Beta-binomial Bayesian reliability calculator

RELIABILITY · EVIDENCE & UNCERTAINTY

Bayesian reliability updating

Update a probability of success with an explicit Beta prior and Bernoulli observations. Inspect the posterior, uncertainty and predictions for future 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

Let $p$ be the probability of success on the same mission. In the $Beta (α, β)$ prior, $α$ is the success-side shape and $β$ is the failure-side shape; both must be positive. The $s$ successes and $f$ failures are conditionally independent Bernoulli observations sharing $p$ . Conjugacy keeps the posterior in the Beta family.

$p \sim Beta (α, β); L (p) \propto p^{s} (1 - p)^{f}$ $p | data \sim Beta (α + s, β + f) = Beta (a, b)$ $E [p | data] = \frac{a}{a + b}$ $[I^{- 1} (\frac{1 - γ}{2}; a, b), I^{- 1} (\frac{1 + γ}{2}; a, b)]$ $P (p \geq r | data) = 1 - I_{r} (a, b)$ $K | data \sim Beta-Binomial (m, a, b)$ $P (K = k | data) = (\binom{m}{k}) \frac{B (a + k, b + m - k)}{B (a, b)}$ $P (K = m | data) = \prod_{j = 0}^{m - 1} \frac{a + j}{a + b + j}$

$I$ is the regularized incomplete beta function; $B$ is the beta function. The interval is equal-tailed, not highest-density. It contains posterior probability mass $γ$ conditional on the chosen prior and likelihood. This credible interval has a different interpretation from frequentist confidence. Future trials are independent given $p$ , but their joint predictions are dependent after integrating uncertainty in $p$ . Thus all-success probability generally differs from the posterior mean raised to $m$ .

Worked example · invented test evidence

Observe 9 successes and 1 failure in an invented mission test. Choose a uniform $Beta (1,1)$ prior for illustration; this is not universally “uninformative” or correct.

$Beta (1,1) + (s = 9, f = 1) \to Beta (10,2)$
$E [p | data] = \frac{10}{12} = 0.83333333$
95% equal-tail credible interval = $[0.58722008, 0.97716880]$ .
$P (p \geq 0.8 | data) = 1 - (11 \times {0.8}^{10} - 10 \times {0.8}^{11}) = 0.67787745$
$P (K = 5 | data) = \frac{10}{12} \frac{11}{13} \frac{12}{14} \frac{13}{15} \frac{14}{16} = 0.45833333$
$(\frac{10}{12})^{5} = 0.40187757; E [K | data] = 5 \times \frac{10}{12} = 4.1666667$

The observed success fraction is $\frac{9}{10} = 0.9$ , whereas the posterior mean is 0.8333 because the prior contributes. With no observations, the posterior equals the prior. Repeat with alternative defensible priors to inspect sensitivity; do not count the same evidence in both the prior and likelihood.

Horizontal axis: success probability

p

from 0 to 1. Vertical axis: cumulative probability P(reliability ≤

p

). Solid teal: posterior; dashed amber: prior. The band marks the equal-tail posterior credible interval.

PosteriorPriorCredible interval

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.