Binomial Distribution Calculator
Exact binomial probabilities P(X = k), cumulative P(X ≤ k), P(X ≥ k), plus mean, variance and standard deviation — with a full distribution chart and step-by-step working.
The binomial distribution explained, step by step
The binomial distribution models successes in a fixed number of independent yes/no trials with a constant success probability. Each card below covers one piece of the formula this calculator uses.
1. The four binomial conditions
A binomial random variable counts successes across a fixed number of independent trials where each trial has the same success probability p and only two outcomes (success or failure). If any condition breaks, the binomial model no longer fits.
2. The PMF — count arrangements × per-sequence probability
P(X = k) has two parts. C(n, k) counts how many orderings of k successes and n − k failures exist. Multiplied by p^k · (1 − p)^(n − k) — the probability of any one such sequence — you get the probability of exactly k successes.
3. Mean, variance and standard deviation
Because a binomial is a sum of n independent Bernoulli trials, its mean and variance are the per-trial values scaled by n. Variance peaks at p = 0.5 (maximum uncertainty per trial) and vanishes as p → 0 or 1.
4. When to switch models — hypergeometric, Poisson, normal
Sampling without replacement from a small population breaks independence — use hypergeometric. Very large n with tiny p is well-approximated by Poisson (λ = np). Very large n with p away from 0 and 1 is well-approximated by Normal(np, np(1 − p)).
Features of this calculator
- Exact probabilities: P(X = k), P(X ≤ k), P(X < k), P(X ≥ k), P(X > k) — all in one calculation.
- Uses log-gamma internally, so it stays numerically stable for large n (up to n = 5000).
- Full probability bar chart across every possible value of X from 0 to n, with the bar for your chosen k highlighted.
- Mean, variance and standard deviation reported alongside the probability results.
- Show/hide step-by-step working with the binomial coefficient, the p^k and (1 − p)^(n−k) terms and the multiplication laid out.
Frequently asked questions
+Can I use this for very large n?
Yes — the calculator uses log-gamma internally, so probabilities stay stable up to n = 5000. For larger n you can safely use the normal approximation with μ = n·p and σ = √(n·p·(1 − p)).
+What if p is unknown?
Estimate p from data as the sample proportion (successes ÷ trials), then use it here. If you're testing a hypothesised p, see the P-Value Calculator or a proportion Z-test.
+How is P(X ≥ k) different from P(X > k)?
P(X ≥ k) includes X = k itself, so it equals P(X = k) + P(X > k). If you want strictly more than k successes, use P(X > k).
+What's the connection to the Bernoulli distribution?
A single trial (n = 1) is a Bernoulli trial. Adding up n independent Bernoulli(p) trials gives Binomial(n, p).
+When should I use Poisson instead?
Use Poisson when you count events over a fixed interval and the event rate is known, but there is no natural fixed n. If n is large and p is small, Binomial(n, p) is very well approximated by Poisson with λ = n·p.