Binomial Probability Calculator
Compute the probability of getting exactly, at most, or at least k successes in n independent Bernoulli trials, along with the distribution mean and variance.
Mode
Probability of the selected outcome
- 1
Binomial coefficient C(n, k)
C(10, 3) = 120Number of ways to arrange k successes among n trials. - 2
pᵏ
0.5 ^ 3 = 0.125 - 3
(1 − p)ⁿ⁻ᵏ
0.5 ^ 7 = 0.007813 - 4
P(X = k)
120 × 0.125 × 0.007813 = 0.1172
How does this calculator work?
Binomial probability is P(X = k) = C(n, k)·pᵏ·(1 − p)ⁿ⁻ᵏ for k successes in n independent trials with success probability p. Use cumulative sums for at most or at least k. The distribution has mean np, variance np(1 − p), and standard deviation √(np(1 − p)).
Formula
How this is calculated
The binomial distribution models the number of successes X in n independent trials, where each trial succeeds with probability p. Enter n (the number of trials), k (the number of successes of interest), and p (the per-trial success probability between 0 and 1). The point probability is P(X = k) = C(n, k) · pᵏ · (1 − p)ⁿ⁻ᵏ, where C(n, k) is the binomial coefficient "n choose k".
The Mode selector chooses what to report. "Exactly" returns the single-point probability P(X = k). "At most" returns the cumulative probability P(X ≤ k) = Σ P(X = i) for i = 0..k. "At least" returns P(X ≥ k) = 1 − P(X ≤ k − 1). The binomial coefficient is built with an iterative product loop (multiplying n − i and dividing by i + 1) so large factorials never overflow.
The distribution summary uses mean μ = np, variance σ² = np(1 − p), and standard deviation σ = √(σ²). Inputs must satisfy n ≥ 0 and integer, 0 ≤ k ≤ n and integer, and 0 ≤ p ≤ 1; otherwise the result is undefined. The bar chart shows the probability mass for values of X in a window around k so you can see where k sits in the distribution.
Frequently asked questions
"Exactly k" is the probability of precisely k successes. "At most k" sums the probabilities of 0 through k successes. "At least k" sums k through n successes, computed as 1 − P(X ≤ k − 1).
There must be a fixed number of trials n, each trial independent, exactly two outcomes (success/failure), and a constant success probability p across all trials.
The mean np gives the expected number of successes, while the variance np(1 − p) and standard deviation measure how much the count typically varies around that average.
Also known as
TG we-Calculate Editorial Team. (2026). Binomial Probability Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/binomial-probability-calculator
TG we-Calculate Editorial Team. "Binomial Probability Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/binomial-probability-calculator.
TG we-Calculate Editorial Team, "Binomial Probability Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/binomial-probability-calculator
@misc{wecalculate_binomial_probability_calculator, title = {Binomial Probability Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/binomial-probability-calculator}}, year = {2026}, note = {TG we-Calculate} }
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