Statistics
Binomial Probability Calculator
Compute P(X = k), P(X ≤ k), or P(X ≥ k) for a binomial random variable with parameters n and p.
binomial distributionprobabilitydiscrete
Binomial Probability
Compute exact, cumulative, or tail probabilities for a binomial experiment.
Probability
0.2508
Expected value
4
Variance
2.4
Binomial distribution
P(X = k) = C(n, k) pᵏ (1 − p)ⁿ⁻ᵏ
The calculator also returns the distribution’s expected value (np) and variance (np(1 − p)).
How to use
- Provide the number of trials n and success probability p.
- Enter the number of successes k and choose exact, at-most, or at-least mode.
- Read the probability, expected value, and variance.
Example
Input: n = 10, p = 0.4, k = 4, mode = exact
Output: P(X = 4) ≈ 0.2508
FAQ & notes
How does rounding work?
Probabilities are displayed with up to four decimal places, but calculations use double precision internally.
Is cumulative probability inclusive?
Yes. P(X ≤ k) includes the k term, and P(X ≥ k) includes k as well.