1. **State the problem:** We need to find the conditional probability $P(W|C)$, which is the probability of event $W$ given that event $C$ has occurred. 2. **Recall Bayes' formula:** $$ P(W|C) = \frac{P(C|W) \cdot P(W)}{P(C)} $$ where $P(C)$ is the total probability of $C$ occurring. 3. **Calculate $P(C)$ using the law of total probability:** $$ P(C) = P(C|U)P(U) + P(C|V)P(V) + P(C|W)P(W) $$ Given: - $P(U) = 0.1$, $P(V) = 0.4$, $P(W) = 0.5$ - $P(C|U) = 0.1$, $P(C|V) = 0.8$, $P(C|W) = 0.9$ Calculate: $$ P(C) = (0.1)(0.1) + (0.8)(0.4) + (0.9)(0.5) = 0.01 + 0.32 + 0.45 = 0.78 $$ 4. **Apply Bayes' formula:** $$ P(W|C) = \frac{P(C|W)P(W)}{P(C)} = \frac{(0.9)(0.5)}{0.78} $$ 5. **Simplify the fraction:** $$ P(W|C) = \frac{0.45}{0.78} $$ 6. **Calculate the decimal value:** $$ P(W|C) \approx 0.577 $$ **Final answer:** $$ P(W|C) \approx 0.577 $$