Binomial Distribution Calculator

It models the number of successes in a fixed number of independent trials where each trial has the same probability of success — like counting heads in 10 coin flips, or defective items in a batch. The distribution is defined by n (number of trials) and p (success probability per trial), and this calculator also reports the mean np and variance np(1−p).

P(X = k) is the probability of exactly k successes, while P(X ≤ k) is the probability of k or fewer successes. For example, in 10 coin flips the chance of exactly 5 heads is 24.6%, but the chance of 5 heads or fewer is 62.3%. The calculator shows both, plus P(X ≥ k) which is 1 − P(X ≤ k−1).

Yes. Any repeated independent trial with two outcomes fits the binomial model. Rolling a specific face on a die has p = 1/6, a yes/no survey response has p equal to the expected yes rate, and a quality check counting defects uses p as the defect rate. Just set n as the number of repetitions and k as the count you are interested in.

Enter the probability as a whole percentage — for example 50 for a fair coin or 30 for a 30% defect rate. The calculator converts it to a fraction automatically. If you enter a probability outside 0–100%, it is clamped to the nearest valid value so the results always stay meaningful.

For large n (roughly above 20 with p near 0.5), the binomial distribution looks almost identical to a normal curve, so the normal distribution calculator gives a close match. This calculator sums exact probabilities, so it stays exact for n up to about 2000; beyond that it automatically falls back to a normal approximation with continuity correction.