GCSE Maths Practice: venn-diagrams

Using Venn Diagrams for “OR” Probability

Venn diagrams are a key GCSE Maths tool for solving probability questions involving overlapping groups. In this question, the groups are people who like tea and people who like coffee. Because some people like both drinks, the two groups overlap. If this overlap is not handled correctly, some people will be counted twice, leading to an incorrect probability.

A Venn diagram helps organise the information visually so that each person is counted exactly once. This makes probability questions involving the word or much easier to understand and solve.

Filling the Venn Diagram Step by Step

Start by drawing a rectangle to represent the whole group of 60 people. Inside the rectangle, draw two overlapping circles. Label the left circle Tea and the right circle Coffee.

Step 1: Fill the overlap first. The question states that 20 people like both tea and coffee. Write 20 in the overlapping region of the circles.

Step 2: Find the “tea only” region. A total of 35 people like tea, but 20 of these are already included in the overlap. Subtract the overlap:

Tea only = 35 − 20 = 15.

Write 15 in the part of the tea circle that does not overlap with coffee.

Step 3: Find the “coffee only” region. A total of 30 people like coffee, and again 20 of these are already in the overlap. Subtract the overlap:

Coffee only = 30 − 20 = 10.

Write 10 in the coffee-only region.

Finding the Required Probability

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

Add the numbers inside the circles:

15 + 20 + 10 = 45.

So, 45 people like at least one of the two drinks.

Calculating and Simplifying the Probability

Probability is calculated as:

Probability = favourable outcomes ÷ total outcomes

Here, the favourable outcomes are the 45 people who like tea or coffee, and the total outcomes are the 60 people in the group:

\(\frac{45}{60}\)

GCSE answers should always be simplified. Divide the numerator and denominator by 15:

\(\frac{45}{60} = \frac{3}{4}\)

Worked Examples

Example 1: In a group of 40 people, 22 like apples, 18 like bananas, and 10 like both. Apples only = 22 − 10 = 12. Bananas only = 18 − 10 = 8. Apples or bananas = 12 + 10 + 8 = 30. Probability = \(\frac{30}{40} = \frac{3}{4}\).

Example 2: In a class of 50 students, 28 like football, 20 like basketball, and 8 like both. Football only = 28 − 8 = 20. Basketball only = 20 − 8 = 12. Football or basketball = 20 + 8 + 12 = 40. Probability = \(\frac{40}{50} = \frac{4}{5}\).

Common Mistakes

• Adding 35 and 30 without subtracting the overlap of 20.
• Writing the totals directly into the circles instead of finding the “only” regions.
• Forgetting to simplify the final fraction.

Real-Life Applications

Venn diagrams are commonly used in surveys and market research. For example, cafés may analyse how many customers like tea, coffee, or both drinks. Venn diagrams help ensure customers who like both are not counted twice.

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 drink?
No, not for this question. However, it can be used to check your work. Here, 60 − 45 = 15 people like neither.

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

Study Tip

If a probability question mentions two groups and the word “both”, draw a Venn diagram straight away. This prevents double counting and makes the method clear.