Binomial Probability Calculator

The binomial distribution counts the number of successes in n independent Bernoulli trials, each with success probability p. Jacob Bernoulli derived its algebra in Ars Conjectandi (published posthumously in 1713) and also proved the law of large numbers in the same volume. Abraham de Moivre (1733) and Pierre-Simon Laplace (1812) showed that for large n the binomial converges to a normal with mean np and variance np(1 − p), establishing the foundation of all normal-approximation inference for proportions. The probability mass function P(X = k) = C(n, k) p^k (1 − p)^(n − k) is the canonical model for count-of-success data: responders in a trial, heads in coin flips, defective items in a batch, correct answers on a multiple-choice quiz.

Picking which probability you want hinges on the question. P(X = k) gives the chance of an exact value and is the building block of every other binomial probability. P(X ≤ k) is the lower-tail CDF and answers at-most questions. P(X ≥ k) is the upper-tail survival probability and is the standard one-sided p-value when you want to know how unusual an observation is. The between form P(a ≤ X ≤ b) covers range questions like "what is the probability of 4 to 6 successes in 10 attempts." The calculator's probability tab exposes all six query types so you can answer any of these without reaching for a separate formula.

When you observe k successes in n trials and want to test whether the true success probability differs from a null value p0, the exact binomial test is the gold standard. Its two-sided version follows R's binom.test convention: sum the probability mass over every outcome with probability no greater than the observed, the "doubled-tail" method introduced by Yates and adopted by Sterne (1954). For one-sided tests the p-value is simply the upper or lower tail of the CDF. The exact test is exact in the sense that its p-value is computed directly from the binomial PMF rather than from a chi-square or normal approximation, so it remains valid for any n and p, including the small-sample, extreme-p regimes where the chi-square and Wald tests fail.

For a confidence interval on a proportion there are four standard methods, each with a different trade-off between simplicity, coverage, and bound behavior. Wald, the normal-approximation interval p̂ ± z × √(p̂(1 − p̂) / n), is the simplest and the one most introductory texts teach, but it has erratic coverage and can extend below 0 or above 1, so modern guidance (Agresti and Coull 1998, Newcombe 1998, Brown, Cai, and DasGupta 2001) is to avoid it. Wilson (1927) inverts the score test and has excellent coverage for any n and p. Agresti-Coull (1998) is the "add 2 successes and 2 failures" shortcut to Wilson and is the default in many modern style guides. Clopper-Pearson (1934) inverts the exact binomial test and guarantees ≥ 95% coverage at the cost of being slightly conservative (its intervals are wider than Wilson or Agresti-Coull). The calculator returns all four so you can pick by audience: Clopper-Pearson for regulatory submissions and guideline panels, Wilson or Agresti-Coull for journal articles, Wald only when required for backward compatibility.

Binomial results feed directly into downstream analyses. To combine many binomial outcomes into a single pooled proportion across studies, use the proportion meta-analysis calculator. To translate a binomial test statistic into a confidence interval (or vice versa) for any distribution, use the p-value to CI converter. To compare two independent proportions in a 2×2 table, use the chi-square calculator. To plan sample size for a single-proportion study, use the sample size calculator. For data extraction, exact-test reporting, and APA-formatted manuscripts from a PhD statistician, our statistical analysis service handles everything end-to-end.