Binomial Probability Calculator
Exact P(X = k), P(X ≤ k), and P(X ≥ k) for a binomial experiment with n trials, success probability p, and k successes. n is capped at 1000 on this page.
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How to evaluate a binomial PMF and CDF
- Enter trials n as an integer from 0 to 1000. Each trial is a success or failure with fixed p.
- Enter successes k as an integer from 0 through n.
- Enter success probability p between 0 and 1 inclusive.
- Read P(X = k), the left and right cumulative tails, mean np, variance, and SD. Nothing is uploaded.
Binomial distribution PMF and CDF
Exact discrete binomial probabilities for n, k, p
| PMF | P(X=k) = C(n,k) p^k (1−p)^(n−k) |
|---|---|
| CDF rows | P(X ≤ k) and P(X ≥ k) |
| Moments | Mean np, variance np(1−p), SD |
| n limit | Integer trials from 0 to 1000 |
| Defaults | n = 10, k = 3, p = 0.5 |
| Engine path | Log-choose PMF sum for the CDF |
What a binomial model assumes
A binomial count is the number of successes in n independent Bernoulli trials that share one success chance p. This page returns the exact point mass at k, the cumulative probability through k, and the upper tail from k upward. It also lists mean, variance, and standard deviation. It is not a normal approximation tool and not a negative-binomial waiting-time calculator.
How PMF and CDF are built
P(X = k) uses C(n, k) p^k (1 − p)^(n − k). The engine evaluates the choose term in log space, then sums PMF values from 0 through k for the left CDF. The right cumulative is one minus the left CDF plus the exact mass at k, so k is counted once. Edge cases p = 0 and p = 1 collapse to deterministic 0 or n successes.
Bernoulli trials, choose coefficients, and related jobs
Entities: Bernoulli trial, binomial random variable, combination coefficient C(n, k), PMF, and CDF. Two-event and/or helpers without a trial count live on the probability calculator. The choose count alone is combinations. Ordered selections without replacement are permutations, which do not define this PMF by themselves.
Use cases
Coin-flip homework with fair p = 0.5. Quality checks that model defect chance per item. Classroom demos that compare exact P(X = 3) for n = 10 against a rough normal guess. Not a clinical trial power calculator and not a beta-binomial hierarchical model.
Default n = 10, k = 3, p = 0.5
With ten fair trials, the PMF at exactly three successes is C(10, 3) / 1024 = 120 / 1024. The page also stacks every mass from 0 through 3 for P(X ≤ 3) and reports the matching upper tail, plus mean 5 and variance 2.5.
Limits
n cannot exceed 1000. Trials must be independent with constant p. No continuity correction table. No Poisson limit switch. Educational floating-point arithmetic in the browser. CZNull does not receive the numbers.