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Statistics — Probability
Binomial Distribution Calculator
Enter the number of trials, successes and the per-trial probability to get P(X = k), the cumulative probabilities, mean and variance.
A number between 0 and 1.
P(X = k) — exactly k
—
Probability of exactly k successes.
| P(X ≤ k) at most k | — |
|---|---|
| P(X ≥ k) at least k | — |
| Mean n × p | — |
| Variance n × p × (1 − p) | — |
| Std. deviation | — |
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Enter your figures above to see the step-by-step working.
For a fixed number of independent, two-outcome trialsEverything computed client-side — nothing leaves this page
How the Binomial Distribution Calculator Works
The binomial distribution gives the probability of getting exactly k successes in n independent trials, when each trial succeeds with the same probability p. The calculator uses the binomial formula — the number of ways to choose k successes, times the chance of those successes, times the chance of the remaining failures — and also sums it to give the cumulative probabilities.
Enter the number of trials, the number of successes you are interested in, and the per-trial probability. Alongside the exact and cumulative probabilities, the desk reports the distribution’s mean (n×p) and variance (n×p×(1−p)).
P(X = k) = C(n, k) × pk × (1 − p)n−k
Frequently Asked Questions
When can I use the binomial distribution?
When you have a fixed number of independent trials, each with only two outcomes (success or failure) and the same probability of success. Coin flips, pass/fail tests and yes/no survey answers are typical examples.
What is the difference between P(X = k) and P(X ≤ k)?
P(X = k) is the chance of exactly k successes. P(X ≤ k) is the chance of k or fewer — the sum of the probabilities for 0, 1, up to k successes. The calculator shows both, plus P(X ≥ k) for k or more.
What are the mean and variance of a binomial distribution?
The mean is n times p, the expected number of successes. The variance is n times p times (1 minus p), and the standard deviation is its square root. For 10 trials at p = 0.5 the mean is 5 and the variance is 2.5.
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