1. **Problem Statement:** A company has three machines A1, A2, and A3 producing 20%, 35%, and 45% of total output respectively. Defective rates are 2% for A1, 3% for A2, and 5% for A3. A randomly chosen product is defective. Find the probability it was produced by A3. 2. **Formula Used:** We use Bayes' Theorem to find the conditional probability: $$P(A3|D) = \frac{P(D|A3) \times P(A3)}{P(D)}$$ where: - $P(A3|D)$ is the probability product is from A3 given it is defective. - $P(D|A3)$ is the probability product is defective given it is from A3. - $P(A3)$ is the probability product is from A3. - $P(D)$ is the total probability product is defective. 3. **Calculate $P(D)$ using Total Probability Theorem:** $$P(D) = P(D|A1)P(A1) + P(D|A2)P(A2) + P(D|A3)P(A3)$$ Substitute values: $$P(D) = 0.02 \times 0.20 + 0.03 \times 0.35 + 0.05 \times 0.45$$ $$P(D) = 0.004 + 0.0105 + 0.0225 = 0.037$$ 4. **Calculate $P(A3|D)$:** $$P(A3|D) = \frac{0.05 \times 0.45}{0.037} = \frac{0.0225}{0.037} \approx 0.6081$$ 5. **Interpretation:** There is approximately a 60.81% chance that a defective product was produced by machine A3.