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 Video by Zeeshan Zamurred Probability

Edexcel AS Level Maths: 5.2 Venn DiagramsWatch the full walkthrough 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 “ $A$ $and$ $B$ ”; $A \cup B$ (union) means “ $A$ $or$ $B$ (or both)”; $A^{'}$ (complement) means “ $not$ $A$ ”.

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

The three symbols you must know.

Filling in the diagram

Always start in the $middle$ (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.

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.

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

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