Binomial Distribution Calculator
The binomial distribution answers a precise question: in n independent trials, each succeeding with probability p, what is the chance of exactly k successes? This calculator evaluates that probability exactly, along with the cumulative probability of k or fewer successes and the distribution's mean, variance, and standard deviation. Enter the number of trials, the success probability between 0 and 1, and the number of successes you are interested in. Use it for coin-flip problems, quality-control sampling, survey responses, and any setting that fits the binomial model.
Binomial distribution formula
P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)
C(n, k) = n! / (k! * (n - k)!)
P(X <= k) = sum of P(X = i) for i = 0 to k
Mean = n * p
Variance = n * p * (1 - p)
Standard deviation = sqrt(variance)
The combination term counts the ways k successes can be arranged among n trials, and the power terms give the probability of any one such arrangement. Summing from 0 to k gives the cumulative probability.
Probability context
- The four conditions are: fixed n, independent trials, two outcomes per trial, and constant p.
- When p equals 0.5 the distribution is symmetric about its mean of n over 2.
- The mean n p is the long-run average number of successes over many repetitions of the n trials.
- For large n the binomial is well approximated by a normal distribution with the same mean and standard deviation.
- Probabilities always lie between 0 and 1, and all n plus 1 individual probabilities sum to exactly 1.
Binomial distribution: frequently asked questions
What is the binomial distribution?
The binomial distribution gives the probability of getting exactly k successes in n independent trials, where each trial succeeds with the same probability p. It applies whenever you repeat a fixed number of independent yes-or-no experiments, such as coin flips, with a constant success probability.
What is the binomial probability formula?
The probability of exactly k successes is C(n,k) times p to the power k times (1 minus p) to the power (n minus k), where C(n,k) is the number of ways to choose k items from n. C(n,k) equals n factorial divided by the product of k factorial and (n minus k) factorial.
What is the difference between P(X=k) and P(X<=k)?
P(X=k) is the probability of exactly k successes. P(X<=k), the cumulative probability, is the chance of k or fewer successes, found by summing the individual probabilities for 0, 1, up to k. This calculator reports both so you can answer at-most and exactly questions.
What are the mean and standard deviation of a binomial?
The mean (expected number of successes) is n times p. The variance is n times p times (1 minus p), and the standard deviation is the square root of the variance. These summarise the centre and spread of the distribution without listing every probability.
What are the requirements for using this distribution?
There must be a fixed number of trials n, each trial must be independent, each must have only two outcomes (success or failure), and the success probability p must be the same on every trial. If trials are not independent or p changes, the binomial model does not apply.
Official sources
- NIST/SEMATECH e-Handbook of Statistical Methods: Binomial distribution.
- NIST Digital Library of Mathematical Functions: DLMF.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.