Probability Trees and Venn Diagrams

Structure

Branches show outcomes with probabilities. Each set of branches sums to 1.

Rules

• AND: multiply along branches.
• OR: add the final probabilities.

With/Without Replacement

• With replacement: probabilities stay the same.
• Without replacement: probabilities change (conditional).

Example

Bag: 3 red, 2 blue. Pick two without replacement.

$P(\text{both red}) = \frac{3}{5} \times \frac{2}{4} = \frac{6}{20} = \frac{3}{10}$

$P(\text{one of each}) = \frac{3}{5} \times \frac{2}{4} + \frac{2}{5} \times \frac{3}{4} = \frac{12}{20} = \frac{3}{5}$

Two Sets

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

Regions

• $A$ only: $P(A) - P(A \cap B)$
• $B$ only: $P(B) - P(A \cap B)$
• Neither: $1 - P(A \cup B)$

Example

30 students: 18 play football, 12 play tennis, 6 play both.

Football only: 12. Tennis only: 6. Both: 6. Neither: 6.