Binomial Distribution Calculator

In short The binomial distribution answers "how likely is exactly k successes in n independent tries". Five heads in ten coin flips is 24.61% — which is 252 of the 1,024 possible sequences, and notably NOT the near-certainty people expect from the most likely single outcome. That is worth sitting with: the most probable result of ten flips still happens under a quarter of the time. The formula is C(n,k) × pᵏ × (1−p)ⁿ⁻ᵏ: the number of orderings, times the chance of each. Four assumptions have to hold — a fixed number of trials, two outcomes each, the same probability every time, and independence — and it is the last two that real data breaks. Drawing cards without putting them back changes the probability as you go, which makes it a hypergeometric problem instead, and the card page handles that. The most useful form in practice is "at least one", which is 1 − (1−p)ⁿ: the chance of at least one six in four rolls of a die is 51.77%, just over half, which is the bet that reputedly started the mathematical study of probability. This page computes the exact sum in logarithms rather than offering a normal approximation. Approximations existed because the exact calculation used to be expensive; it is not expensive now, so there is nothing here to approximate away.