Binomial Formula Calculator
Last updated: July 2026 · Free · No sign-up required
Enter values above and click Calculate to see your result instantly.
Inside the Binomial Formula Calculator
Every result on this page comes from a real formula — P(X = k) = C(n,k) × pᵏ × (1−p)ⁿ⁻ᵏ; μ = np; σ² = np(1−p) — computed live in your browser the moment you press the button.
The reference table below covers the most common binomial formula calculator cases at a glance, the methodology section breaks the calculation into verifiable steps, and the FAQ tackles the edge cases.
Quick reference
| n, k, p | P(X = k) |
|---|---|
| 10, 3, 0.5 | 0.1172 |
| 10, 5, 0.5 | 0.2461 |
| 20, 10, 0.5 | 0.1762 |
| 100, 50, 0.5 | 0.0796 |
functions Shows the working, not just the answer
For students and teachers, the final number is only half the value. Every math tool here exposes the formula it applied, the intermediate steps, and the rounding rule, so you can follow along, check your homework, or use the answer in a proof or report with confidence.
calculate Accurate to the spec
Calculations use 64-bit floating point with sensible rounding for the domain (currency to 2 decimals, percentages to 4 decimals, algebra to 6 significant figures). Where exact rational arithmetic matters — fractions, factorials, simplification — we use a dedicated BigNumber path so 1/3 + 1/6 returns ½, not 0.49999.
school Free for classroom use
Educators are welcome to link to any math calculator on CalcaTools from a class site, Google Classroom, or worksheet. The pages are mobile-friendly, free, ad-supported (so we can keep them free) and have no sign-up wall — students just click and use them in class or at home.
tips_and_updates Pair with the spoke articles
Below the calculator we link a small set of plain-English explainer pages — "What is a percentage?", "Why does PEMDAS matter?", and so on. They cover the underlying concept in 4–6 short paragraphs. Read those before the calculator if the topic is new, or after if you want the extra context.
Interpretation guide
| Output | Question it answers |
|---|---|
| P(X = k) | Exactly k successes |
| P(X ≤ k) | At most k successes |
| P(X ≥ k) | At least k successes |
| Mean np, variance npq | Centre and spread of the distribution |
Formula & methodology
Formula: P(X = k) = C(n,k) × pᵏ × (1−p)ⁿ⁻ᵏ; μ = np; σ² = np(1−p)
- Enter the number of independent trials n, the successes k, and the per-trial probability p.
- The tool computes the binomial coefficient C(n,k) via log-gamma (no factorial overflow).
- It multiplies by pᵏ(1−p)ⁿ⁻ᵏ for the exact PMF, then sums terms for the ≤ and ≥ tails.
Example: n = 10 coin flips, k = 3 heads, p = 0.5 → C(10,3) = 120, P = 120÷1024 = 0.117188.
Frequently asked questions
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