GCSE Maths Practice: venn-diagrams

GCSE Maths (Higher): Probability with Overlapping Groups

In survey-based probability questions, people can belong to more than one group. Venn diagrams are used to organise this information clearly.

What does “likes dogs or cats” mean?

The word or means we include everyone who:

• Likes dogs only
• Likes cats only
• Likes both dogs and cats

The only people excluded are those who like neither animal.

Why double-counting happens

If we simply add the number of people who like dogs and the number who like cats, anyone who likes both animals is counted twice. To correct this, the overlap must be subtracted once.

The inclusion–exclusion rule

For two overlapping sets A and B:

n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

This rule applies to all GCSE Higher questions involving two sets.

Worked example (different data)

In a group of 400 people:

• 260 like tea
• 190 like coffee
• 120 like both drinks

Step 1: Add the two totals:

\(260 + 190 = 450\)

Step 2: Subtract the overlap:

\(450 - 120 = 330\)

Step 3: Divide by the total number of people:

\(\frac{330}{400} = \frac{33}{40}\)

Why this is Higher tier

• Requires careful interpretation of overlapping information
• Students must avoid common counting mistakes
• This method is often extended to neither or conditional probability questions

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 two groups overlap, always ask: “Who would be counted twice if I just added the totals?”