📊 Binomial Distribution Calculator – PMF, CDF & Statistics
The binomial distribution is one of the most widely used probability distributions in statistics. It describes the number of successes in a sequence of n independent Bernoulli trials, where each trial has the same constant probability of success p. Whether you are analysing quality control data, modelling polling results, or solving statistics problems, this calculator gives you exact probabilities, cumulative probabilities, and full distribution statistics in seconds.
🔢 The Binomial Formula
The probability of observing exactly k successes in n trials is given by the Probability Mass Function (PMF):
P(X = k) = C(n, k) × p^k × (1 − p)^(n − k)where C(n, k) = n! / (k! × (n − k)!) is the binomial coefficient counting the number of ways to choose k items from n. This calculator evaluates the formula in log-space to stay accurate even for large values of n.
📐 Calculation Modes Explained
The tool supports five modes to cover every common probability query:
- Exact P(X = k) — the probability of getting precisely k successes.
- Cumulative P(X ≤ k) — the probability of at most k successes; equals the sum of all PMF values from 0 to k.
- Upper Tail P(X ≥ k) — the probability of at least k successes; computed as 1 − P(X ≤ k − 1).
- Interval P(a ≤ X ≤ b) — the probability that successes fall within a range; equals P(X ≤ b) − P(X ≤ a − 1).
- Distribution Summary — shows the full PMF/CDF table and descriptive statistics for the chosen n and p.
📈 Key Distribution Statistics
For a Binomial(n, p) distribution, the following statistics are always displayed:
- Mean (μ): The expected number of successes,
μ = np. - Variance (σ²): A measure of spread,
σ² = np(1 − p). - Standard Deviation (σ):
σ = √(np(1 − p)). - Skewness: How asymmetric the distribution is;
(1 − 2p) / σ. Positive when p < 0.5, negative when p > 0.5, and zero (symmetric) when p = 0.5.
🔍 Normal Approximation
When both np ≥ 5 and n(1 − p) ≥ 5, the binomial distribution is approximately bell-shaped and can be approximated by a Normal distribution with the same mean and standard deviation. The calculator flags this condition automatically so you can decide whether the approximation is appropriate for your use case.
🏫 Common Use Cases
- Quality Control — computing the probability that a batch contains fewer than k defects.
- Education & Exams — solving textbook statistics and probability exercises.
- Polling & Surveys — estimating the likelihood of a certain number of respondents choosing an option.
- Reliability Engineering — modelling the chance that at least k out of n components are functioning.
- Clinical Trials — calculating the probability of observing a given number of positive outcomes in a fixed-size trial.
- Finance & Risk — pricing binary options or modelling default events in a portfolio of independent loans.
⚠️ Assumptions & Limitations
The binomial model is valid only when these four conditions hold:
- Fixed number of trials — n is determined in advance.
- Binary outcome — each trial results in success or failure.
- Independence — the outcome of one trial does not affect others.
- Constant probability — p remains the same across all trials.
If trials are dependent or the probability changes between draws (as with sampling without replacement from a small population), consider the hypergeometric distribution instead. For rare events with very large n and very small p, the Poisson distribution with λ = np is a useful approximation.