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Binomial Probability Calculator

Find the binomial probability of k successes.

How it is calculated

The tool gives the probability of exactly k successes: P(X = k) = C(n, k) × p^k × (1 − p)^(n − k), where n is the number of independent trials, k the number of successes and p the success probability of one trial, written as a decimal between 0 and 1. The coefficient C(n, k) = n! / (k! (n − k)!) counts the orders in which the successes can occur.

n and k are rounded down to whole numbers. The result is shown as a percentage, together with the expected number of successes n × p. No result is given if k is larger than n or p is outside 0 to 1. It is a single-value probability, not a cumulative one. Large n works too, since it is computed with logarithms (n = 2000, k = 1000, p = 0.5 gives 1.78%).

Worked example

Flip a fair coin 10 times and ask for exactly 3 heads: n = 10, k = 3, p = 0.5. C(10, 3) = 120. The rest is 0.5³ × 0.5⁷ = 0.5¹⁰ = 1/1024. So P = 120 / 1024 ≈ 0.1172, which the tool shows as 11.72%. The expected number of heads is 10 × 0.5 = 5.00.

Try this example →

Common mistakes

  • Entering p as a percentage (50) instead of a decimal (0.5); the tool then returns no result.
  • Reading the answer as "3 or fewer" or "at least 3". It is the chance of exactly 3; for ranges, add the probabilities for each k.
  • Using the model when trials are not independent or p changes, such as drawing cards without putting them back.
  • Confusing the expected value (5 successes) with the most probable outcome for non-integer n × p; they can differ.

About this calculator

This calculator computes the binomial probability formula, P(X=k) = C(n,k) × pᵏ × (1−p)ⁿ⁻ᵏ, the probability of getting exactly k successes in n independent trials each with success probability p — the classic "coin flip" or "pass/fail" probability model.

It's used in statistics coursework, by quality control teams estimating defect rates in a batch, and in genetics for calculating the probability of a specific number of offspring inheriting a trait, or in polling analysis for estimating outcome probabilities from a known success rate.

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Comments (1)

  • Editorial team

    The percentage shown is for exactly k successes, not for k or more and not for k or fewer. For an at-least question, run each k from your threshold up to n and add the results, or subtract the cases below it from 100%. The mean is simply n times p. The model also assumes every trial is independent with the same p. Which at-least or at-most question would you like a ready answer for?

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