How it works

Venn diagrams show how sets overlap using circles inside a rectangle representing the universal set. The intersection (A ∩ B) contains elements in both sets, and the union (A ∪ B) contains elements in either set. Probabilities can be calculated by writing the number of outcomes in each region and dividing by the total number in the universal set.

The method

1. 1.Draw a rectangle for the universal set and circles for each set
2. 2.Fill in the intersection value first
3. 3.Fill in the remaining parts of each circle by subtraction
4. 4.Use the diagram to read off or calculate probabilities

Worked example

In a class of 30, 18 like Maths, 15 like Science, 8 like both. Find P(likes neither).

1. Only Maths=10, only Science=7, both=8
2. Total liking at least one = 25, neither = 5
3. P(neither) = 5/30 = 1/6

Worked example

Find P(A ∩ B) if 12 out of 40 are in both sets.

1. P(A ∩ B) = 12/40 = 3/10

Common mistakes

• Double counting elements that belong in the intersection
• Confusing union (∪, either set) with intersection (∩, both sets)
• Forgetting elements outside both sets when using the universal set