1. **State the problem:** We have two plants producing parts. Plant 1 produces 1500 parts with 150 defective, and Plant 2 produces 2800 parts with 260 defective. A part is randomly selected and found defective. We want the probability it came from Plant 2. 2. **Identify the formula:** We use Bayes' theorem for conditional probability: $$P(\text{Plant 2} | \text{Defective}) = \frac{P(\text{Defective} | \text{Plant 2}) \times P(\text{Plant 2})}{P(\text{Defective})}$$ 3. **Calculate each probability:** - Total parts = $1500 + 2800 = 4300$ - $P(\text{Plant 1}) = \frac{1500}{4300}$ - $P(\text{Plant 2}) = \frac{2800}{4300}$ - $P(\text{Defective} | \text{Plant 1}) = \frac{150}{1500} = \frac{1}{10}$ - $P(\text{Defective} | \text{Plant 2}) = \frac{260}{2800} = \frac{13}{140}$ 4. **Calculate total defective probability:** $$P(\text{Defective}) = P(\text{Defective} | \text{Plant 1}) \times P(\text{Plant 1}) + P(\text{Defective} | \text{Plant 2}) \times P(\text{Plant 2})$$ $$= \frac{1}{10} \times \frac{1500}{4300} + \frac{13}{140} \times \frac{2800}{4300}$$ 5. **Simplify:** $$= \frac{1500}{43000} + \frac{13 \times 2800}{140 \times 4300} = \frac{1500}{43000} + \frac{36400}{602000}$$ 6. **Convert to common denominator and add:** $$\frac{1500}{43000} = \frac{1500 \times 14}{43000 \times 14} = \frac{21000}{602000}$$ $$P(\text{Defective}) = \frac{21000}{602000} + \frac{36400}{602000} = \frac{57400}{602000}$$ 7. **Calculate numerator for Bayes' theorem:** $$P(\text{Defective} | \text{Plant 2}) \times P(\text{Plant 2}) = \frac{13}{140} \times \frac{2800}{4300} = \frac{13 \times 2800}{140 \times 4300} = \frac{36400}{602000}$$ 8. **Apply Bayes' theorem:** $$P(\text{Plant 2} | \text{Defective}) = \frac{\frac{36400}{602000}}{\frac{57400}{602000}} = \frac{36400}{\cancel{602000}} \times \frac{\cancel{602000}}{57400} = \frac{36400}{57400}$$ 9. **Simplify fraction:** $$\frac{36400}{57400} = \frac{364}{574} = \frac{182}{287}$$ 10. **Final answer:** $$P(\text{Plant 2} | \text{Defective}) = \frac{182}{287} \approx 0.634$$ So, the probability that a defective part came from Plant 2 is approximately 0.634 or 63.4%.