1. **State the problem:** Calculate the probabilities of reaching each final outcome in a branching tree diagram with given branch probabilities. 2. **Formula and rules:** The probability of reaching a final outcome is the product of the probabilities along the branches leading to it. 3. **Calculate each final outcome probability:** - For path to A then A: $$\frac{3}{7} \times \frac{1}{3} = \frac{3 \times 1}{7 \times 3} = \frac{3}{21} = \frac{\cancel{3}^1}{7 \times \cancel{3}^1} = \frac{1}{7}$$ - For path to A then B: $$\frac{3}{7} \times \frac{2}{3} = \frac{3 \times 2}{7 \times 3} = \frac{6}{21} = \frac{\cancel{3}^2 \times 2}{7 \times \cancel{3}^2} = \frac{2}{7}$$ - For path to B then A: $$\frac{4}{7} \times \frac{1}{2} = \frac{4 \times 1}{7 \times 2} = \frac{4}{14} = \frac{\cancel{2}^2}{7 \times \cancel{2} \times 1} = \frac{2}{7}$$ - For path to B then B: $$\frac{4}{7} \times \frac{1}{2} = \frac{4}{14} = \frac{2}{7}$$ 4. **Interpretation:** The probabilities of the final outcomes are: - A then A: $\frac{1}{7}$ - A then B: $\frac{2}{7}$ - B then A: $\frac{2}{7}$ - B then B: $\frac{2}{7}$ 5. **Check sum:** $$\frac{1}{7} + \frac{2}{7} + \frac{2}{7} + \frac{2}{7} = \frac{7}{7} = 1$$ This confirms the probabilities are consistent and complete.