Binomial Distribution Calculator
Model a fixed number of independent Bernoulli trials with one constant success probability. Set equal minimum and maximum counts for one exact count, or enter different bounds for an inclusive range.
How to use the Binomial Distribution Calculator
Enter trial count, success probability, and inclusive minimum and maximum success counts.
Review independence, fixed trial count, two-outcome definition, constant probability, and whether sampling without replacement violates the model.
Read inclusive range probability, probability percentage, mean, variance, and standard deviation.
Formula
For X ~ Binomial(n,p), P(X=k) = C(n,k)p^k(1−p)^(n−k). Mean = np and standard deviation = √(np(1−p)); cumulative probabilities sum the relevant integer k values.
Example
For n = 10 and p = 0.5, P(X = 5) = C(10,5) × 0.5¹⁰ = 0.24609375.
Assumptions and limitations
- Trials are independent, each has the same success probability, and the random variable counts successes across a fixed number of trials.
- Binomial Distribution Calculator treats the entered values as estimates and does not infer missing project, account, or personal details.
- Calculations retain working precision except where the cited method requires an intermediate rounding step; displayed values are then formatted separately. Source rules and dated rates must be rechecked when they change.
Sources
- OpenStax: Introductory Statistics 2eMean, deviation, probability, and binomial-distribution formulas. Reviewed 2026-08-14.
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Frequently asked questions
How does the Binomial Distribution Calculator work?
For X ~ Binomial(n,p), P(X=k) = C(n,k)p^k(1−p)^(n−k). Mean = np and standard deviation = √(np(1−p)); cumulative probabilities sum the relevant integer k values.
What information should I enter in the Binomial Distribution Calculator?
Enter trial count, success probability, and inclusive minimum and maximum success counts, then verify independence, fixed trial count, two-outcome definition, constant probability, and whether sampling without replacement violates the model.
What should I verify before using this Binomial Distribution Calculator result?
Trials are independent, each has the same success probability, and the random variable counts successes across a fixed number of trials.
Reviewed 2026-08-14 · binomial-pmf-v1 · content-2026-08-14 · Trusty Calculator Editorial Team