GCSE Maths Practice: venn-diagrams

Question 1 of 10

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

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\( \begin{array}{l}\textbf{A Venn diagram shows information about people who like} \\ \textbf{football and basketball in a set of 100 people.} \\ \textbf{70 people like football, 50 like basketball, and 40 like both.} \\ \textbf{Find the probability that a randomly chosen person likes} \\ \textbf{football or basketball. Give your answer in its simplest form.}\end{array} \)

Diagram

Choose one option:

Always fill the overlap first, then subtract it from each total before adding inside the circles.

Using Venn Diagrams to Find “OR” Probabilities

Venn diagrams are an important GCSE Maths tool for working with probability questions that involve overlapping groups. In this question, the two groups are people who like football and people who like basketball. Some people enjoy both sports, so the groups overlap. If this overlap is not handled correctly, some people will be counted twice, which leads to an incorrect probability.

A Venn diagram separates the information into clear regions so that each person is counted exactly once. This makes probability questions involving the word or much easier to understand and solve.

Step-by-Step Venn Diagram Method

Start by drawing a rectangle to represent the whole group of 100 people. Inside the rectangle, draw two overlapping circles. Label one circle Football and the other Basketball.

Step 1: Fill the overlap first. The question states that 40 people like both football and basketball. Write 40 in the overlapping region of the two circles.

Step 2: Find the “football only” region. A total of 70 people like football, but 40 of these are already included in the overlap. Subtract the overlap:

Football only = 70 − 40 = 30.

Write 30 in the part of the football circle that does not overlap with basketball.

Step 3: Find the “basketball only” region. A total of 50 people like basketball, and 40 of these are already included in the overlap. Subtract again:

Basketball only = 50 − 40 = 10.

Write 10 in the basketball-only region.

Finding the Required Probability

The question asks for the probability that a randomly chosen person likes football or basketball. In GCSE Maths, “or” means football only, basketball only, or both. This corresponds to everything inside the two circles.

Add all values inside the circles:

30 + 40 + 10 = 80.

So, 80 people like at least one of the two sports.

Calculating and Simplifying the Probability

Probability is calculated as:

Probability = favourable outcomes ÷ total outcomes

Here, the favourable outcomes are the 80 people who like football or basketball, and the total outcomes are the 100 people in the set. This gives:

\(\frac{80}{100}\).

GCSE answers should always be simplified. Dividing the numerator and denominator by 20 gives:

\(\frac{80}{100} = \frac{4}{5}\).

Common Mistakes

  • Adding 70 and 50 without subtracting the overlap of 40.
  • Writing totals directly inside the circles instead of splitting into regions.
  • Forgetting to simplify the final fraction.

Real-Life Applications

Venn diagrams are often used in surveys and sports participation studies. For example, schools and clubs may want to know how many people are interested in at least one sport when planning facilities or events.

FAQ

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

Do I need to find the number who like neither sport?
No, not for this question. However, it can be used as a check. Here, 100 − 80 = 20 people like neither.

What should I always do first?
Always place the overlap first, then subtract it from each total.

Study Tip

Whenever a probability question involves two sports and the word “both”, sketch a Venn diagram immediately to avoid double counting.

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