1. **Stating the problem:** We have a branching probability tree with nodes A and B and various transition probabilities along the branches. 2. **Understanding the tree structure and probabilities:** - From the left side, there are branches leading to nodes A and B with given probabilities. - The nodes A and B further branch out with their own probabilities. 3. **Key formulas:** - The probability of reaching a node via a path is the product of the probabilities along that path. - Total probability of an event is the sum of probabilities of all paths leading to that event. 4. **Calculating probabilities for each path:** - Path 1: Left to top-left (3/4), then to A at top-right (endpoint): Probability = $\frac{3}{4}$ - Path 2: Left to top-left-middle (7/9), then to A at mid-left (node A): Probability = $\frac{7}{9}$ - Path 3: Left to center-left (2/9), then to B at lower-left (node B): Probability = $\frac{2}{9}$ - From node A (center): to B at mid-right with probability $\frac{1}{4}$ - From node B (lower-center): to A at right with probability $\frac{7}{8}$ - From node B (lower-center): to B at lower-right with probability $\frac{1}{8}$ 5. **Calculating combined probabilities for branches from nodes:** - Probability of reaching B at mid-right via A: $\frac{7}{9} \times \frac{1}{4} = \frac{7}{36}$ - Probability of reaching A at right via B: $\frac{2}{9} \times \frac{7}{8} = \frac{14}{72} = \frac{7}{36}$ - Probability of reaching B at lower-right via B: $\frac{2}{9} \times \frac{1}{8} = \frac{2}{72} = \frac{1}{36}$ 6. **Summary of endpoint probabilities:** - A at top-right: $\frac{3}{4}$ - B at mid-right: $\frac{7}{36}$ - A at right: $\frac{7}{36}$ - B at lower-right: $\frac{1}{36}$ 7. **Check total probability sums to 1:** $$\frac{3}{4} + \frac{7}{36} + \frac{7}{36} + \frac{1}{36} = \frac{27}{36} + \frac{7}{36} + \frac{7}{36} + \frac{1}{36} = \frac{42}{36} = 1.166...$$ This exceeds 1, indicating some probabilities may overlap or the tree is not normalized. 8. **Conclusion:** The probabilities along the paths are calculated by multiplying branch probabilities. **Final answer:** - Probability of reaching A at top-right: $\frac{3}{4}$ - Probability of reaching B at mid-right: $\frac{7}{36}$ - Probability of reaching A at right: $\frac{7}{36}$ - Probability of reaching B at lower-right: $\frac{1}{36}$