GCSE Maths Practice: venn-diagrams

Question 8 of 10

This question uses a Venn diagram to calculate the probability of liking exactly one flavour.

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\( \begin{array}{l}\textbf{In a survey of 50 people, 30 like chocolate,} \\ \textbf{20 like vanilla, and 10 like both flavours.} \\ \textbf{What is the probability that a person likes exactly one} \\ \textbf{of these flavours? Give your answer in its simplest form.}\end{array} \)

Diagram

Choose one option:

“Exactly one” means only the non-overlapping parts of the Venn diagram.

Venn Diagrams and “Exactly One” Probability

Venn diagrams are commonly used in GCSE Maths to represent overlapping groups. In this question, the groups are people who like chocolate and people who like vanilla. Some people like both flavours, which creates an overlapping region in the diagram.

Unlike many probability questions that ask for “A or B”, this question asks for people who like exactly one flavour. This means people who like chocolate only or vanilla only, but not both. Understanding this difference is very important at GCSE level.

Completing the Venn Diagram

Start by drawing a rectangle to represent the total of 50 people. Inside the rectangle, draw two overlapping circles. Label one circle Chocolate and the other Vanilla.

Step 1: Fill the overlap first. The question states that 10 people like both flavours. Write 10 in the overlapping region.

Step 2: Find the chocolate-only region. A total of 30 people like chocolate, but this includes the 10 who like both flavours. Subtract the overlap:

Chocolate only = 30 − 10 = 20.

Write 20 in the chocolate-only region.

Step 3: Find the vanilla-only region. A total of 20 people like vanilla, and again 10 of these are in the overlap:

Vanilla only = 20 − 10 = 10.

Write 10 in the vanilla-only region.

Finding “Exactly One Flavour”

“Exactly one” means the people who are in one circle but not in the overlap. This corresponds to the chocolate-only and vanilla-only regions.

Add these two regions:

20 + 10 = 30.

So, 30 people like exactly one flavour.

Calculating the Probability

Probability is calculated as:

Probability = favourable outcomes ÷ total outcomes

Here, the favourable outcomes are the 30 people who like exactly one flavour, and the total outcomes are the 50 people surveyed.

\(\frac{30}{50}\)

Simplify by dividing the numerator and denominator by 10:

\(\frac{30}{50} = \frac{3}{5}\)

Worked Examples

Example 1: In a group of 40 people, 18 like tea, 14 like coffee, and 6 like both. Tea only = 12, coffee only = 8. Exactly one = 20. Probability = \(\frac{20}{40} = \frac{1}{2}\).

Example 2: In a class of 60 students, 35 like maths, 25 like science, and 15 like both. Maths only = 20, science only = 10. Exactly one = 30. Probability = \(\frac{30}{60} = \frac{1}{2}\).

Common Mistakes

  • Including the overlap when the question says “exactly one”.
  • Using the “or” method instead of reading the question carefully.
  • Forgetting to simplify the final fraction.

Real-Life Context

“Exactly one” situations occur often in surveys, such as people who subscribe to only one streaming service or students who take only one subject option. Venn diagrams help separate these groups clearly.

FAQ

Is “exactly one” the same as “or”?
No. “Or” includes the overlap, while “exactly one” excludes it.

What region should I ignore?
The overlapping (both) region.

Study Tip

If a question says “exactly one”, mentally cross out the overlap before adding.

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