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Binomial Distribution Calculator

Enter independent trials, successes, and a fixed success probability to get exact and cumulative binomial probabilities.

Calculate a Binomial Probability

Enter trials, successes, and a fixed success probability, then choose exact, at-most, or at-least probability.

The selected probability, exact probability, cumulative probabilities, mean, and standard deviation will appear here.

How to Use Binomial Distribution Calculator

1

Enter Trials and Successes

Use whole numbers, with successes from zero through the number of trials.

2

Enter Success Probability

Provide the fixed probability for one trial as a percentage from 0% to 100%.

3

Choose the Probability Event

Select exactly x, at most x, or at least x and review all three values for comparison.

Binomial Distribution Calculator Features

  • Exact P(X = x)
  • At-most and at-least probabilities
  • Expected successes and standard deviation
  • Boundary support for 0% and 100% probability

Common Use Cases

Model repeated yes-or-no trials

Calculate the chance of a chosen number of successes when every independent trial has the same success probability.

Check cumulative homework answers

Compare exactly x, at most x, and at least x without manually adding many probability terms.

Limits and Important Notes

Result type: Exact calculation

Calculation Details

  • Convert the entered percentage to probability p from 0 to 1.
  • Evaluate the binomial probability mass function for each integer outcome needed.
  • Sum outcomes through x for at most, or from x through n for at least.

Formula

P(X=x) = C(n,x)p^x(1−p)^(n−x); mean = np; standard deviation = √[np(1−p)].

Rounding

The engine retains floating-point precision and displays probabilities with up to eight decimal percentage places.

Default Assumptions

  • Trials are independent.
  • Each trial has exactly two mutually exclusive outcomes.
  • The probability of success stays fixed for every trial.

Verified Example

Inputs: 10 trials, 3 successes, and 50% success probability.

Calculation: P(X=3) = C(10,3) × 0.5^3 × 0.5^7 = 120/1024.

Result: Exactly 3 has probability 11.71875%; at most 3 is 17.1875%; at least 3 is 94.53125%.

Sources and Version

Version: NIST binomial PMF and CDF formulas

Last verified: 2026-08-15

Update responsibility: OnlineTool.me

  • Do not use the model when trials affect each other or probability changes between trials.
  • The calculator supports up to 1,000 trials and requires a whole-number success count.
  • Very small probabilities may display as 0% at the shown precision even though the internal result is positive.

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Frequently Asked Questions