GCSE Maths Practice: venn-diagrams

GCSE Maths (Higher): Interpreting “At Least One”

In probability questions, phrases such as at least one, either, or or all describe the same situation when dealing with two overlapping sets.

What does “at least one” include?

It includes everyone who is:

• In the first group only
• In the second group only
• In both groups

The only people excluded are those who belong to neither group.

The inclusion–exclusion principle

When two groups overlap, simply adding their totals will count some people twice. To correct this, use:

n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

This formula is essential for all GCSE Higher Venn-diagram problems involving two sets.

Worked example (different data)

In a group of 500 customers:

• 280 buy tea
• 320 buy coffee
• 150 buy both

Step 1: Add the two totals:

\(280 + 320 = 600\)

Step 2: Subtract the overlap:

\(600 - 150 = 450\)

Step 3: Divide by the total number of customers:

\(\frac{450}{500} = \frac{9}{10}\)

Why this is Higher tier

• Students must correctly interpret overlapping information
• Large numbers increase the chance of arithmetic mistakes
• This structure is often extended to neither or conditional probability

Common mistakes to avoid

• Adding both totals without subtracting the overlap
• Subtracting the overlap more than once
• Dividing by the wrong total

Study tip

Whenever you see at least one or or, check carefully whether an overlap is mentioned before calculating.