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Mihsaba›Binomial Distribution Calculator

Binomial Distribution Calculator

Calculate exact, at-most and at-least binomial probabilities for k successes in n independent trials with constant success probability p, plus mean, variance and standard deviation.

How to use

The binomial distribution models the number of successes in a fixed number of independent trials when the success probability stays constant.

  1. Enter integer trials n from 0 to 1000.
  2. Enter success probability p from 0 to 100%, then successes k.
  3. Calculate P(X=k), P(X≤k), P(X≥k), mean and variance.

How it is calculated

P(X=k)=C(n,k)pᵏ(1−p)ⁿ⁻ᵏ; μ=np; σ²=np(1−p)

Example

Example: n=10, p=50%, k=5 gives an exact-five probability of about 24.61%.

Important notes

The model assumes independent trials, constant p and two outcomes per trial. Sampling without replacement from a small population may call for a hypergeometric model instead.

Frequently asked questions

Exact versus at most?

Exact k is one outcome count; at most k sums outcomes from 0 through k.

Can p be 0% or 100%?

Yes. The distribution becomes deterministic and the calculator handles both endpoints directly.

Is this calculator free?

Yes. It is free to use and requires no account.