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Probability · Distributions

Binomial & Normal Distribution Calculator

Calculate binomial point and cumulative probabilities or normal interval probabilities with distribution parameters and standardized bounds.

Formula shown Runs locally Reviewed Aug 11, 2026
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Equation guide

Discrete binomial trials and continuous normal probability

Binomial mass

P(X=k)=C(n,k)pᵏ(1−p)ⁿ⁻ᵏ

Binomial center

μ=npσ²=np(1−p)

Normal standardization

z=(x−μ)/σ
Probability · Distributions

Evaluate a binomial or normal model

Binomial trials are independent with constant p. The normal model is continuous and uses σ > 0.

Inputs and displayed decimal results use at most three decimal places. Full internal precision is retained during calculation.

P(X=k)=C(n,k)pᵏ(1−p)ⁿ⁻ᵏ

Choose a distribution and enter its parameters.

Using the model

Symbols, assumptions and limitations

Interpretation notes

The binomial model assumes n independent trials with constant success probability p. The normal model is continuous, requires σ>0, and uses a numerical approximation to the standard normal cumulative distribution.

Worked examples

See the method in practice

01

Binomial successes

For n=10, p=0.5, the probability of exactly 5 successes is approximately 0.246.

02

Standard normal interval

For μ=0 and σ=1, the probability between −1 and 1 is approximately 0.683.

Questions

Frequently asked

When is a binomial model appropriate?

When there is a fixed number of independent trials, two outcomes per trial, and constant success probability.

Why is normal point probability zero?

A continuous distribution assigns probability to intervals; any single exact point has probability zero.

What is a z-score?

It expresses how many standard deviations a value lies above or below the mean.

Sources & review

Equations and examples are checked against the references below. Results are educational and should be independently verified for safety-critical work.

OpenStax Introductory Statistics — The Binomial Distribution OpenStax Introductory Statistics — The Standard Normal Distribution

Written by the STEM Hub editorial team · Reviewed August 11, 2026 · Review methodology