Calculators
Binomial Distribution Calculator
Calculate binomial probabilities, mean, variance, and standard deviation for a given number of trials and success probability locally in your browser.
Enter the number of trials, probability of success, and number of successes. Calculations run locally in your browser; your values are not uploaded or stored.
Find exact and cumulative probabilities for binomial experiments.
A binomial experiment has a fixed number of independent trials n, each with two outcomes (success or failure) and a constant success probability p. Enter n, p, and a target number of successes k. The calculator returns the mean n×p, variance n×p×(1−p), standard deviation, and exact probability P(X=k), plus cumulative probabilities P(X≤k), P(X<k), P(X>k), and P(X≥k). All calculations use stable logarithmic arithmetic and run locally in your browser; your values are not uploaded or stored.
Frequently Asked Questions
Everything you need to know about this tool, how it works, and privacy.
What is a binomial distribution?
A binomial distribution models the number of successes in a fixed number of independent trials, where each trial has exactly two outcomes (success or failure) and a constant probability of success p. It is defined by two parameters: n (number of trials) and p (probability of success per trial).
What formulas does this calculator use?
The calculator uses: Mean = n × p, Variance = n × p × (1 − p), Standard deviation = √Variance. The probability mass function is P(X = k) = C(n, k) × p^k × (1 − p)^(n − k). Cumulative probabilities sum the PMF: P(X ≤ k) = Σ_{i=0}^{k} P(X = i). All calculations use logarithmic arithmetic to handle large n without overflow.
What is the difference between P(X = k) and P(X ≤ k)?
P(X = k) is the exact probability of observing exactly k successes. P(X ≤ k) is the cumulative probability of observing k or fewer successes, which is the sum of P(X = i) for all i from 0 to k.
When should I use a binomial distribution?
Use it when you have a fixed number of independent trials, each with the same success probability, and you want to know the probability of a certain number of successes. Common examples: coin flips, quality control sampling, survey responses with yes/no answers, and clinical trial outcomes.
Do my values stay private?
Yes. All calculations run locally in your browser. CodeASystem does not upload or store the values you enter.