Addition and Multiplication Rules of Probability with Venn and Tree Diagrams

What it shows

Two events A and B are drawn as regions inside a sample space of area 1, so each area equals a probability. Moving B changes the overlap P(AB), and the union follows the addition rule P(A ∪ B) = P(A) + P(B) − P(AB). With no overlap the events are mutually exclusive; when P(AB) = P(A)·P(B) they are independent, and the multiplication rule holds. A two-step tree diagram shows the same situation with conditional probabilities P(B|A) and P(B|Ā). Random points dropped on the diagram give relative frequencies that approach the theoretical values.

How to use

Set P(A) and P(B), then drag B sideways to change P(AB) and read the addition rule at each position, including when the regions separate. Press Set independent and compare P(B|A) with P(B) on the Tree diagram. Use Random toss several times, increasing the Number of random tosses per button press, to see the frequencies settle near the theory. Ask students to decide whether each arrangement is exclusive, independent or neither.

Parameters you can change

• P(A) 0.05–0.95
• P(B) 0.05–0.95
• P(AB) = P(A ∩ B) (automatically clamped to a valid range) 0–0.95
• Show two-step tree diagram
• Number of random tosses per button press 100–3000

Questions to explore

1. Can two mutually exclusive events with positive probabilities be independent, and what does the diagram show?
2. When is P(A ∪ B) exactly equal to P(A) + P(B)?
3. If P(B|A) = P(B), what does the tree diagram tell you about events A and B?