What is a Venn Diagram?
A Venn diagram is a visual tool used to represent relationships between sets (or events) as overlapping circles within a rectangle that represents the entire sample space (universal set).
In probability, each circle represents an event, and the overlapping region represents outcomes that belong to both events simultaneously. The rectangle surrounding all circles is the sample space — it contains every possible outcome.
The key regions in a two-event Venn diagram are:
- Inside A only: outcomes in A but not B
- Inside B only: outcomes in B but not A
- Intersection (A ∩ B): outcomes in both A and B
- Outside both circles: outcomes in neither A nor B
The probabilities of all regions in a Venn diagram must sum to 1 (or 100%), since they together cover the entire sample space.
Reading and Filling in a Venn Diagram
When you are given information about a situation involving two events, your first task is to correctly place values into the four regions of the Venn diagram.
Always start from the innermost region (the intersection) and work outwards. This prevents double-counting and avoids needing to apply inclusion-exclusion at each step.
- Fill in first.
- Subtract it from to get the A only region.
- Subtract it from to get the B only region.
- Subtract all three regions from 1 to get the neither region.
In a class, 65% of students play soccer, 45% play basketball, and 25% play both.
Step 1: Intersection = 25% ✓
Step 2: Soccer only = 65% − 25% = 40%
Step 3: Basketball only = 45% − 25% = 20%
Step 4: Neither = 100% − 40% − 25% − 20% = 15%
The four regions (40%, 25%, 20%, 15%) sum to 100%. ✓
From the diagram, we can quickly read off: