StatReasonWhat this answers
This calculator answers "what is the probability of getting k or fewer successes" or "k or more successes" across a fixed number of independent trials, each with the same success probability. This differs from the exact single-count question the Binomial Probability Calculator answers, since it sums across a whole range of possible outcomes rather than one specific count.
How it is calculated
Rather than summing individual binomial probabilities term by term (which can lose precision for large n), this calculator computes the cumulative probability directly via its relationship to the regularized incomplete beta function, the same numerically stable approach used throughout this project's distribution engines.
Worked example
For n=3 trials with p=.5, the probability of at least 2 successes works out to exactly .5: summing the probabilities of exactly 2 successes (.375) and exactly 3 successes (.125) gives .5, matching the direct cumulative calculation here.
Assumption audit
Common mistakes
A common mistake is confusing "at least k" with "more than k"; this calculator's "at least" option includes k itself, so double-check whether your intended threshold is inclusive or exclusive before comparing to a published table. Another is entering a threshold k greater than n, which is an impossible event and correctly throws an error rather than silently returning a meaningless number. A third is treating a small cumulative probability as automatically "significant" in a hypothesis-testing sense; this calculator reports a probability under an assumed model, not a formal hypothesis test with a stated null and alternative.
Limitations
This calculator assumes a fixed number of trials, a constant success probability, and independence between trials, exactly the same assumptions behind any binomial model; it does not check whether your actual process satisfies them.