GCSE Maths Practice: venn-diagrams

Using Venn Diagrams to Find “OR” Probabilities

Venn diagrams are a key GCSE Maths tool for solving probability questions that involve two groups with shared members. In this question, the two groups are students who like Science and students who like Math. Because some students like both subjects, the groups overlap. If this overlap is ignored, students are easily counted twice, leading to an incorrect probability.

A Venn diagram visually separates the class into regions so that each student is counted exactly once. This makes it much easier to find probabilities involving the word or.

Step-by-Step Venn Diagram Method

Start by drawing a rectangle to represent the whole class of 60 students. Inside the rectangle, draw two overlapping circles. Label the left circle Science and the right circle Math.

Step 1: Fill the overlap first. The question states that 15 students like both science and math. Write 15 in the overlapping region of the two circles.

Step 2: Find the Science-only region. A total of 35 students like science, but this includes the 15 students already placed in the overlap. Subtract the overlap to avoid double counting:

Science only = 35 − 15 = 20.

Write 20 in the science-only region.

Step 3: Find the Math-only region. A total of 25 students like math, and again 15 of these are already in the overlap:

Math only = 25 − 15 = 10.

Write 10 in the math-only region.

Finding the Required Probability

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

Add all values inside the circles:

20 + 15 + 10 = 45.

So, 45 students like science or math.

Calculating and Simplifying the Probability

Probability is calculated as:

Probability = favourable outcomes ÷ total outcomes

Here, the favourable outcomes are 45 students, and the total number of students is 60. This gives:

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

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 class of 40 students, 18 like biology, 22 like chemistry, and 10 like both. Biology only = 18 − 10 = 8. Chemistry only = 22 − 10 = 12. Biology or chemistry = 8 + 10 + 12 = 30. Probability = \(\frac{30}{40} = \frac{3}{4}\).

Example 2: In a group of 50 students, 28 like physics, 20 like computing, and 8 like both. Physics only = 28 − 8 = 20. Computing only = 20 − 8 = 12. Physics or computing = 20 + 8 + 12 = 40. Probability = \(\frac{40}{50} = \frac{4}{5}\).

Common Mistakes

• Adding 35 and 25 without subtracting the overlap of 15.
• Writing the totals directly in the circles instead of splitting them into regions.
• Forgetting to simplify the final fraction.

Real-Life Applications

Venn diagrams are widely used to analyse survey data, such as students’ subject preferences or customers’ interests. They ensure that individuals who belong to multiple groups are counted correctly, which is essential for accurate conclusions.

FAQ

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

Do I need to find the number who like neither subject?
No, not for this question. However, it can be used to check your work. Here, 60 − 45 = 15 students like neither.

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

Study Tip

Whenever you see two groups and the word “both”, draw a Venn diagram straight away. It makes “OR” probability questions much easier and helps avoid common mistakes.