Calculate binomial probability, the probability of exactly k successes (and the cumulative probability of k or fewer) across n independent trials, each with the same probability of success.
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For n independent trials, each with the same probability p of success, the probability of exactly k successes is P(X=k) = C(n,k) × p^k × (1−p)^(n−k), where C(n,k) is the combination 'n choose k'. This calculator also sums P(X=i) for every i from 0 to k to give the cumulative probability P(X≤k), the chance of getting k or fewer successes.
The binomial distribution applies when you have a fixed number of independent trials, each with only two possible outcomes (success or failure), and the same probability of success on every trial, classic examples include counting heads in a series of coin flips, or counting defective items in a batch where each item independently has the same defect probability.
Exact probability, P(X=k), is the chance of getting precisely k successes. Cumulative probability, P(X≤k), is the chance of getting k or fewer successes, the sum of the exact probabilities for every outcome from 0 up to k.
The mean (n × p) is the expected number of successes on average across many repetitions. The standard deviation measures how much the actual number of successes typically varies from that average, a larger standard deviation means more variability from trial set to trial set.
No, the binomial distribution specifically assumes the same success probability on every trial. If probabilities vary trial to trial, a different (non-binomial) model is needed.
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