Defect Attribution and Screening Cost
A small electronics workshop uses three soldering stations. Station A makes 50% of the boards, station B makes 30%, and station C makes 20%. Historical records show defect rates of 2% for A, 3% for B, and 5% for C. A finished board is selected at random and found defective. The manager must decide which station to investigate first. A trainee argues for C because it has the highest defect rate; another argues for A because it produces the most boards. The workshop is also considering a 100-yen-per-board screening test. The test detects a defective board with probability 0.90 and gives a false positive on a good board with probability 0.05. A defective board that escapes detection costs 5,000 yen in warranty work, while a false positive costs 500 yen in unnecessary rework. The manager wants the posterior probability that the defective board came from each station, the expected number of defective boards in a 1,000-board production run, and a comparison of expected costs with and without the screening test, assuming all defective boards not caught would otherwise incur the warranty cost.
1. Find the posterior probability that a defective board came from each station. Which station should the manager investigate first?
Read the worked answer
By Bayes,
P(D)=0.5·0.02+0.3·0.03+0.2·0.05=0.029.
Then
P(A|D)=0.010/0.029=10/29≈34.5%, P(B|D)=0.009/0.029=9/29≈31.0%, P(C|D)=0.010/0.029=10/29≈34.5%.
A and C tie. C's higher defect rate is offset by A's larger production share. The trainee who picks C from defect rate alone ignores base rates, and the trainee who picks A from volume alone ignores rates. The manager should investigate A and C together, not just one.
2. Find the expected number of defective boards in a 1,000-board run, broken down by station.
Read the worked answer
In 1,000 boards, A makes 500, B makes 300, and C makes 200. Expected defects:
A: 500·0.02=10, B: 300·0.03=9, C: 200·0.05=10.
The total is 29, which matches 1000·0.029.
3. Compare the expected cost of no screening with the expected cost of screening all 1,000 boards under the stated assumptions. Is the test worthwhile? Mention one assumption that could reverse the decision.
Read the worked answer
No screening: 29 defects × 5,000 = 145,000 yen.
Screening:
- test cost: 1,000×100=100,000 yen;
- missed defects: 29×0.10=2.9, cost 14,500 yen;
- false positives among 971 good boards: 971×0.05=48.55, cost 24,275 yen.
Total screening cost = 100,000+14,500+24,275=138,775 yen. Screening saves 6,225 yen, so it is worthwhile under these numbers. The margin is small, however. If a false positive costs more than about 628 yen, the test loses money: solve 114,500+48.55c=145,000. Alternatively, if the test cost exceeds about 106.2 yen per board, the conclusion reverses. The result also assumes detected defects incur no extra repair cost.