Binomial Probability Calculator
For n independent trials that each succeed with probability p, this calculator gives the exact probability of exactly k successes — P(X = k) — together with all four cumulative variants: at most k, at least k, fewer than k and more than k. The distribution's mean (np) and standard deviation √(np(1−p)) are included. Probabilities are computed in log-space with the same accuracy as R's dbinom/pbinom, so large n does not overflow. Typical uses: quality control (defect counts), A/B sanity checks, genetics problems and exam questions of the "probability of at least 8 heads in 10 tosses" type.
Frequently asked questions
What is the binomial probability formula?
P(X = k) = C(n, k) · pᵏ · (1−p)ⁿ⁻ᵏ, where C(n, k) is the number of ways to choose which k of the n trials succeed. It applies when trials are independent and the success probability p is the same on every trial.
What is the difference between P(X = k), P(X ≤ k) and P(X ≥ k)?
P(X = k) is the chance of exactly k successes. P(X ≤ k) adds up everything from 0 to k ("at most"), while P(X ≥ k) sums from k to n ("at least"). Exam questions usually hinge on this wording — "at least" and "more than" differ by exactly P(X = k).
When can the normal approximation be used instead?
The classic rule is np ≥ 5 and n(1−p) ≥ 5 (stricter: ≥ 10), using mean np and SD √(np(1−p)) with a 0.5 continuity correction. This calculator computes the exact sum, so you do not need the approximation — but it is what a z-based hand calculation would use.