Venn Diagrams for Probability — GCSE Maths

A Venn diagram uses overlapping circles inside a rectangle to display how items or events are distributed across categories. At GCSE, Venn diagrams are used to calculate probabilities by counting or reading frequencies from the diagram's regions, including unions, intersections, and conditional probabilities.

What do the regions in a Venn diagram represent?

For two events A and B inside a universal set ξ:

• A only (left circle, not overlapping): outcomes in A but not B.
• A ∩ B (the overlap): outcomes in both A and B (the intersection).
• B only (right circle, not overlapping): outcomes in B but not A.
• Outside both circles: outcomes in neither A nor B.
• A ∪ B (everything inside either circle): outcomes in A or B or both (the union).

The rectangle represents the universal set — every possible outcome.

How do you fill in a Venn diagram from given information?

Step 1 — Fill the intersection (A ∩ B) first, as it is usually given directly.
Step 2 — Find A only = total for A − (A ∩ B).
Step 3 — Find B only = total for B − (A ∩ B).
Step 4 — Find outside = universal total − all values placed so far.

Worked example 1

In a class of 30 students, 18 study French, 12 study Spanish, and 7 study both. How many study neither?

Step 1 — Intersection: 7 study both. Place 7 in the overlap.
Step 2 — French only: 18 − 7 = 11.
Step 3 — Spanish only: 12 − 7 = 5.
Step 4 — Neither: 30 − 11 − 7 − 5 = 7 students.

Diagram summary:

How do you calculate probabilities from a Venn diagram?

Once the diagram is filled in, probability = (frequency in the region) ÷ (total frequency).

Using the example above (total = 30):

• P(French) = 18/30 = 3/5
• P(French ∩ Spanish) = 7/30
• P(French ∪ Spanish) = (11 + 7 + 5)/30 = 23/30
• P(neither) = 7/30
• P(French only, not Spanish) = 11/30

How do you find conditional probability from a Venn diagram?

Conditional probability P(A | B) means "the probability of A given that B has already occurred." You restrict your sample space to B, then ask what fraction of B is also A.

P(A | B) = P(A ∩ B) ÷ P(B)

Worked example 2

Using the class of 30 above: a student is chosen at random from those who study Spanish. What is the probability they also study French?

P(French | Spanish) = P(French ∩ Spanish) ÷ P(Spanish)
= (7/30) ÷ (12/30)
= 7/12

Answer: 7/12

On the Venn diagram, you are looking only at the 12 students in the Spanish circle — of those, 7 are in the overlap. So 7 out of 12.

What is the addition rule for probability?

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

This formula corrects for double-counting: P(A) and P(B) each include the intersection, so subtract it once.

Worked example 3

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

P(A ∪ B) = 0.5 + 0.4 − 0.2 = 0.7

Frequently asked questions

What is the difference between A ∩ B and A ∪ B?

A ∩ B (intersection, read "A and B") is the overlap region — outcomes in both events. A ∪ B (union, read "A or B") is everything inside either circle — outcomes in at least one event. In everyday language: intersection = AND, union = OR.

What if the total is given as a probability rather than a frequency?

Work directly with the probabilities. The method is identical — the four regions must sum to 1 rather than to the total count. Fill in P(A ∩ B) first, then find the remaining regions so they add to 1.

How do you handle a three-circle Venn diagram?

Fill in the central region (all three events) first, then the three pairwise overlaps (each minus the centre), then each individual circle, then the outside. Always work from the innermost region outward. Three-circle diagrams appear at GCSE Higher and follow the same logical sequence.

What does it mean if P(A ∩ B) = 0?

The events are mutually exclusive — they cannot happen at the same time. On the Venn diagram, the circles do not overlap. In this case, P(A ∪ B) = P(A) + P(B) with no subtraction needed, since there is nothing to double-count.

Professor Pi is ready to walk you through GCSE probability step by step — visit aitutors.me.