Binomial Distribution Calculator

Exact binomial probability P(X = k) plus cumulative P(X ≤ k), P(X ≥ k) and tails, with mean, variance and SD for n trials at probability p.

The Binomial Distribution Calculator computes P(X = k) = C(n, k)·pᵏ·(1 − p)ⁿ⁻ᵏ for k successes in n independent trials, along with the cumulative and tail probabilities and the distribution's mean np, variance np(1 − p) and standard deviation.

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How to use it

Enter n, k and p.
Read exact and cumulative probabilities.

What you enter

n, k, p

What you get

Exact and cumulative probabilities
Mean, variance, SD

Formula

P(X = k) = n! / [k!(n − k)!] · pᵏ · (1 − p)ⁿ⁻ᵏ

Each of the C(n, k) arrangements of k successes among n trials has probability pᵏ(1 − p)ⁿ⁻ᵏ. Cumulative probabilities sum the exact terms.

Worked example

Input: n = 20, k = 0, p = 0.1

Result: P(X = 0) = 0.1216

A 12 % chance of zero positives among 20 samples when the true prevalence is 10 %.

Questions

How do I get the probability of at least k successes?

Read P(X ≥ k) directly; it equals 1 − P(X ≤ k − 1).

Limitations

Requires independent trials with constant p; for sampling without replacement from a small population use the hypergeometric distribution.

References

Ross SM (2014). A First Course in Probability, 9th ed. Pearson.

More Statistics

All free tools · By Dr. Zubair Khalid, DVM, MS, PhD.