How Do You Use a Venn Diagram to Calculate Probability?

Use a Venn diagram by placing outcomes into the correct regions, then adding the regions that represent the required event. The key is to fill the intersection first so that outcomes belonging to both events are not counted twice.

This is part of Mathematics: Analysis and Approaches SL 4.6: Combined events and probability diagrams, which is core content for both SL and HL.

Suppose a class contains 40 students. Of these, 22 study Biology, 18 study Chemistry, and 8 study both. Let the events be $B$ and $C$.

1. Put 8 in the overlap, representing $B \cap C$.
2. The Biology-only region is $22-8=14$.
3. The Chemistry-only region is $18-8=10$.
4. The number studying at least one subject is $14+8+10=32$.
5. The number studying neither is $40-32=8$.

Each region is disjoint: a student belongs to exactly one of Biology only, Chemistry only, both, or neither. Therefore, adding selected regions does not double-count any student. For conditional probability, the word “given” changes the denominator: once Chemistry is known, the relevant sample space contains only the 18 Chemistry students. Of these, 8 also study Biology, so $P(B\mid C)=8/18$, not $8/40$. This is the restricted sample space instead.

Probabilities are found by dividing the relevant frequency by the total frequency, 40.

The addition rule confirms the union:

$$P(B \cup C)=P(B)+P(C)-P(B \cap C).$$

A common misconception is adding $P(B)+P(C)$ without subtracting the intersection. This counts students in both subjects twice.

Exam technique: For a “calculate” or “find” question, label events clearly, fill the overlap first, show the relevant region or formula, and give an exact fraction or a decimal to 3 significant figures where appropriate. Do not confuse mutually exclusive events, whose intersection is zero, with independent events, whose intersection satisfies $P(B \cap C)=P(B)P(C)$.