S5.2StatisticsCore

Venn Diagrams

Venn diagrams organise probabilities of overlapping events into regions, making “and”, “or” and “not” problems visual. Master the notation and the diagrams become a reliable problem-solving tool.

25 min Probability

Edexcel AS Level Maths: 5.2 Venn DiagramsCurated by Kepler, not produced by us. Watch it before the notes below.

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What you'll be able to do

Interpret intersection, union and complement notation
Fill in a Venn diagram from given probabilities
Read probabilities off a Venn diagram
Use the addition formula

Set notation

$A \cap B$ (intersection) means “ $A$ $\textbf{and}$ $B$ ”; $A \cup B$ (union) means “ $A$ $\textbf{or}$ $B$ (or both)”; $A'$ (complement) means “ $\textbf{not}$ $A$ ”.

A \cap B = \text{and}, \quad A \cup B = \text{or}, \quad A' = \text{not}

The three symbols you must know.

Filling in the diagram

Always start in the $\textbf{middle}$ (the intersection) and work outwards, so each region is counted once. The numbers in all regions must add to $1$ (or to the total frequency).

Tip — Fill the overlap first, then subtract to get the “only A” and “only B” regions.

The addition formula

The probability of $A$ or $B$ adds the two probabilities and subtracts the overlap (so it is not double-counted).

P(A \cup B) = P(A) + P(B) - P(A \cap B)

Subtract the intersection to avoid double-counting.

Example 1P(A) = 0.5, P(B) = 0.4, P(A∩B) = 0.2. Find P(A∪B).

0.5 + 0.4 - 0.2

Answer

0.7

Formula recap

A \cap B,\ A \cup B,\ A'

And / or / not.

P(A \cup B) = P(A) + P(B) - P(A \cap B)

Addition formula.

\text{all regions sum to } 1

Completeness check.

Common mistakes to avoid

Forgetting to subtract the intersection in the addition formula.

P(A∪B) = P(A) + P(B) − P(A∩B); the overlap must not be double-counted.

Filling “A” with the whole circle instead of “A only”.

Work from the intersection outwards so each region is counted once.

Key takeaways

∩ = and, ∪ = or, A′ = not.
Fill Venn diagrams from the middle out; regions sum to 1.
P(A∪B) = P(A) + P(B) − P(A∩B).

Test yourself

Answer a few questions, then review anything you missed.

Next step

Check your understanding

Work through questions on Venn Diagrams. Use the feedback to decide what needs another look.

7 past-paper questions · 14 marks

Practise this topic

Related lessons

All 4 lessons

Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.