Binomial Probability Calculator

The binomial distribution gives the probability of getting exactly k successes in n independent trials that each succeed with the same probability p: P(X = k) = C(n,k) × pᵏ × (1−p)ⁿ⁻ᵏ, where C(n,k) is the number of ways to choose which k of the n trials succeed. This calculator evaluates that formula in log space (summing log-factorials rather than multiplying raw factorials) so it stays numerically accurate even for a large number of trials, rather than overflowing.

"At least k" and "at most k" sum that exact-probability formula across every outcome from k to n, or from 0 to k respectively — the standard cumulative binomial probability. The expected number of successes is n × p, and the standard deviation is √(n × p × (1−p)), both closed-form properties of the distribution rather than separate calculations.

This applies to any fixed-probability repeated trial with only two outcomes per trial — coin flips, pass/fail quality checks, free-throw attempts — as long as each trial is independent and shares the same success probability. It's a different question from a dice-sum calculator, which convolves multiple dice faces rather than counting successes across repeated identical trials.