Binomial Probability Calculator

Calculate exact, cumulative or tail probability for a fixed number of independent Bernoulli trials

Your result will appear here.

About the Binomial Probability Calculator

Binomial Probability Calculator finds the probability of a specified number of successes in a fixed number of independent trials when every trial has the same success probability. It is useful for repeated yes/no outcomes.

The binomial model is only appropriate when trial conditions remain consistent. Changing success probability or dependence between trials requires another model.

How to Use the Binomial Probability Calculator

  1. Enter the number of trials n.
  2. Enter the success probability p for one trial.
  3. Enter the desired number of successes k or a supported cumulative range.
  4. Calculate the probability and verify that the independence and constant-probability assumptions fit the situation.

How the Calculation Works

For exactly k successes, P(X=k) = C(n,k)p^k(1−p)^(n−k), where n is the number of trials, k is the number of successes and p is the constant success probability for each trial.

Cumulative probabilities are found by summing the exact binomial probabilities across the requested values of k. The combination term C(n,k) counts the distinct arrangements that contain exactly k successes.

Worked example

Binomial Probability Calculator Worked Example

Exactly 3 heads in 5 fair coin flips

P = C(5,3)×0.5³×0.5² = 10×0.5⁵ = 10/32 = 0.3125, or 31.25%.

Result guide

Understanding Your Binomial Probability Calculator Result

The result is the model probability for the defined repeated-trial process. It does not mean that exact proportion must appear in one small batch of trials. When n is large and p is very small or very large, direct arithmetic can become numerically delicate. Reliable implementations avoid unnecessary factorial overflow and use stable probability calculations.

Before you rely on the answer

Assumptions and Limitations

  • Number of trials is a non-negative integer.
  • Each trial has the same success probability.
  • Trials are independent under the model.

Practical tip

Using This Result in Practice

Before using a binomial formula, ask whether one trial changes the next. Sampling without replacement from a small population, for example, can violate independence.

Related calculations

Useful Calculations to Continue With

If the problem does not have a repeated identical-trial structure, use the general Probability Calculator.

For large-sample approximations under appropriate conditions, the Normal Distribution Calculator can support comparison with a normal model.

Common questions

Binomial Probability Calculator FAQ

What makes a problem binomial?

A fixed number of independent trials, two outcome categories and a constant success probability.

Can p change between trials?

Not in the standard binomial model.

What does C(n,k) do?

It counts how many different trial arrangements contain exactly k successes.