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Binomial Distribution Calculator

Math Calculators

P(X=k) & cumulative · mean/SD · approximations

Quick Answer

Binomial Distribution Calculator: P(X=k) & cumulative · mean/SD · approximations. Binomial Distribution Calculator uses standard math formulas to compute accurate results instantly.

%
P(X = 5)24.61%
P(X ≤ 5)62.3%
P(X ≥ 5)62.3%
P(X = 5) as decimal0.246094
P(X < 5)37.7%
P(X > 5)37.7%
Expected value E = np5.000
Std dev √(np(1−p))1.581

How the binomial distribution works

  1. 1Model: n = 10 independent trials, each succeeds with probability p = 50.0% — X ~ B(10, 0.5).
  2. 2Single point: P(X = k) = C(n,k)·p^k·(1−p)^(n−k) = C(10,5)·0.5^5·0.50^5.
  3. 3Cumulative: P(X ≤ k) sums all probabilities from 0 to k; P(X ≥ k) = 1 − P(X ≤ k−1).
  4. 4Moments: E(X) = np = 5.000, Var(X) = np(1−p) = 2.500.

Model quick reference

Scenarionsuccess
Quality inspection100item is defective
Free throws10shot scores
Guessing MCQs20a random guess is right
Vaccine trial200infection occurs
  • Requirements: fixed number of trials, independent repeats, constant success probability, binary outcome (success/failure).
  • Poisson approximation: when n ≥ 20, p ≤ 0.05 and np ≤ 10, use λ = np. Normal approximation: np ≥ 5 and n(1−p) ≥ 5.
  • Binomial distribution counts successes in n trials; the binomial theorem (a+b)^n expansion is a different topic.
  • The mode is ⌊(n+1)p⌋; when (n+1)p is an integer both (n+1)p and (n+1)p−1 are modes.

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How to Use

  1. Enter values or expressions: type directly into the Binomial Distribution Calculator — keyboard friendly.
  2. Instant results: everything computes live as you type, no button needed.
  3. Follow the steps: the result panel shows full working and formula substitution.
  4. Copy or recompute: one-click copy; change any input to recalculate instantly.

Key Points

  • Binomial Distribution Calculator is a free online math tool — open and use, no sign-up.
  • All calculations run locally in your browser; your data never leaves the device.
  • Key capability: P(X=k) & cumulative · mean/SD · approximations.
  • Results come with formulas and reference tables so you can verify every number.

Formula & How It Works

Binomial Distribution Calculator formula

Binomial Distribution Calculator uses standard math formulas to compute accurate results instantly.

FAQ

How do I use the Binomial Distribution Calculator?
Open the page and fill in the input fields — results update automatically. P(X=k) & cumulative · mean/SD · approximations.
Are the results accurate?
Yes. All calculations use standard mathematical formulas and are verified for accuracy.
Is it free? Do I need to sign up?
Completely free. No registration, login or download required — just open and use.
Does it work on mobile?
Yes. The responsive design works on phones, tablets and desktops.
Is my data saved?
No. All calculations run locally in your browser. Nothing is uploaded or stored.

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Source & Updates

Standard mathematical formulas. All logic verified. Last updated: 2025-07

Last updated:

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Cite this page: YiJianSuan. Binomial Distribution Calculator[EB/OL]. https://www.yijiansuan.com/en/c/math/binomial-distribution

Results provided by this tool are for reference only and do not constitute professional advice.