What Is a Binomial Distribution?

A binomial distribution calculator computes the probability of getting exactly k successes in n independent trials, each with a fixed success probability p. This is one of the most widely used discrete distributions in statistics — it powers quality control, A/B testing, and any analysis of yes/no outcomes repeated a fixed number of times.

The Binomial Formula

P(X = k) = C(n, k) × p^k × (1−p)^(n−k). The combination term C(n, k) counts how many ways k successes can be arranged among n trials, and the two powers weight the arrangement by the probability of that exact sequence of successes and failures.

Reading the Results

P(X = k) is the exact probability of your target count. P(X ≤ k) is the cumulative chance of at most k successes — the natural value for 'how likely is a defect rate this low or lower?' The expected value np tells you the average number of successes you would see over many repetitions, and the standard deviation measures how much the count typically varies around that average.