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Binomial Distribution Calculator

Find binomial probabilities — exactly k, at most k and at least k successes in n independent trials — plus the mean and standard deviation.

Number of trials (n)

Number of successes (k)

Probability of success (p)

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How it works

P(X = k) = C(n, k) · p^k · (1 − p)^(n − k) · P(X ≤ k) = Σ · mean = np · σ = √(np(1 − p))

The binomial distribution gives the probability of getting exactly k successes in n independent trials that each succeed with probability p. The binomial coefficient C(n, k) counts the number of ways those successes can be arranged, and the two powers give the probability of any one such arrangement. This calculator computes the coefficient through log-gamma functions so it stays accurate even for large n, and sums the mass function to get the cumulative "at most" and "at least" probabilities.

Worked example

Flipping a fair coin 10 times (n = 10, p = 0.5), the chance of exactly 3 heads is C(10, 3) · 0.5³ · 0.5⁷ = 120 · 0.5¹⁰ ≈ 0.1172, or 11.72%. Summing k = 0 to 3 gives P(at most 3) ≈ 17.19%, and the mean is np = 5 with a standard deviation of √2.5 ≈ 1.58.

Sources: Binomial distribution — Wikipedia

Frequently asked questions

When does the binomial distribution apply?

It applies when you have a fixed number of independent trials, each with only two outcomes (success or failure) and the same success probability p. Coin flips, pass/fail tests and yes/no survey answers all fit; situations where the probability changes between trials do not.

What is the difference between "at most" and "at least" k?

P(at most k) is the cumulative probability of k or fewer successes — the sum from 0 up to k. P(at least k) is the probability of k or more, from k up to n. Together with P(exactly k) they cover the three questions people usually ask of a binomial.

How are the mean and standard deviation found?

The mean number of successes is np and the variance is np(1 − p), so the standard deviation is √(np(1 − p)). For 10 fair coin flips that is a mean of 5 and a standard deviation of about 1.58.

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For education — check assumptions (sample size, distribution) before using statistics in real decisions.