

Venn diagrams are a visual tool used in GCSE Maths to represent groups of items and how those groups overlap. They are especially useful in probability questions where people, objects, or outcomes belong to more than one category. In this question, the categories are students who like swimming and students who like tennis.

A Venn diagram uses overlapping circles to show shared members. Each circle represents a group, and the overlapping region represents members that belong to both groups. This helps avoid a very common mistake in probability: double counting.

Step-by-Step Method Using a Venn Diagram

Start by drawing two overlapping circles inside a rectangle. The rectangle represents all 30 students in the class. One circle is labelled Swimming, and the other is labelled Tennis.

First, place the number of students who like both sports in the overlapping region. In this case, 8 students like both swimming and tennis, so write 8 in the overlap.

Next, calculate how many students like only swimming. Since 12 students like swimming in total and 8 of them also like tennis, subtract the overlap: 12 − 8 = 4. Write 4 in the swimming-only region.

Then calculate how many students like only tennis. Since 18 students like tennis in total and 8 like both sports, subtract again: 18 − 8 = 10. Write 10 in the tennis-only region.

Now the Venn diagram is complete. To find how many students like swimming or tennis, add all values inside the circles: 4 + 8 + 10 = 22.

Finally, divide by the total number of students to calculate the probability: 22 out of 30. Simplifying this fraction gives \( \frac{11}{15} \).

Worked Example

If a class has 25 students, 9 like football, 11 like basketball, and 5 like both, the number who like football or basketball is found in the same way. Football only = 9 − 5 = 4, basketball only = 11 − 5 = 6. Total = 4 + 5 + 6 = 15. The probability is \( \frac{15}{25} = \frac{3}{5} \).

Common Mistakes to Avoid

  • Adding both groups without subtracting the overlap.
  • Placing total numbers directly into the circles instead of splitting them.
  • Forgetting to simplify the final fraction.

Real-Life Applications

Venn diagrams are used in surveys, marketing research, sports participation studies, and even medical testing. Anytime data overlaps, a Venn diagram helps organise information clearly and prevents errors.

FAQ

Do I always need a Venn diagram?
For simple questions, you might solve it mentally, but drawing a diagram is strongly recommended at GCSE to avoid mistakes.

What does “or” mean in probability?
In maths, “or” means one group, the other group, or both.

Study Tip

If a probability question mentions two groups and shared members, always sketch a Venn diagram before calculating.

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