Calculate binomial distribution probabilities step by step. Enter the number of trials, the success probability and k to get P(X = k), P(X ≤ k), the expected value, the variance and a full probability bar chart.
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The Binomial Distribution Calculator evaluates any binomial experiment. Enter the number of trials n, the probability of success p and the number of successes k to get P(X = k), the cumulative probability P(X ≤ k), the expected value E(X) = np and the variance Var(X) = np(1 − p).
Every answer comes with a worked step-by-step breakdown of the binomial experiment formula plus a bar chart of the whole probability mass function, so you can see how likely every outcome from 0 to n really is.
A binomial distribution models the number of successes in a fixed number of independent trials that each have the same probability of success. It is written X ~ B(n, p), where n is the number of trials and p the success probability.
P(X = k) = C(n, k) · p^k · (1 − p)^(n − k), where C(n, k) counts the ways to choose k successes from n trials. Adding those terms from 0 to k gives the cumulative probability P(X ≤ k).
The expected value is E(X) = np and the variance is Var(X) = np(1 − p). For ten fair coin flips that is a mean of 5 successes and a variance of 2.5.
You need a fixed number of trials, two outcomes per trial, a constant success probability and independent trials. If the probability changes between trials or trials influence each other, a binomial model is not appropriate.
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