Binomial Distribution Calculator

Calculate PMF, CDF, mean and variance for the binomial distribution B(n,p).

About Binomial Distribution Calculator

The binomial distribution B(n, p) models the number of successes in n independent Bernoulli trials, each with success probability p. It returns the exact probability of k successes (PMF) and the cumulative probability up to k (CDF).

Formula

P(X=k) = C(n,k) × p^k × (1−p)^(n−k)

n: Number of trials
k: Number of successes
p: Probability of success per trial

Worked example

With the default values below, Binomial Distribution Calculator returns P(X = k) of 0.24609375. Change any input above and every figure here is recalculated in your browser.

Inputs used

Number of Trials (n)accepts 1–1,000: 10
Success Probability (p)accepts 0–1: 0.5
Number of Successes (k)minimum 0: 5

Results

P(X = k): 0.24609375
P(X ≤ k): 0.62304688
Mean (np): 5.0000
Variance (np(1−p)): 2.5000

Step by step

1Values used
2Formula applied
3P(X = k)= 0.24609375
4P(X ≤ k)= 0.62304688
5Mean (np)= 5.0000
6Variance (np(1−p))= 2.5000

Assumptions and limits

The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

When is the binomial distribution appropriate?

When you have a fixed number of independent trials, each with the same probability of success, and you want the probability of a specific number of successes.

What is the relationship to the normal distribution?

For large n, the binomial distribution can be approximated by a normal distribution with mean np and variance np(1−p).