Free calculator
Binomial distribution calculator
Work with X ~ Binomial(n, p) in one place: P(X = k) (exactly k successes), P(X ≤ k) (at most), P(X ≥ k) (at least), or P(a ≤ X ≤ b) with integer endpoints (inclusive). We also show mean np, variance np(1 − p), and SD. Match BINOM.DIST / BINOM.DIST.RANGE in Google Sheets and Excel. Educational discrete‑probability math—not inverse‑binomial software, not Poisson or hypergeometric engines in v1.
When to use this calculator
Textbook binomial checks before you mirror the same idea in Sheets or Excel—transparent formulas, not a full statistics suite.
- Homework or specs that ask for exactly k successes, at most k, at least k, or a count between two integers after n independent trials with the same success probability p.
- Read off np and np(1 − p) for the same n and p when you are comparing spread to a normal approximation (see the normal distribution tool for continuous tails).
- Match
BINOM.DISTandBINOM.DIST.RANGEcells against this page when auditing a template row. - Planning a Wald proportion interval or sample size? Use the confidence interval and sample size tools—this page stays on Binomial(n, p) mass and cumulative probabilities only.
A random variable X has a binomial distribution Binomial(n, p) when it counts the number of successes in n independent trials, each with the same success probability p.
The four usual assumptions
Fixed n; independent trials; two outcomes per trial (success / failure); the same p on every trial. When any assumption fails badly, the binomial model may not match your real process.
Probability mass **P(X = k)**
P(X = k) = C(n, k) p^k (1 − p)^(n − k) for integers k in [0, n], with C(n, k) = n! / (k! (n − k)!).
Cumulative tails and between
P(X ≤ k) sums P(X = i) for i = 0 … k. P(X ≥ k) = 1 − P(X ≤ k − 1) when k ≥ 1, and 1 when k = 0. For integers a ≤ b, P(a ≤ X ≤ b) = P(X ≤ b) − P(X ≤ a − 1).
We do not implement inverse binomial quantiles, Poisson or negative binomial calculators, hypergeometric counts without replacement, or Wilson / exact two‑sided proportion intervals on this v1—use dedicated statistics software when you need those models.
For N(μ, σ²) continuous tails and percentiles, open the normal distribution calculator.
For two‑sided Wald proportion intervals (not exact binomial), open the confidence interval calculator.
FAQs cover inclusive between, TI / Casio wording, BINOM.DIST vs BINOM.DIST.RANGE, and what we intentionally exclude in v1.
Google Sheets & Excel
In English function names, BINOM.DIST(k, n, p, FALSE) is P(X = k) and BINOM.DIST(k, n, p, TRUE) is P(X ≤ k) for X ~ Binomial(n, p). BINOM.DIST.RANGE(n, p, a, b) returns P(a ≤ X ≤ b) when available. Replace cell references with your values. On localized Excel, use Insert function to find the same patterns under your language pack.
=BINOM.DIST(A2, B2, C2, FALSE)A2 = k, B2 = n, C2 = p. FALSE = probability mass (not cumulative).
=BINOM.DIST(A2, B2, C2, TRUE)Same k, n, p as above. TRUE = cumulative P(X ≤ k).
=BINOM.DIST.RANGE(B2, C2, D2, E2)B2 = n, C2 = p, D2 = a, E2 = b (requires BINOM.DIST.RANGE in your build).
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Frequently asked questions
What does this calculator compute?
Probabilities for X ~ Binomial(n, p) on integers: P(X = k), P(X ≤ k), P(X ≥ k), or P(a ≤ X ≤ b) with inclusive endpoints. It also shows np, np(1 − p), and the standard deviation √(np(1 − p)).
Are a and b inclusive in “between”?
Yes. P(a ≤ X ≤ b) includes both endpoints a and b. You may enter a larger than b; the tool uses the ordered interval.
Which Google Sheets or English Excel functions match this page?
BINOM.DIST(k, n, p, FALSE) gives P(X = k). BINOM.DIST(k, n, p, TRUE) gives P(X ≤ k). When your build supports it, BINOM.DIST.RANGE(n, p, a, b) matches P(a ≤ X ≤ b) inclusive.
What are the German Excel names for BINOM.DIST?
Typical Excel (Deutsch) names include BINOM.VERT for BINOM.DIST and BINOM.VERT.BEREICH for BINOM.DIST.RANGE—confirm in your function library; argument separators may be semicolons.
What are the French Excel names for BINOM.DIST?
Typical Excel (français) names include LOI.BINOMIALE for BINOM.DIST and LOI.BINOMIALE.SERIE for BINOM.DIST.RANGE—verify on your install.
Is this the same as Binomial PD / CD on a Casio or TI calculator?
Same discrete ideas: PD‑style readouts line up with P(X = k); CD‑style cumulative readouts line up with P(X ≤ k) or complements for at least questions. Menus differ by model—use your course’s steps for exams.
Do you include a Poisson calculator here?
No. Poisson models rare events with a rate parameter λ and is a different distribution family. Keep Binomial(n, p) on this page and use dedicated Poisson tools when your problem is rate‑based.
Can I type a target probability and get k?
Not in v1. Inverse binomial (smallest k with P(X ≤ k) ≥ p) is useful but needs a different UX and edge‑case policy—we may add it later.
When is a normal approximation reasonable?
Some courses use N(np, np(1 − p)) to approximate Binomial(n, p) when n is large and p is not too close to 0 or 1. This page always uses the exact binomial mass; use the normal distribution tool when you truly want continuous N(μ, σ²) tails.
Is this professional statistics advice?
No. It is a free educational calculator. For regulated reporting, clinical trials, or production quality systems, follow qualified professionals and institutional methods.