P(X≤k) — at most k — is the probability of k or fewer successes. P(X≥k) — at least k — is the probability of k or more successes. These are cumulative probabilities, summing the exact-k formula across the relevant range, and are usually more useful in practice than a single exact-k probability.
What conditions does my situation need to meet to use this calculator?+−
Four conditions: a fixed number of trials (n), each trial has only two possible outcomes, trials are independent of each other, and the probability of success (p) stays the same on every trial. See our binomial distribution guide for details on each condition.
Is there a limit on how large n can be?+−
This calculator supports n up to 1000, computed with a numerically stable log-gamma method to avoid overflow. For very large n, consider whether the normal approximation (covered in our binomial distribution article) is accurate enough for your purposes and faster to reason about.
What do the mean and standard deviation tell me?+−
The mean (μ = np) is the expected number of successes across many repetitions. The standard deviation (σ = √[np(1−p)]) measures how much the actual count of successes typically varies from that expected value from one set of n trials to the next.