Factory Defect Tracing and Predictive Inference

A factory has three production lines with the following output shares and defect rates:

| Line | Share of output | Defect rate | |------|----------------|-------------| | 1 | 50% | 2% | | 2 | 30% | 3% | | 3 | 20% | 5% |

An item is selected at random from today's output and found to be defective.

(a) What is the posterior probability that the item came from each production line?

(b) A second item is now drawn independently from the same (unknown) production line. What is the conditional probability that this second item is also defective, given that the first was defective?