Calculate binomial probabilities, cumulative distribution, mean, variance, and confidence intervals for any number of trials and success probability.
The binomial distribution calculator computes exact and cumulative probabilities for a fixed number of independent Bernoulli trials, each with the same success probability. Whether you need P(X = k), P(X ≤ k), or P(X ≥ k), this tool gives the result directly.
The distribution fits yes/no situations such as coin flips, pass/fail inspections, free throws, survey responses, and medication response rates. Its two parameters, n and p, fully determine the probability model.
Beyond the probability itself, the calculator also shows the full distribution table, the mean, the standard deviation, skewness, and an approximate confidence interval for the underlying proportion.
Binomial probability is the right model whenever you count successes in a fixed number of independent trials. That covers a lot of common work: product defect checks, election polling, A/B tests, and basic clinical or lab-style yes/no outcomes.
Putting the exact probability, the full distribution, and summary statistics on the same page makes it easier to see both the single event you care about and the wider shape of the trial outcomes.
P(X = k) = C(n, k) × p^k × (1−p)^(n−k). Mean μ = np. Variance σ² = np(1−p). CDF: P(X ≤ k) = Σ P(X = i) for i = 0 to k.
Result: P(X = 5) ≈ 0.2461 (24.61%)
With 10 fair coin flips, the probability of exactly 5 heads is C(10,5) × 0.5^10 = 252/1024 ≈ 24.61%.
Use binomial when trials are independent with fixed p. Use hypergeometric for sampling without replacement. Use Poisson when n is very large and p is very small (rare events). Use negative binomial when you're counting trials until a fixed number of successes.
The binomial can be approximated by a normal distribution when both np ≥ 5 and n(1−p) ≥ 5. Apply a continuity correction: P(X ≤ k) ≈ Φ((k + 0.5 − np) / √(np(1−p))). This is useful for quick calculations when exact computation is impractical.
In acceptance sampling, a manufacturer tests a sample of n items. If the number of defects k is below a threshold c (acceptance number), the lot is accepted. The operating characteristic curve plots the acceptance probability vs. true defect rate — this is directly computed from the binomial CDF.
Last updated:
Fixed number of trials (n), exactly two outcomes per trial, constant probability (p), and independent trials. If any of those assumptions fail, another distribution is usually a better fit.
The calculator handles up to n = 1000. For larger values, use the normal approximation: X ≈ N(np, np(1−p)).
Binomial assumes independent trials (sampling with replacement). Hypergeometric is for sampling without replacement from a finite population, where each draw changes the remaining probabilities.
Yes. Set n = number of items tested, p = expected defect rate, and compute P(X ≤ k) for the probability of at most k failures.
A single experiment with exactly two outcomes: success (probability p) or failure (probability 1−p). A binomial variable is the sum of n independent Bernoulli trials.
The normal approximation CI can slightly exceed [0, 1] with small samples. In practice, clip to [0, 1]. For exact intervals, use the Clopper-Pearson method.