Binomial Distribution Calculator
The probability of getting exactly, at most, or at least a given number of successes in a fixed number of independent trials.
The binomial distribution counts successes in n independent trials that each succeed with the same probability p — coin flips, free throws, defective parts.
How the binomial distribution calculator works
The binomial distribution counts successes in n independent trials that each succeed with the same probability p — coin flips, free throws, defective parts. The probability of exactly k successes is C(n,k) × pᵏ × (1−p)ⁿ⁻ᵏ: the number of ways to arrange k successes among n trials, multiplied by the probability of any one such arrangement.
The two conditions matter. Trials must be independent, so drawing cards without replacement is not binomial, and p must be the same each time, so a shooter who tires is not either.
Formula: P(X = k) = C(n,k) pᵏ (1−p)ⁿ⁻ᵏ
Worked examples
| Inputs | P(X = k) | Note |
|---|---|---|
| 6 heads in 10 fair coin flips | 20.5078% | 20.51% |
| Exactly 5 heads in 10 | 24.6094% | 24.61% — the most likely single outcome |
| No sixes in 6 dice rolls | 33.4898% | 33.5% |
FAQFrequently asked questions
When is a distribution binomial?
When there is a fixed number of independent trials, each with the same probability of success, and you are counting successes.
What is the probability of 6 heads in 10 flips?
20.51%. Five heads is the single most likely outcome at 24.61%, though it is still less likely than not.
Why is C(n,k) in the formula?
Because there are many orders in which k successes can occur among n trials, and each is equally likely.
When is it not binomial?
When trials are not independent — drawing cards without replacement — or the probability changes between trials.
What is the mean and spread?
The mean is n × p and the standard deviation √(n p (1−p)).
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- Common Core State Standards — Mathematics — the terms and methods taught in US schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.