Venn diagrams and set notation
Objective. Connect set regions with probability statements.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
A Venn diagram uses overlapping circles within a rectangle (the universal set) to represent events and their relationships, where the overlapping region shows the intersection A∩B and the combined shaded area of both circles shows the union A∪B. Regions outside both circles but inside the rectangle represent neither event occurring, written as (A∪B)'. Probabilities are often written directly inside each region so that calculations such as P(A|B) or P(A') can be read straight from the diagram once the regions sum correctly to 1. For example, if P(A only) = 0.2, P(B only) = 0.3, and P(A∩B) = 0.1, then P(A∪B) = 0.2 + 0.3 + 0.1 = 0.6, leaving 0.4 for neither.
Worked example
In a Venn diagram, P(A) = 0.5, P(B) = 0.4 and P(A∩B) = 0.2. Find P(neither A nor B occurs).
- 1Find P(A∪B) using the addition law: 0.5 + 0.4 − 0.2 = 0.7.
- 2Recall P(neither) = P((A∪B)') = 1 − P(A∪B).
- 3Substitute: 1 − 0.7.
- 4Compute: 1 − 0.7 = 0.3.
Answer: 0.3
Concept mastery
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Practice
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Start practisingCommon mistake. Students sometimes write the full P(A) value inside the overlapping intersection region instead of just the intersection part, double-counting it when regions are summed. Only the pure intersection probability belongs in the overlapping area; the rest of P(A) goes in the non-overlapping part of A's circle.
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