GCSE Maths Practice: venn-diagrams

Question 10 of 10

This Venn diagram question asks for the probability that a student likes neither football nor basketball.

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\( \begin{array}{l}\textbf{In a survey of 80 students, 45 like football,} \\ \textbf{30 like basketball, and 20 like both sports.} \\ \textbf{What is the probability that a student likes neither} \\ \textbf{football nor basketball? Give your answer in its simplest form.}\end{array} \)

Diagram

Choose one option:

To find “neither”, subtract the total inside the circles from the overall total.

Venn Diagrams and “Neither” Probability

Venn diagrams are a powerful GCSE Maths tool for working with probability questions that involve overlapping groups. In this question, the two groups are students who like football and students who like basketball. Some students like both sports, which creates an overlap between the two groups.

Unlike many questions that ask for students who like one sport or the other, this question asks for students who like neither football nor basketball. This means students who are outside both circles in the Venn diagram. To find this correctly, the diagram must be fully completed first.

Completing the Venn Diagram Step by Step

Start by drawing a rectangle to represent the total number of students, which is 80. Inside the rectangle, draw two overlapping circles. Label one circle Football and the other Basketball.

Step 1: Place the overlap first. The question states that 20 students like both football and basketball. Write 20 in the overlapping region.

Step 2: Find the football-only region. A total of 45 students like football, but 20 of these are already included in the overlap. Subtract the overlap:

Football only = 45 − 20 = 25.

Write 25 in the football-only region.

Step 3: Find the basketball-only region. A total of 30 students like basketball, and again 20 of these are already in the overlap:

Basketball only = 30 − 20 = 10.

Write 10 in the basketball-only region.

Finding the Number Who Like Neither Sport

Students who like football or basketball are represented by everything inside the circles. Add these regions together:

25 + 20 + 10 = 55.

This means 55 students like at least one of the two sports.

To find how many students like neither sport, subtract this from the total:

80 − 55 = 25.

Calculating the Probability

Probability is calculated using:

Probability = favourable outcomes ÷ total outcomes

Here, the favourable outcomes are the 25 students who like neither football nor basketball, and the total number of students is 80:

\(\frac{25}{80}\).

This fraction can be simplified by dividing the numerator and denominator by 5:

\(\frac{25}{80} = \frac{5}{16}\).

Worked Examples

Example 1: In a class of 40 students, 18 like drama, 14 like music, and 6 like both. Drama or music = 18 + 14 − 6 = 26. Neither = 40 − 26 = 14. Probability = \(\frac{14}{40} = \frac{7}{20}\).

Example 2: In a group of 60 people, 35 like swimming, 25 like running, and 15 like both. Swimming or running = 35 + 25 − 15 = 45. Neither = 60 − 45 = 15. Probability = \(\frac{15}{60} = \frac{1}{4}\).

Common Mistakes

  • Forgetting to subtract the overlap before finding totals.
  • Trying to calculate “neither” without completing the Venn diagram.
  • Using the “or” formula instead of subtracting from the total.

Real-Life Applications

“Neither” probabilities are common in surveys, such as finding how many people do not take part in any sports or activities. Schools and organisations use this information when planning clubs or events.

FAQ

Where are the “neither” values on a Venn diagram?
They are outside both circles but still inside the rectangle.

Do I always need to find “or” first?
Yes. To find “neither”, you must first know how many are inside the circles.

Study Tip

If a question asks for “neither”, complete the entire Venn diagram and subtract the total inside the circles from the overall total.

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