Calculate exact and cumulative binomial probabilities for pass/fail processes — with a full probability mass function chart, distribution table, and Six Sigma acceptance sampling context.
Define the distribution, then choose the probability type to calculate.
Enter the number of trials, probability of success, and your target number of successes — then choose exact, cumulative, or range probability.
| k | P(X=k) | P(X≤k) | P(X≥k) |
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5% defect rate, batch of 20, inspector checks them all. Enter the lab and find the probability of exactly 2 defectives appearing.
Use the calculator above to compute exact and cumulative binomial probabilities for pass/fail processes, with a full probability mass function chart, a distribution table and an acceptance-sampling context view. The binomial distribution is the right tool whenever each trial has two outcomes and the probability of success stays constant.
The binomial distribution describes the probability of obtaining exactly k successes in n independent trials, where each trial has the same probability p of success. It is the foundation of attribute-sample acceptance plans, defect-rate inference and many quality control techniques.
P(X = k) = C(n,k) × pᵏ × (1−p)^(n−k), where C(n,k) is the number of ways to choose k from n. The cumulative version sums probabilities for X ≤ k or X ≥ k. The calculator handles both, plus the practical question "what is the chance of zero defects given this defect rate?".
A supplier claims a 2% defect rate. You inspect 100 units. The probability of finding zero defects: P(X = 0) = C(100,0) × 0.02⁰ × 0.98¹⁰⁰ ≈ 0.133, or 13%. There is an 87% chance of seeing at least one defect — useful context for acceptance decisions.
If the actual rate is 5%, P(X = 0) drops to 0.6%. A zero-defects result from 100 units is very strong evidence the rate is well below 5% — but not below 1%. The binomial lets you calculate exactly what the data says.
Binomial probabilities turn defect counts into defensible statements about underlying defect rates. They make acceptance sampling work rigorously rather than by gut feel.
Use the binomial for any pass/fail data with a constant success probability — supplier audits, sample acceptance, A/B test results, machine reliability and survey response.
The binomial sits behind attribute sample size, p-chart and np-chart control charts, and defect-rate inference. It is also the foundation for the rare-event Poisson approximation.
Use the binomial output to set acceptance criteria, validate supplier defect-rate claims, or design pass/fail audits with the right power. Pair with attribute sample size calculations.
Combine the binomial with attribute sample size, p-charts and DPMO for a complete pass/fail inferential workflow.
The probability distribution of the number of successes in n independent trials, where each trial has the same probability p of success.
When trials are independent, each has two outcomes, and the probability of success stays constant. Examples: supplier defect rates, A/B test conversions, pass/fail audits.
Exact: P(X = k) — the chance of exactly k successes. Cumulative: P(X ≤ k) or P(X ≥ k) — the chance of at most or at least k. Acceptance plans usually need cumulative.
When trials are not independent, when p changes during sampling (use hypergeometric), or when n is large and p is small (the Poisson approximation may be simpler).
Acceptance plans use cumulative binomial probabilities to set the maximum allowable defects in a sample at a chosen confidence level. Zero-acceptance plans (c = 0) are a special case.
Want to understand how binomial distributions apply to pass/fail and defect analysis? The Green Belt covers this in full.
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