Success-Run Sample Size (Zero-Failure Test)

A success-run test puts n units through the test and accepts the design only if every unit passes. Suppose each unit works with probability R and the units behave independently. The chance that all n pass is then Rⁿ.

If the true reliability were only R, a clean run of n units would happen with probability Rⁿ. When that probability is no more than 1 − C, a clean run is unlikely enough that you can say, at confidence C, the reliability is at least R. Setting Rⁿ ≤ 1 − C and taking logarithms gives the sample size.

1. Convert confidence and reliability from percent to fractions: C = confidence / 100, R = reliability / 100.
2. Work out ln(1 − C) / ln(R). Both logarithms are negative, so the ratio is positive.
3. Round up to the next whole number. A fraction of a unit cannot be tested, and rounding down would fall short of C. The tool subtracts 10⁻⁹ before rounding up so that a ratio that is a whole number is not pushed up by floating-point error.
4. The same relation solved the other ways gives the reliability that n passing units demonstrate, R = (1 − C)^1/n, and the confidence that n passing units give for a reliability R, C = 1 − Rⁿ.

For example, 90 % reliability at 90 % confidence needs ln(0.10) / ln(0.90) = 21.85, so 22 units must all pass. With 22 units the confidence actually reached is 1 − 0.90²² = 90.15 %.

• Assumptions. The units are independent and come from the same population, each one either passes or fails, and the test conditions represent the conditions of use (load, environment and duration).
• No failures allowed. The plan holds only if every unit passes. A single failure means the test does not demonstrate R at C; a plan that tolerates failures needs a different calculation.
• Large samples at high reliability. As R approaches 100 % the sample size grows quickly: 99 % reliability at 90 % confidence already needs 230 units.
• A lower bound, not an estimate. A passed test shows that reliability is at least R at confidence C. It does not estimate the actual reliability, which may be much higher.
• This calculator demonstrates reliability at a confidence level; to accept or reject a lot by sample, use the AQL Sampling Plan.

• Reading R as the expected reliability. R is the lowest reliability the test supports at confidence C, not a measured value.
• Continuing after a failure. Replacing a failed unit, retesting it or adding units to reach the planned count does not keep the zero-failure plan valid.
• Testing under easier conditions. A shorter test, a lighter load or a milder environment than in use demonstrates reliability for those easier conditions only.
• Rounding down. 21.85 units means 22; 21 units reach only 1 − 0.90²¹ = 89.06 % confidence.
• Mixing percent and fractions. The tool takes percent; 0.9 entered as confidence means 0.9 %, not 90 %.