Binomial Probability Calculator
Find the probability of k successes in n independent trials.
Results update as you type.
About this calculator
The binomial distribution models the number of successes in a fixed number of independent trials, each with the same success probability. This calculator returns the probability of exactly k successes, P(X = k) = C(n, k)·pᵏ·(1 − p)ⁿ⁻ᵏ, along with the cumulative probabilities of at most k and at least k successes, and the distribution’s mean (n·p) and standard deviation.
Frequently asked questions
When can I use the binomial distribution?
Use it when there is a fixed number of independent trials, each trial has just two outcomes (success or failure), and the success probability is the same every time — like flipping a coin 10 times or counting defective items in a batch.
What is the difference between P(X = k) and P(X ≤ k)?
P(X = k) is the chance of exactly k successes, while P(X ≤ k) adds up the chances of 0, 1, … up to k successes. The cumulative version answers questions like "what is the probability of at most 3 successes?"
Results are estimates for general guidance only, not financial, medical or tax advice.