Binomial Distribution Calculator
Compute P(X=k) and cumulative probability for a binomial distribution.
Input Data
Number of independent trials.
Number of successes (0≤k≤n).
Probability of success per trial.
%
Results
0.117187
0.171875
At a glance:P(X=k) = C(n,k)·pᵏ·(1−p)ⁿ⁻ᵏ, where C(n,k) is the binomial coefficient.
Formula
P(X = k) = C(n, k) · pᵏ · (1 − p)ⁿ⁻ᵏ
$$P(X = k) = \binom{n}{k} p^{k} (1-p)^{n-k}$$How to Use
- Enter the number of trials n and the number of successes k (both integers).
- Enter the per-trial success probability p (in percent).
- The calculator outputs the probability of exactly k successes P(X=k).
- It also gives the cumulative probability P(X ≤ k).
FAQ
How is the binomial related to the normal distribution?
When n is large and p is not near 0 or 1, the binomial approximates a normal distribution (mean np, variance np(1−p)); this follows from the central limit theorem.
When should I use the binomial distribution?
When you have a fixed number of independent trials, each with only success/failure outcomes — e.g. exactly k good items among n samples.
References
Content reviewed by the Calculatorism editorial team. Results are for reference only; please refer to the relevant authorities for the official figures.