How this calculation works
A run of successful trials does not prove perfect reliability. Under an independent binomial mission model, the number of zero-failure trials needed depends on both the target success probability and the confidence requirement. This page plans that number and evaluates the one-sided reliability lower bound for an entered all-successful test count. It keeps mission success separate from time-based failure-rate inference.
Required n = ceiling[ln(1−confidence)/ln(Rtarget)]; zero-failure lower bound = (1−confidence)^(1/n)A worked example
For target reliability 0.9 and confidence 0.95, ln(0.05)/ln(0.9) is about 28.433. The required whole number of independent successful missions is twenty-nine. At twenty-eight successes the target’s all-success probability is still above 0.05; at twenty-nine it is below. This neighboring-integer check verifies why the ceiling matters.
Good to know
Independent identical mission trials and a binomial success probability are assumed. Shared environments, repeated dependent trials and unrepresentative test conditions can overstate effective evidence. Any observed failure requires a different interval or test-planning method. This calculation does not certify a safety level, convert accelerated exposure into field reliability or guarantee future mission success.
Reference: NIST — binomial confidence intervals (opens in a new tab) ↗
What this calculation answers
A run of successful trials does not prove perfect reliability. Under an independent binomial mission model, the number of zero-failure trials needed depends on both the target success probability and the confidence requirement.
This page plans that number and evaluates the one-sided reliability lower bound for an entered all-successful test count. It keeps mission success separate from time-based failure-rate inference.
The Failure count exposure rate and Poisson bounds calculator evaluates event rate per exposure. Zero-failure reliability demonstration instead assumes every recorded trial completed the full mission.
Use the right input conventions
Both reliability and confidence use fractions between zero and one. Every completed trial is assumed to represent the same complete mission with an observed success. The page has no failure-count field because this model specifically assumes zero failures.
Censored operating hours are not interchangeable with completed successful missions. If a unit has not finished the mission, its evidence needs a model that represents that partial exposure.
Interpret the result
The required count is the smallest integer making the probability of all successes at the target reliability no greater than one minus confidence. Rounding down can therefore invalidate the planned inequality.
The lower bound is not the observed fraction, which equals one for any zero-failure dataset. It expresses what the finite all-successful sample supports under the specified one-sided procedure.
Work through the example
For target reliability 0.9 and confidence 0.95, ln(0.05)/ln(0.9) is about 28.433. The required whole number of independent successful missions is twenty-nine.
At twenty-eight successes the target’s all-success probability is still above 0.05; at twenty-nine it is below. This neighboring-integer check verifies why the ceiling matters.
Where the model stops
Independent identical mission trials and a binomial success probability are assumed. Shared environments, repeated dependent trials and unrepresentative test conditions can overstate effective evidence.
Any observed failure requires a different interval or test-planning method. This calculation does not certify a safety level, convert accelerated exposure into field reliability or guarantee future mission success.
A few questions, answered
Can I use this after observing one failure?
No. The zero-failure formula must be replaced by an inference method that includes the failure count.
What changes in the comparison?
Raise target reliability from ninety to ninety-nine percent at unchanged confidence.