GCSE Maths Practice: venn-diagrams

Venn Diagrams and “But Not” Probability

Venn diagrams are a useful way to organise information when groups overlap. In this question, the groups are students who like apples and students who like oranges. Some students like both fruits, which creates an overlapping region in the diagram.

This question is different from many others because it asks for students who like apples but not oranges. The words “but not” are very important. They mean that we must only count students who are in the apples group and exclude anyone who is also in the oranges group.

Completing the Venn Diagram

Start by drawing a rectangle to represent the total number of students, which is 40. Inside the rectangle, draw two overlapping circles. Label one circle Apples and the other Oranges.

Step 1: Place the overlap first. The question states that 15 students like both apples and oranges. Write 15 in the overlapping region.

Step 2: Find the apples-only region. A total of 25 students like apples, but 15 of these also like oranges. Subtract the overlap to find how many students like only apples:

Apples only = 25 − 15 = 10.

Write 10 in the apples-only region.

Step 3: Find the oranges-only region (not required but helpful). A total of 20 students like oranges, and 15 like both fruits:

Oranges only = 20 − 15 = 5.

Finding the Required Probability

The question asks for the probability that a student likes apples but not oranges. This corresponds to the apples-only region of the Venn diagram.

There are 10 students who like apples only, out of a total of 40 students.

Calculating the Probability

Probability is calculated using:

Probability = favourable outcomes ÷ total outcomes

Here, the favourable outcomes are the 10 students who like apples but not oranges, and the total outcomes are the 40 students surveyed:

\(\frac{10}{40}\)

This fraction can be simplified by dividing the numerator and denominator by 10:

\(\frac{10}{40} = \frac{1}{4}\)

Worked Examples

Example 1: In a class of 50 students, 28 like maths, 18 like science, and 10 like both. Maths only = 18. Probability of liking maths but not science = \(\frac{18}{50}\).

Example 2: In a group of 60 people, 35 like tea, 25 like coffee, and 15 like both. Tea only = 20. Probability = \(\frac{20}{60} = \frac{1}{3}\).

Common Mistakes

• Including the overlap when the question says “but not”.
• Using the “or” method instead of reading the wording carefully.
• Forgetting to subtract the overlap.

Real-Life Applications

Questions like this appear in surveys when analysing preferences, such as people who prefer one product but not another. Venn diagrams make it clear which group is being counted.

FAQ

Is “but not” the same as “exactly one”?
No. “But not” focuses on one specific group, while “exactly one” includes both single-group regions.

Which region should I look at?
The part of the circle that does not overlap with the other group.

Study Tip

When you see “but not” in a question, cross out the overlapping region and only count the single part of the circle.