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 delivers precise results.
The binomial distribution models scenarios with a fixed number of yes/no trials: coin flips, quality inspections, free throws, survey responses, and medication success rates. Its two parameters — n (trials) and p (success probability) — fully determine the distribution.
Beyond probabilities, the calculator shows the complete distribution table with visual bars, computes mean, standard deviation, skewness, and provides a confidence interval for the true proportion using the normal approximation. Presets let you quickly explore classic scenarios. Check the example with realistic values before reporting. Use the steps shown to verify rounding and units. Cross-check this output using a known reference case. Use the example pattern when troubleshooting unexpected results. Validate that outputs match your chosen standards.
The binomial distribution is one of the most widely used probability distributions in science, engineering, business, and medicine. From clinical trials to quality control to A/B testing, any scenario involving counts of successes in fixed trials is binomial.
This calculator provides exact results (no approximations needed), a visual distribution, and practical statistics — everything you need for homework, research, or professional analysis.
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.
Fixed number of trials (n), each trial has exactly two outcomes (success/failure), constant probability (p) across trials, and trials are independent. Use this as a practical reminder before finalizing the result.
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.