A conduit plant fills orders from three extrusion lines. Line 1 supplies 50% of output at a 2% defect rate, Line 2 supplies 30% at a 4% defect rate, and Line 3 supplies 20% at a 5% defect rate. A unit drawn at random from finished stock is found to be defective. The probability that it was produced on Line 2 is most nearly:
- (A)0.040
- (B)0.300
- (C)0.313
- (D)0.375
Show worked solution
Answer: (D)
Line 2 accounts for 0.375 of all defective units, more than its 30% share of production because its defect rate runs above the plant average.
Total probability applies because the three lines partition the finished stock, so the plant-wide defect rate is the share-weighted sum of the line rates.
Bayes' theorem then rescales the 30% prior share of Line 2 by how readily that line generates the observed defect.
FE Reference Handbook — Engineering Probability and Statistics: Bayes' Theorem
Why the other choices appear
- (A)Reports the conditional defect rate of Line 2 itself, , rather than the reverse conditional probability.
- (B)Reports the prior output share , ignoring the defect evidence entirely.
- (C)Uses the Line 1 (or Line 3) numerator by mistake, .