# Binomial Distribution Probability Calculator Compute binomial probability mass function, cumulative distribution, mean, and variance instantly with this free statistics calculator. --- - **Canonical URL:** https://dothecalculation.com/calculators/binomial-distribution-calculator - **Category:** Math calculators - **Publisher:** Do The Calculation (https://dothecalculation.com) - **Cost:** Free, no account or sign-up required - **Privacy:** Runs entirely in the browser; inputs are never sent to a server - **Methodology:** https://dothecalculation.com/methodology --- ## Binomial Distribution Calculator Calculate binomial probability mass (PMF), cumulative probabilities (CDF), mean, and variance for independent trials. - Independent trials probability (PMF) - Cumulative range probabilities (CDF) - Probability mass histogram chart ## Introduction to the Binomial Probability Distribution In probability theory and statistics, the binomial distribution is a discrete probability distribution that models the number of successes in a sequence of \(n\) independent trials, each yielding a binary outcome: success (with probability \(p\)) or failure (with probability \(q = 1 - p\)). A classic example is flipping a coin multiple times and counting the number of heads. The binomial distribution is fundamental for quality control, medical trials (determining recovery rates), spam filters, and games of chance. For a distribution to be binomial, four conditions must be met (often remembered by the acronym BINS): 1. **Binary**: Trials have only two possible outcomes (success or failure). 2. **Independent**: The outcome of one trial does not affect the others. 3. **Number**: There is a fixed, pre-determined number of trials \(n\). 4. **Same**: The probability of success \(p\) is constant for each trial. This calculator evaluates binomial probabilities. By entering the trials \(n\), success probability \(p\), and target success count \(k\), it computes the individual PMF, cumulative CDF, mean, and variance, plotting a complete probability distribution histogram. ## Probability Mass (PMF) and Combinatorial Combinations The Probability Mass Function (PMF) calculates the exact probability of obtaining exactly \(k\) successes in \(n\) independent trials: $$P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}$$ Where \(\binom{n}{k}\) is the binomial coefficient (the number of combinations, read as "n choose k"), calculated as: $$\binom{n}{k} = \frac{n!}{k!(n - k)!}$$ The term \(p^k\) represents the probability of \(k\) successes, and \((1 - p)^{n-k}\) represents the probability of the remaining \(n - k\) failures. ## Cumulative Distributions, Mean, and Variance The Cumulative Distribution Function (CDF) calculates the probability of obtaining at most \(k\) successes: $$P(X \leq k) = \sum_{i=0}^{k} \binom{n}{i} p^i (1 - p)^{n - i}$$ The mean (expected value) and variance of a binomial random variable are calculated using simple formulas: $$\text{Mean } \mu = np$$ $$\text{Variance } \sigma^2 = np(1 - p)$$ $$\text{Standard Deviation } \sigma = \sqrt{np(1 - p)}$$ ## How to Use This Calculator Enter the number of trials \(n\) and the success probability \(p\), choose an operator (exact \(P(X=k)\), cumulative \(P(X \leq k)\), or at-least \(P(X \geq k)\)), and set the target success count \(k\). The calculator returns the probability, the mean, variance, and standard deviation, plus a histogram of the full probability mass function. ## Worked Example: 10 Coin Flips, p = 0.5, Exactly 5 Heads Trials \(n = 10\), success probability \(p = 0.5\), target \(k = 5\). $$\binom{10}{5} = 252$$ $$P(X = 5) = 252 \times (0.5)^5 \times (0.5)^5 = 0.246094 \; (24.61\%)$$ $$\mu = np = 10 \times 0.5 = 5, \qquad \sigma^2 = np(1-p) = 2.5, \qquad \sigma = \sqrt{2.5} \approx 1.5811$$ So in 10 fair coin flips, the probability of landing exactly 5 heads is about 24.6%, and the expected number of heads is 5 with a standard deviation of roughly 1.58. ## Real-World Uses for the Binomial Distribution Anywhere you're counting successes across a fixed number of independent yes/no trials, the binomial applies: quality control (how many of 200 units off the line are defective, given a known defect rate), clinical trials (how many of 50 patients respond to treatment), A/B testing (how many of 1,000 visitors convert), or polling (how many of a sample agree with a statement, given an assumed true proportion). It's a useful sanity check even outside formal statistics: if you expect a 5% failure rate and you're seeing 15 failures out of 100 attempts, the binomial PMF around \(k=15\), \(n=100\), \(p=0.05\) will tell you just how unlikely that run is under your assumed rate — a quick way to notice something's actually gone wrong rather than just gotten unlucky. ## Related Calculators For continuous data instead of discrete trial counts, use the [normal distribution calculator](/calculators/normal-distribution-calculator) — the binomial distribution approaches it as \(n\) grows large. To update a probability with new evidence rather than compute a fixed distribution, see the [Bayes' theorem calculator](/calculators/bayes-theorem-calculator). ## Frequently asked questions ### What is a binomial distribution? A discrete probability distribution that models the number of successes in a fixed number of independent trials with a constant success probability. ### What is the PMF of a binomial distribution? The Probability Mass Function, P(X = k), calculates the probability of getting exactly k successes. ### What is the CDF of a binomial distribution? The Cumulative Distribution Function, P(X ≤ k), calculates the probability of getting k or fewer successes. ### How do you calculate "n choose k"? Use the combination formula: n! / (k!(n-k)!) where "!" represents factorials. ### What is the mean of a binomial distribution? The mean (expected successes) is calculated as μ = np. ### What is the variance of a binomial distribution? The variance is calculated as σ² = np(1-p). ### What are the criteria for a binomial experiment? Fixed number of trials, independent trials, binary outcomes (success/failure), and a constant probability of success. ### How is the probability of failure calculated? The probability of failure is simply q = 1 - p, where p is the probability of success. ### What happens to the binomial distribution as the number of trials gets very large? According to the De Moivre-Laplace theorem, it approaches a normal distribution, which allows normal approximations to be used for large n. ### Can the success probability p be greater than 1? No, probabilities must always be between 0 and 1 (inclusive). ### What does P(X ≥ k) mean? It represents the probability of obtaining k or more successes, calculated as 1 - P(X < k). ### What is the difference between binomial and Poisson distributions? The binomial distribution has a fixed number of trials n. The Poisson distribution models the number of events occurring in a fixed interval of time or space with no upper limit. ## Related concepts - **Binomial Coefficient** — The combinatorial formula representing ways to choose items. - **Discrete Probability** — Probability distributions representing countable outcomes. - **Expected Value** — The long-term average outcome of a random variable. ## Related guides - [Standard Deviation Guide: Measure Data Spread Clearly](https://dothecalculation.com/blog/math/standard-deviation-explained) — Understand variance, population vs sample formulas, and data spread with worked examples tied to the live DTC standard deviation calculator. ## Related calculators - [Standard Deviation Calculator](https://dothecalculation.com/calculators/standard-deviation-calculator) — Calculate mean, variance, and standard deviation for any dataset instantly with this free step-by-step statistics calculator. - [Descriptive Statistics & Outlier Detector](https://dothecalculation.com/calculators/descriptive-statistics-calculator) — Calculate variance, standard deviation, quartiles, IQR, and detect statistical outliers with box plots using this free tool. - [Normal Distribution & Z-Score Calculator](https://dothecalculation.com/calculators/normal-distribution-calculator) — Compute Z-scores and normal distribution (CDF) probabilities for left-tail, right-tail, and between-bounds ranges with a visual bell curve shown instantly. - [Average Calculator](https://dothecalculation.com/calculators/average-calculator) — Use our free average calculator to find the mean, weighted average, and GPA. Shows step-by-step work for any set of numbers. - [Confidence Interval Calculator](https://dothecalculation.com/calculators/confidence-interval-calculator) — Calculate confidence intervals for population means and proportions, including margin of error, with this free statistics calculator. - [Markov Chain Transition & Steady State Calculator](https://dothecalculation.com/calculators/markov-chain-calculator) — Compute multi-step transition distributions and steady-state probabilities for Markov chains instantly with this free calculator. --- _This mathematical solver is designed for academic, engineering, and educational analysis. Rounding errors, numerical tolerances, or algorithm constraints might apply near poles, boundary conditions, or complex coordinate spaces. Always verify critical computations independently._ --- _Source: [Do The Calculation](https://dothecalculation.com/calculators/binomial-distribution-calculator). Quote freely with attribution and a link to this page._