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) |
|---|
P(X=k) = C(n,k) ⋅ p² ⋅ (1−p)^(n−k). This gives the probability of exactly k successes (or defects) in n independent trials when each trial has probability p of success. The binomial requires: fixed n, independent trials, constant p, binary outcomes.
P(X ≤ k) is the sum of all PMF values from 0 to k — the probability of at most k successes. P(X ≥ k) = 1 − P(X ≤ k−1). These are essential for acceptance sampling decisions: "what is the probability of accepting a lot with c or fewer defects?"
Attribute sampling uses the binomial to evaluate acceptance plans. If you inspect n items from a lot with defect rate p, the probability of finding k or fewer defects is P(X ≤ k). Setting k=0 gives the c=0 plan probability. Compare this to the producer's risk (α) and consumer's risk (β) to evaluate plan quality.