Binomial Cumulative Probability Calculator

About this calculator

For a series of n independent trials that each succeed with probability p (like flipping a biased coin n times), the binomial cumulative probability P(X ≤ k) gives the chance of getting at most k successes in total. It's calculated by summing the individual binomial probabilities C(n, i) × p^i × (1−p)^(n−i) for every count i from 0 up to k, where C(n, i) is the number of ways to choose which i trials succeeded.

Cumulative binomial probabilities are the standard way to answer “at least” or “at most” questions in quality control (the probability that no more than k defective items appear in a batch), clinical trials and medical statistics (the chance that at most k patients respond to a treatment), and everyday probability problems like the odds of winning at least a certain number of games or coin flips out of a fixed number of attempts, where the distribution's mean is simply n × p.

Enter the number of trials n, the success probability p, and a count k, and this calculator returns the cumulative probability P(X ≤ k), the exact probability at k, and the mean.