This binomial distribution calculator computes exact and cumulative probabilities for a binomial experiment — n independent trials, success probability p — plus the distribution’s mean, standard deviation and most likely outcome.
The Formula Behind It
P(X = k) = C(n,k) · pᵏ · (1−p)ⁿ⁻ᵏ — choose which trials succeed (the binomial coefficient, computed on our Combination Calculator), weight by the probabilities of that exact pattern. Mean = np, variance = np(1−p). The conditions are strict: fixed n, independent trials, constant p, two outcomes. When trials sample without replacement from a small population, the hypergeometric distribution owns the problem; when n is large and p small, the Poisson (next door on this hub) takes over elegantly. Part of our Statistics Calculators collection.
Worked Example (Verified)
10 fair coin flips, k = 6 heads (the widget’s defaults):
- P(X = 6) = C(10,6) ÷ 2¹⁰ = 210 ÷ 1024 = 0.2051 — six heads is the second-most-likely count yet happens only a fifth of the time
- P(X ≤ 6) = 0.8281 · P(X ≥ 6) = 0.3770 (= 1 − P(X ≤ 5))
- Mean 5.0, SD = √2.5 = 1.581 — so 6 heads is a thoroughly ordinary outcome, 0.63 SDs out; getting suspicious needs counts near the edges (9–10 heads: 1.07% combined)
- The mode: at p = 0.5 the likeliest counts are 5 (24.6%) — even-money intuition should center there, not at “about half” vaguely
Everyday Binomial Territory
- Quality control: probability a batch of 50 with 2% defect rate holds ≤ 1 defective (92.6%? compute it: that’s a shipping decision).
- A/B testing and click rates: conversions per visitor are Bernoulli trials; significance tests for them lean on the binomial (or its normal approximation when np and n(1−p) both exceed ~10).
- Genetics: each child independently inheriting a recessive allele set — Punnett fractions become binomial predictions across a family.
- The approximation ladder: binomial → normal for large n (that’s how z-tests for proportions arise), binomial → Poisson for rare events.
Related Tools
- Poisson Distribution Calculator — the rare-events sibling
- Combination Calculator — the C(n,k) inside the formula
- Statistics Calculators — the full collection
Frequently Asked Questions
How do you calculate a binomial probability?
P(X = k) = C(n,k) pᵏ (1−p)ⁿ⁻ᵏ. For 6 heads in 10 flips: 210 × 0.5¹⁰ = 0.2051. Cumulative probabilities sum the exact values over the range you care about — the cards here do that summing for you.
What are the conditions for a binomial distribution?
Four: a fixed number of trials, each trial independent, exactly two outcomes per trial, and the same success probability throughout. Violate independence (drawing without replacement) and you need the hypergeometric instead.
What is the mean and standard deviation?
Mean μ = np; SD σ = √(np(1−p)). Ten flips at p = 0.5: expect 5 ± 1.58 heads in the long-run-average sense — a built-in plausibility gauge for any observed count.
When can I approximate the binomial with a normal curve?
The common rule: np ≥ 10 and n(1−p) ≥ 10. Then mean np and SD √(npq) plug straight into z-score machinery (with a 0.5 continuity correction for careful work). Below that, use exact binomial math — this page.
Why is the probability of the “expected” outcome so low?
Because probability is spread across many near-tied outcomes: at n = 10 the top few counts share most of the mass (5: 24.6%, 6 and 4: 20.5% each). Single-point probabilities shrink as n grows even as the proportion concentrates — a distinction that unlocks most distribution intuition.
Is this binomial calculator free?
Yes — free, instant, no sign-up.