Binomial Distribution Calculator
Use this tool to compute exact and cumulative binomial probabilities for Bernoulli trials.
What is a binomial distribution?
The binomial distribution models the number of successes in a fixed number of independent trials, where each trial has only two outcomes:
success or failure. If each trial has the same probability of success p, and you run n trials, then the random variable
X (number of successes) follows a binomial distribution.
In plain language, this is the right model when you ask questions like: “Out of 20 customers shown an ad, how many click?”, or “Out of 12 manufactured parts, how many pass inspection?”
Binomial formula used in this calculator
For an exact probability, the calculator uses:
P(X = k) = C(n, k) · pk · (1 - p)n-k
- n = total number of trials
- k = number of successes
- p = probability of success on each trial
- C(n, k) = number of combinations (
n choose k)
For cumulative probabilities such as P(X ≤ k), P(X ≥ k), or range probabilities, the calculator sums the
exact probabilities over the relevant values of k.
How to use the online binomial distribution calculator
1) Enter your model inputs
- Set n (number of trials).
- Set p between 0 and 1.
- Select the probability type: exact, at most, at least, or between.
2) Set the success count
Use k for exact and one-sided cumulative probabilities. If you select range mode, enter k₁ and k₂ for the inclusive interval.
3) Click calculate
You’ll get the requested probability plus summary statistics:
mean np, variance np(1-p), and standard deviation.
Practical examples
Marketing conversion
Suppose a landing page has a conversion rate of 8% and 50 visitors arrive today.
Set n = 50, p = 0.08, and choose P(X ≥ 5) to estimate the probability of getting at least five conversions.
Quality assurance
If each component has a 97% pass probability and you test 30 components,
use P(X ≤ 27) to estimate the chance of having 27 or fewer passing units.
Education and exam scoring
For a true/false quiz with random guessing at p = 0.5, this calculator can estimate the likelihood
of scoring exactly or at least a certain number of correct answers.
Common mistakes to avoid
- Using different success probabilities across trials (that breaks the binomial assumption).
- Forgetting that trials should be independent.
- Entering percentages as whole numbers (use 0.35, not 35).
- Mixing “at most” (
≤) and “at least” (≥) interpretations.
Quick reference
- Expected value:
E[X] = np - Variance:
Var(X) = np(1-p) - Standard deviation:
sqrt(np(1-p))
This online binomial distribution calculator is ideal for statistics homework, data science prep, A/B testing analysis, and operational decision-making where outcomes are binary.