Binomial Distribution Calculator

The binomial distribution models the number of successes in a fixed number of independent trials. Every trial must have two relevant outcomes, commonly called success and failure, and the probability of success must remain constant from trial to trial. The parameters are n, the number of trials, and p, the probability of success on one trial. The random variable counts how many successes occur, so its possible values are the whole numbers from zero through n. The probability of exactly k successes is the binomial coefficient n choose k multiplied by p raised to k and one minus p raised to n minus k. The coefficient counts the different orders in which k successes can occur. This calculator evaluates that probability along with two cumulative results. At most k adds probabilities from zero through k. At least k adds probabilities from k through n, calculated here using the complement of fewer than k successes. These cumulative probabilities are useful for threshold questions that exact probability alone cannot answer. The distribution's mean is n times p, representing the long-run average success count over many repetitions of the entire experiment. Its variance is n times p times one minus p, and the standard deviation would be the square root of that variance. The mean does not need to be a possible whole-number outcome; it describes the center of the distribution rather than a guaranteed count. Spread is largest near p equal to one-half and becomes smaller as p approaches zero or one. Common applications include coin flips, manufacturing defect counts, survey responses, conversions, pass-fail tests, and treatment outcomes. Before using the model, check its assumptions carefully. Trials should be independent, the number of trials should be fixed in advance, and each trial should use the same success probability. Sampling without replacement from a small population may violate independence and is often better modeled by the hypergeometric distribution. Events with changing probabilities also require another model. Enter p as a decimal between zero and one, such as 0.25 for 25%. Use exact probability for one count, at-most probability for upper-bound questions, and at-least probability for meeting or exceeding a target.