12. Company A produces \( 10 \% \) defective products, Company B produces \( 20 \% \) defective products and C produces \( 5 \% \) defective products. If choosing a company is an equally likely event, then what is the probability that the product chosen is defective?

To find the overall probability that the product chosen is defective, we can use the law of total probability. Since each company is equally likely to be chosen, we can take the average of the probabilities of defectiveness from each company. Let’s denote: - Company A: \( P(D|A) = 0.10 \) - Company B: \( P(D|B) = 0.20 \) - Company C: \( P(D|C) = 0.05 \) The probability that a company is chosen is \( P(A) = P(B) = P(C) = \frac{1}{3} \). Now we calculate the total probability of selecting a defective product: \[ P(D) = P(D|A) \cdot P(A) + P(D|B) \cdot P(B) + P(D|C) \cdot P(C) \] \[ P(D) = (0.10 \cdot \frac{1}{3}) + (0.20 \cdot \frac{1}{3}) + (0.05 \cdot \frac{1}{3}) \] \[ P(D) = \frac{0.10 + 0.20 + 0.05}{3} = \frac{0.35}{3} \approx 0.1167 \] Thus, the probability that the product chosen is defective is approximately \( 0.1167 \) or \( 11.67\% \).