Zero-failure reliability demonstration calculator

RELIABILITY · EVIDENCE & UNCERTAINTY

Zero-failure demonstration

How many independent trials with no failures support a target reliability at a chosen confidence? Compare a planned test with the exact one-sided lower confidence bound from completed zero-failure trials.

Educational model only. No safety, class, regulatory, certification or equipment-acceptance approval. Reliability here means success on one consistently defined mission or demand, not availability or a time-to-failure rate.

Model: the same success definition and conditions, a shared fixed $p$ , and trials independent given $p$ . Do not discard failures or restart the counter to obtain a failure-free streak.

Current results

Enter inputs, then calculate.

Calculations run in your browser without upload or persistent storage. CSV download occurs only on request. Changing language or reloading resets inputs. Display: 8 significant digits; CSV retains numeric precision. · reliability-1.0.1

Method and interpretation

If one trial succeeds with probability $p$ , the probability that all $n$ independent trials succeed is $p^{n}$ . Choose the smallest integer $n$ making the all-success probability at target $R_{0}$ no greater than $1 - C$ . This plan requires zero failures; one observed failure invalidates this calculation.

$P (0 failures | p, n) = p^{n}$ $n \geq \frac{\ln (1 - C)}{\ln (R_{0})}$ $n_{\min} = ⌈ \frac{\ln (1 - C)}{\ln (R_{0})} ⌉$ $R^{L} = (1 - C)^{\frac{1}{n}}$ $C (R_{0}, n) = 1 - R_{0}^{n}$

$R^{L}$ is the exact one-sided lower confidence bound for a binomial sample with zero failures. “95% confidence” describes repeated-sampling coverage of the procedure, not a 95% posterior probability that the fixed unknown $p$ lies in this interval. A two-sided 95% interval is different from this one-sided bound.

Worked example · invented test evidence

For an invented actuator test, each trial is one opening demand at the same load and environment. Define success before testing. Choose $R_{0} = 0.95$ and $C = 0.95$ .

$\frac{\ln (0.05)}{\ln (0.95)} = 58.4039748 \to n_{\min} = 59$
58 trials: $1 - {0.95}^{58} = 94.895313 % < 95 %$
59 trials: $1 - {0.95}^{59} = 95.150547 % \geq 95 %$
After 59 successes and zero failures: $R^{L} = {0.05}^{\frac{1}{59}} = 0.95049239$

Under the stated model, the lower bound exceeds the target. This does not guarantee success on every mission. Repeatedly testing one unit may not create independent or representative evidence.

The curve is

R_{0}

raised to the trial count. The horizontal line is

1 - C

. The required integer sample size is the first trial count at or below that line.

Assumptions and limits

Specify the success criterion, mission duration, load, environment, sampling and stopping rule before testing. Optional stopping after inspecting results can invalidate the stated confidence interpretation.
Trials must share a stable success probability. Common causes, ageing, repairs, design changes and unit heterogeneity can violate this simple model. Partial missions and censored lifetime data cannot simply be treated as these counts.
This is a mission-reliability model. It does not calculate an exponential lifetime model, failure rate, MTBF, availability, SIL/PFD or maintenance policy. Examples are entirely invented.
Input limits are software limits, not engineering acceptance criteria. Extreme probabilities can round to 0 or 1 at floating-point precision. For narrow posteriors the sampled curve is illustrative; use the numeric interval for precise endpoints.

Primary sources

NIST/SEMATECH e-Handbook: confidence limits for a binomial proportion

Current results

Method and interpretation

Worked example · invented test evidence

Assumptions and limits

Primary sources

Related context