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Statistics

Binomial Distribution

P(X = k) for a binomial. Free online Binomial Distribution. Calculate binomial distribution online — fast, accurate, mobile-friendly, no signup needed.

P(X = k)
0.246094

Derivation

  1. ├── 01Givenn = 10, p = 0.5, k = 5
  2. ├── 02Formulat};return r(t) / (r(n) × r(t-n)) × (a)^(n) × (1-a)^(t-n)
  3. ├── 03Substitutet};return r(t) / (r(10) × r(t-10)) × (a)^(10) × (1-a)^(t-10)
  4. └── 04Compute P(X = k)0.246094
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§01What is

Understanding the Binomial Distribution

The Binomial Distribution computes P(X = k) from 3 inputs: n (trials), p (success), k (successes). P(X = k) for a binomial.

Statistics is how we make sense of noisy real-world data. Whether you’re analysing survey results, sports scores, or business metrics, a statistics calculator gives you the exact formula-based answer so you can focus on the interpretation. The Binomial Distribution sits in that toolkit — it P(X = k) for a binomial. Enter your numbers above and the result updates instantly; every step of the math is shown in the Derivation panel so you can see exactly how the answer was reached.

§02The Formula

How it’s calculated

t};return r(t) / (r(n) × r(t-n)) × (a)^(n) × (1-a)^(t-n)

Where

n
n (trials)
p
p (success)
k
k (successes)
§03Practical Example

Step-by-step walkthrough

Scenario

Apply the formula to a realistic set of inputs: n (trials) = 10, p (success) = 0.5, k (successes) = 5.

  1. 01Start by noting the input — n (trials): 10.
  2. 02Start by noting the input — p (success): 0.5.
  3. 03Start by noting the input — k (successes): 5.
  4. 04Substitute these values into the formula: t};return r(t) / (r(n) × r(t-n)) × (a)^(n) × (1-a)^(t-n)
  5. 05Compute P(X = k): the calculator returns 0.246094.
  6. 06Cross-check the answer by opening the Derivation panel above — every line of math is shown so you can follow the computation end-to-end.
§04Variants

Common Binomial Distribution Problems

The formula gets rearranged depending on which variable you need. Here are the patterns you’ll run into in the real world — find the one that matches your problem and follow the worked steps.

01 · PATTERN

n (trials) halved

n = 5 (from 10)

Keep every other input at its default and halve the n (trials). See how p(x = k) responds.

  1. 01New n (trials): 5
  2. 02Baseline P(X = k): 0.246094
  3. 03New P(X = k): 0.03125
  4. 04P(X = k) decreases by 87.3% → use this sensitivity to plan for real-world variation.
02 · PATTERN

n (trials) doubled

n = 20 (from 10)

Keep every other input at its default and double the n (trials). See how p(x = k) responds.

  1. 01New n (trials): 20
  2. 02Baseline P(X = k): 0.246094
  3. 03New P(X = k): 0.0147858
  4. 04P(X = k) decreases by 94% → use this sensitivity to plan for real-world variation.
03 · PATTERN

p (success) halved

p = 0.25 (from 0.5)

Keep every other input at its default and halve the p (success). See how p(x = k) responds.

  1. 01New p (success): 0.25
  2. 02Baseline P(X = k): 0.246094
  3. 03New P(X = k): 0.0583992
  4. 04P(X = k) decreases by 76.3% → use this sensitivity to plan for real-world variation.
04 · PATTERN

p (success) doubled

p = 1 (from 0.5)

Keep every other input at its default and double the p (success). See how p(x = k) responds.

  1. 01New p (success): 1
  2. 02Baseline P(X = k): 0.246094
  3. 03New P(X = k): 0
  4. 04P(X = k) decreases by 100% → use this sensitivity to plan for real-world variation.
§05FAQ

Frequently asked questions

Yes. The calculator implements the standard formula as documented and returns exact floating-point results. No approximations are used unless noted in the formula.
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