Binomial Probability Calculator

P(X = k) = C(n,k) p^k (1−p)^(n−k). Probability of exactly k successes in n trials.

Result

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How to use this calculator

Enter n trials, k successes, p success probability.
Read P(X = k), cumulative, and moments.

About this calculator

Binomial distribution models the count of successes in n independent trials, each with probability p. Mean = np; variance = np(1−p). For large n, approximated by normal N(np, np(1−p)). Examples: number of heads in 10 coin flips (n=10, p=0.5), number of defective items, number of yes-responses in a survey. Foundational for hypothesis testing of proportions.

Frequently asked

When is binomial valid?+

Fixed n trials, each with the same p, independent of each other. Two outcomes per trial.

Compare P(X = k) and P(X ≤ k)?+

P(X = k): exact. P(X ≤ k): cumulative (k or fewer). Use ≤ for one-tailed hypothesis tests.

Normal approximation?+

Valid when np ≥ 5 and n(1-p) ≥ 5. Use Z = (X − np) / √(np(1-p)).

Bernoulli?+

Single trial (n = 1). Bernoulli(p) is the building block of Binomial(n, p).

Hypergeometric?+

Sampling without replacement (e.g. card draws). Different formula. Reduces to binomial for large populations.

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Reference Text vs PDF Redline

Paste a reference text (e.g. the canonical doc-template body) and upload a signed PDF — get a redline showing exactly what was changed.