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 two groups are students who like reading and students who like writing. Some students enjoy both activities, which means the groups overlap. If this overlap is not handled carefully, it is easy to make a mistake by counting some students twice.

A Venn diagram solves this problem by splitting the information into clear regions. Each region represents a different group of students, allowing you to see exactly who belongs where before calculating the probability.

Step-by-Step Venn Diagram Method

Begin by drawing a rectangle to represent the whole class of 40 students. Inside the rectangle, draw two overlapping circles. Label one circle Reading and the other Writing.

The first value to place is always the overlap. The question states that 10 students like both reading and writing, so write 10 in the overlapping region of the two circles.

Next, find how many students like only reading. The total number who like reading is 25, but this includes the 10 students already placed in the overlap. Subtract the overlap to avoid double counting:

Reading only = 25 − 10 = 15.

Write 15 in the part of the reading circle that does not overlap with writing.

Now find how many students like only writing. The total number who like writing is 15, and again 10 of these are already in the overlap:

Writing only = 15 − 10 = 5.

Write 5 in the writing-only region.

At this point, the Venn diagram correctly represents all students who like reading, writing, or both.

Finding the Probability

The question asks for the probability that a randomly chosen student likes reading or writing. In probability, “or” means one group, the other group, or both. This corresponds to everything inside the two circles.

Add the numbers inside the circles:

15 + 10 + 5 = 30.

So, 30 students like reading or writing out of a total of 40 students. The probability is therefore \(\frac{30}{40}\). GCSE Maths requires probabilities to be simplified where possible. Dividing the numerator and denominator by 10 gives:

\(\frac{30}{40} = \frac{3}{4}\).

Worked Examples

Example 1: In a class of 50 students, 28 like maths, 22 like science, and 10 like both. Maths only = 28 − 10 = 18. Science only = 22 − 10 = 12. Maths or science = 18 + 10 + 12 = 40. Probability = \(\frac{40}{50} = \frac{4}{5}\).

Example 2: At a club, 36 people like football, 24 like basketball, and 12 like both. Football only = 36 − 12 = 24. Basketball only = 24 − 12 = 12. Football or basketball = 24 + 12 + 12 = 48. If there are 60 people in total, the probability is \(\frac{48}{60} = \frac{4}{5}\).

Common Mistakes

• Adding 25 and 15 without subtracting the overlap of 10.
• Placing the total numbers directly into the circles instead of splitting into regions.
• Forgetting to simplify the final fraction.

Real-Life Applications

Venn diagrams are used in surveys, education, marketing, and data analysis whenever people belong to more than one group. For example, schools might analyse students who enjoy different subjects, and businesses might analyse customers who buy multiple products.

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?
No, not for this question. However, it can be used as a check. Here, 30 students are inside the circles, so 40 − 30 = 10 students like neither.

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

Study Tip

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