Venn Diagrams

A Venn diagram draws sets as overlapping circles inside a rectangle. The rectangle is the universal set — every item under consideration. Each circle is one set, say A and B, and where they overlap holds the items that belong to both.

The overlap is the intersection A \cap B — items in both A and B.
Everything inside either circle is the union A \cup B — items in A or B (or both).
Everything outside A is the complement A' — items not in A.

Writing a number in each region — how many items land there — turns the picture into a counting tool: a probability is just the items in a region divided by the total.

For two sets A and B drawn inside a universal set:

A \cap B (intersection) — the items in both, the overlap;
A \cup B (union) — the items in either circle;
A' (complement) — the items not in A;
read a probability of a region as P(\text{region}) = \frac{\text{items in the region}}{\text{total items}}.

Counting the regions

Step through a worked example. The four regions hold 7, 5, 8 and 4 items, so there are 24 altogether.