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 this 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.
Worked example
For n = 10 and p = 0.5, P(X = 5) = C(10,5) × 0.5¹⁰ = 0.24609375.
Assumptions and limits
- 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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Verification
How this calculator is checked
Trust comes from reproducible evidence, not model confidence. You can inspect the formula, assumptions, worked example, and cited sources on this page.
- Versioned calculationFormula logic is kept separate from the interface and covered by automated registry and behavior checks.
- Cited evidenceSources linked for independent checking: 1.
- Transparent AI useAI may assist drafting or adversarial review. Agreement between models is not proof, and no human expert review is claimed unless a named reviewer is shown.
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.
Last updated 2026-08-25 · binomial-pmf-v1 · content-2026-08-14 · Published by YunFanLabs