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Binomial Formula Calculator | Binomial P(X=k) from n and p

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Calculate binomial probabilities, cumulative probabilities, mean and variance for any number of trials, with log-space arithmetic that handles large n without overflow. Ideal for statistics coursework, quality sampling and probability modeling.

Last updated: July 2026 · Free · No sign-up required

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Results

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, pP(X = k)
10, 3, 0.50.1172
10, 5, 0.50.2461
20, 10, 0.50.1762
100, 50, 0.50.0796

info Binomial Formula Calculator

Free math calculator — enter your numbers and get an instant, accurate result.

info Private by design

Everything runs locally in your browser. No uploads, no accounts, no tracking.

info Works everywhere

Fully responsive and mobile-friendly — calculate on any device, any time.

info Educational

Includes the formula and step-by-step explanation so you understand the math, not just the answer.

Interpretation guide

OutputQuestion 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 npqCentre and spread of the distribution

lightbulb Worked example

Let's say you are using the Binomial Formula Calculator. Calculates binomial formula using P(X = k) = C(n,k) × pᵏ × (1−p)ⁿ⁻ᵏ, instantly in your browser. Enter the values that match your situation into the input fields and press calculate — using realistic numbers makes the result directly useful for you.

Result: The calculator instantly applies the formula P(X = k) = C(n,k) × pᵏ × (1−p)ⁿ⁻ᵏ; μ = np; σ² = np(1−p) and returns the result with appropriate precision.

What this means: Read the result in the context of what you are measuring. The step-by-step breakdown lets you confirm the math and understand which input most affects the outcome.

Formula & methodology

Formula: P(X = k) = C(n,k) × pᵏ × (1−p)ⁿ⁻ᵏ; μ = np; σ² = np(1−p)

The Binomial Formula Calculator is built on a well-established calculation method. It uses the formula P(X = k) = C(n,k) × pᵏ × (1−p)ⁿ⁻ᵏ; μ = np; σ² = np(1−p) to turn your inputs into a reliable result. Calculates binomial formula using P(X = k) = C(n,k) × pᵏ × (1−p)ⁿ⁻ᵏ, instantly in your browser. The steps are shown on the page so you can follow the reasoning from input to output.

  1. Enter the number of independent trials n, the successes k, and the per-trial probability p.
  2. The tool computes the binomial coefficient C(n,k) via log-gamma (no factorial overflow).
  3. 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.

Authoritative source: Wolfram MathWorld

Frequently asked questions

What is the binomial formula?
The Binomial Formula Calculator uses the standard P(X = k) = C(n,k) × pᵏ × (1−p)ⁿ⁻ᵏ; μ = np; σ² = np(1−p). You enter your values, and the calculator applies the formula step by step, showing the working so you can verify the math and understand how the result is derived rather than trusting a black box.
What is the probability of exactly 3 heads in 10 flips?
The Binomial Formula Calculator is free, private, and accurate: it runs entirely in your browser (no uploads, no accounts), applies the standard calculation, and explains each step so you can verify the result. Calculates binomial formula using P(X = k) = C(n,k) × pᵏ × (1−p)ⁿ⁻ᵏ, instantly in your browser. There is no limit on usage, and it works on any device.
When does the binomial distribution apply?
Yes — the Binomial Formula Calculator is fully responsive and works in any modern browser on a phone, tablet, or desktop, with touch-sized inputs and no app to install. The full calculation runs on the device, so you get the same speed and precision on mobile as on a computer, even with a weak connection.
What are the mean and variance?
For a binomial distribution with n trials and success probability p, the mean is np and the variance is np(1−p). For 100 trials with p = 0.3, the mean is 30 successes and the variance is 21, giving a standard deviation of about 4.58. These parameters summarize the distribution and feed the normal approximation for large n.
Can it handle large n without overflow?
Yes — the calculator uses log-space arithmetic for the binomial coefficients, computing nCk and probabilities without overflowing the factorial intermediate values. This lets it handle large n (hundreds or thousands of trials) that would overflow a naive factorial implementation. Results are returned with full precision in the probability.
How do I run a binomial test for significance?
A binomial test asks whether your observed successes could plausibly come from an assumed probability. Example: 34 heads in 50 flips of a supposedly fair coin — the probability of 34 or more is the sum of binomial probabilities P(X ≥ 34) ≈ 0.008, so two-tailed p ≈ 0.016 and the fair-coin hypothesis looks shaky. Compute each term with the binomial formula here and sum the tail.
What is the chance of getting exactly 7 heads in 10 coin flips?
Use P(X = k) = C(n,k) × pᵏ × (1−p)ⁿ⁻ᵏ with n = 10, k = 7, p = 0.5: C(10,7) = 120, so the probability is 120 × 0.5¹⁰ ≈ 11.7%. The tool also gives cumulative probabilities — P(X ≥ 7) is the more useful figure for 'at least' questions. Mean μ = np = 5 and variance σ² = np(1−p) = 2.5 describe the full distribution.
When can I use the binomial formula?
Only when you have a fixed number of independent trials with the same success probability and two outcomes per trial — coin flips, defect sampling, survey yes/no answers. If trials influence each other or p changes between trials, the binomial model does not apply. The tool also handles large n with log-space arithmetic so factorials do not overflow.
What is the difference between P(X = k) and the cumulative probability?
P(X = k) is the chance of exactly k successes; the cumulative P(X ≤ k) adds up all outcomes up to k. For 'at most 3 defects in 50 units' you want the cumulative; for 'exactly 3' you want the point probability. The calculator reports both, plus the mean and variance.
How does it avoid overflow with large n like 10,000?
Direct factorials of 10,000 are astronomically large and would overflow normal floating-point math. The tool computes in log-space — summing logarithms of the terms instead of multiplying giant numbers — so probabilities for n in the thousands stay accurate. Results match published binomial tables to full display precision.

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