Mathematics Behind Failure Rate for Mass Production
I like to make sure my parts have a failure rate below 1 %, how many pieces should I check?
The mathematics behind this is:
Assume that the pass rate for the lot is 99%, if one piece is checked and the chance that you found the unqualified one is 1%, if you check 2 pieces, the chance you found the 2 good ones is, 99% x 99% =0.98（assume the lot is big enough that one piece less would affect the failure rate in the remain parts）, so on and so forth. In general, if the failure rate is f%, check quantity is n, the chance that no wrong piece is found in the check is c%=(1-f%)^n. In a plainer language, if we have checked n samples and found them all good, we have 1-c% level of confidence that this lot is better than failure rate f%.
The following chart gives the parameter relations with a real number.
| Target Failure Rate | Checking Quantity | All good chance | Confidence |
| 1% | 1 | 99.00% | 1.00% |
| 1% | 2 | 98.01% | 1.99% |
| 1% | 3 | 97.03% | 2.97% |
| 1% | 4 | 96.06% | 3.94% |
| 1% | 5 | 95.10% | 4.90% |
| 1% | 10 | 90.44% | 9.56% |
| 1% | 20 | 81.79% | 18.21% |
| 1% | 50 | 60.50% | 39.50% |
| 1% | 100 | 36.60% | 63.40% |
| 0.10% | 1 | 99.90% | 0.10% |
| 0.10% | 2 | 99.80% | 0.20% |
| 0.10% | 10 | 99.00% | 1.00% |
| 0.10% | 100 | 90.48% | 9.52% |
| 0.10% | 1000 | 36.77% | 63.23% |
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