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 determine the probability that a randomly chosen product is defective, we can use the **Law of Total Probability**. Here's a step-by-step breakdown: 1. **Identify the Probabilities:** - **Probability of choosing each company:** - Company A: \( \frac{1}{3} \) - Company B: \( \frac{1}{3} \) - Company C: \( \frac{1}{3} \) - **Probability of a defective product from each company:** - Company A: \( 10\% = 0.10 \) - Company B: \( 20\% = 0.20 \) - Company C: \( 5\% = 0.05 \) 2. **Apply the Law of Total Probability:** \[ P(\text{Defective}) = P(A) \times P(\text{Defective} | A) + P(B) \times P(\text{Defective} | B) + P(C) \times P(\text{Defective} | C) \] Plugging in the values: \[ P(\text{Defective}) = \left(\frac{1}{3} \times 0.10\right) + \left(\frac{1}{3} \times 0.20\right) + \left(\frac{1}{3} \times 0.05\right) \] \[ P(\text{Defective}) = \frac{0.10 + 0.20 + 0.05}{3} = \frac{0.35}{3} \approx 0.1167 \] 3. **Convert to Percentage:** \[ 0.1167 \times 100 \approx 11.67\% \] **Final Answer:** The probability of selecting a defective product is approximately 11.67 %.