Three Suppliers and a Two-Stage Inspection

A firm sources components from three suppliers: $S_1$ (50% of supply, 2% defect rate), $S_2$ (30%, 3% defect rate), $S_3$ (20%, 5% defect rate). A component is selected at random and undergoes two independent inspection stages. Stage 1 detects a defect with probability 0.8 (if defective) and falsely flags a good component with probability 0.05. Stage 2 detects a defect with probability 0.9 (if defective) and falsely flags with probability 0.03. A component is flagged by both stages. Compute:

(a) $P(\text{component is actually defective} \mid \text{flagged by both stages})$.

(b) Given that the component is actually defective and flagged by both stages, what is the probability it came from $S_3$?