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Calculator Answers

Binomial Probability Calculator

Blake Boege
Written by Blake Boege · Founder, Calculator Answers

A binomial probability calculator is a statistical tool that computes the probability of achieving a specific number of successes in a fixed number of independent trials. It uses the binomial distribution formula, which factors in the number of trials, the probability of success in a single trial, and the target number of successes. The calculator outputs the exact probability, cumulative probabilities (such as at least or at most), and the expected mean and variance. Analysts and researchers use this tool to model binary outcomes, such as coin tosses or quality control checks.

Enter trials n, successes k, and per-trial probability p. Pick exactly-k, at-most-k, or at-least-k and the calculator returns the binomial probability as a decimal and a percent.

Quick Answer

Calculate binomial distribution probabilities for independent events. Enter trials, success rates, and successes to find exact and cumulative probabilities.

Probability mode

Whole number. Capped at 1,000 to keep loops responsive. · e.g. 10

Whole number. Cannot exceed n. · e.g. 3

%

Per-trial success probability, between 0 and 100. · e.g. 50

Formula

P(X = k) = C(n, k) × p^k × (1 − p)^(n − k)

The binomial distribution counts successes in n independent trials with the same success probability p on each trial. Each trial outcome must be binary (success or failure).

Binomial probability

Exactly k · n = 10, k = 3, p = 50%

0.117188

11.7188% chance

P(X = k) (exact)0.117188
As percent11.7188%
Trials × p (mean)5
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Examples

n = 10, k = 3, p = 50%, exactly

P(X = 3) ≈ 0.117188

n = 20, k = 10, p = 50%, exactly

P(X = 10) ≈ 0.176197

n = 100, k ≥ 60, p = 50%, at least

P(X ≥ 60) ≈ 0.028444

How it works

Compute the binomial coefficient C(n, k) using the multiplicative form, then multiply by p^k for the success probability and (1 − p)^(n − k) for the failure probability. For cumulative modes, sum the per-k probabilities from 0 to k (at most) or k to n (at least).

P(X = k) · C(n, k) × p^k × (1 − p)^(n − k)

P(X ≤ k) · Σ P(X = i) for i = 0..k

P(X ≥ k) · Σ P(X = i) for i = k..n

Mean of a binomial distribution: n × p. Variance: n × p × (1 − p).

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Binomial Distribution CalculatorCompute individual and cumulative binomial probabilities for any number of trials and success rates.

Frequently asked questions

The binomial distribution counts the number of successes in n independent trials, where each trial has the same probability p of success and only two possible outcomes (success or failure). The binomial probability P(X = k) is the chance of getting exactly k successes in those n trials.

P(X = k) = C(n, k) × p^k × (1 − p)^(n − k). C(n, k) is the number of ways to choose which k of the n trials are successes (the binomial coefficient). p^k is the probability that those specific trials are all successes. (1 − p)^(n − k) is the probability that the other trials are all failures. Multiplying combines them into the probability of any single arrangement plus the count of arrangements.

The binomial model requires four conditions. (1) A fixed number of trials n. (2) Each trial is independent of the others. (3) Each trial has the same probability p of success. (4) Each trial has exactly two outcomes (success or failure). Sampling without replacement from a small population violates independence; flipping a fair coin many times satisfies all four conditions.

P(X ≤ k) is the probability of getting k or fewer successes (the cumulative probability up through k). P(X ≥ k) is the probability of getting k or more successes (the upper-tail cumulative). They are complementary: P(X ≥ k) = 1 − P(X ≤ k − 1). Use 'at most' when asking 'no more than this many', 'at least' when asking 'this many or more', and 'exactly' for a single specific count.

The mean (expected value) of a binomial distribution is n × p. With 100 trials at p = 0.3, you expect 30 successes on average. The variance is n × p × (1 − p), and the standard deviation is the square root of the variance. These help you judge whether a specific result is unusual.

For n above about 30, the normal distribution is often a reasonable approximation, especially when both np and n(1 − p) are at least 5. Use the normal-distribution calculator with mean = np and standard deviation = √(np(1 − p)). For exact binomial probabilities on large n, statistical software handles arbitrary sizes; this calculator caps n at 1,000 to keep the in-browser loop responsive.