1. **State the problem:** We need to find the conditional probability $P(M|B)$, which is the probability of event $M$ given event $B$. 2. **Recall the formula for conditional probability:** $$P(M|B) = \frac{P(M \cap B)}{P(B)} = \frac{P(M \cap B)}{P(M \cap B) + P(N \cap B)}$$ 3. **Identify probabilities from the tree diagram:** - $P(M) = 0.8$ - $P(N) = 0.2$ - $P(B|M) = 0.9$ - $P(B|N) = 0.8$ 4. **Calculate joint probabilities:** $$P(M \cap B) = P(M) \times P(B|M) = 0.8 \times 0.9 = 0.72$$ $$P(N \cap B) = P(N) \times P(B|N) = 0.2 \times 0.8 = 0.16$$ 5. **Calculate $P(M|B)$:** $$P(M|B) = \frac{0.72}{0.72 + 0.16} = \frac{0.72}{0.88}$$ 6. **Simplify the fraction:** $$P(M|B) = \frac{\cancel{0.72}}{\cancel{0.88}} = \frac{72}{88} = \frac{9}{11} \approx 0.8182$$ 7. **Final answer:** $$\boxed{P(M|B) = \frac{9}{11} \approx 0.8182}$$