Binomial Probability Calculator

The binomial distribution models the number of successes in a fixed number of independent trials when each trial has the same probability of success. It applies to settings like coin flips, pass-fail tests, conversions, defect counts, and yes-no survey responses when the assumptions are reasonable.

The two parameters are n, the number of trials, and p, the probability of success on each trial. The result is a probability distribution over possible success counts from 0 through n.

The probability of exactly k successes combines two ideas: the probability of one specific sequence with k successes and n-k failures, and the number of different sequences with that many successes. The combination term counts those arrangements.

This is why the binomial formula includes n choose k. A sequence like success-success-failure may have the same probability as success-failure-success, and all valid arrangements must be counted.

The expected number of successes is n times p. The standard deviation is the square root of n p (1-p). As n grows, the distribution becomes more concentrated in relative terms around its expected proportion, though the absolute count spread can still grow.

When p is near 0.5, variability is highest. When p is near 0 or 1, outcomes are more predictable because most trials tend to have the same result.

The binomial model assumes fixed trials, independent outcomes, constant probability, and two possible result categories. Real data may violate these assumptions through changing user mix, learning effects, clustering, repeated users, or hidden subgroups.

For large n, normal approximations may work well under suitable conditions. For rare events with large n and small p, a Poisson approximation may be useful. The right model depends on how the trials are generated, not only on the count format.