Hypergeometric Distribution (Sampling Without Replacement / Lottery & Audit) Calculator

Compute exact probability P(X=k) when drawing n items without replacement from population N.

Hypergeometric Probability P(X = 2)
27.428 %
Expected Successes in Draw (n·K/N) 1.25 successes
Finite-Population Standard Deviation 0.93 SD
Total Possible Combinations C(N, n) 2,598,960 ways
Inputs
⚡ Scenarios:
N
Total finite population size N (e.g., 52 cards in a deck).
K
Total success items K in the population (e.g., 13 hearts).
n draws
Number of items drawn without replacement (n).
k
Target number of successes k in the sample.
Results & Graph
Hypergeometric Probability P(X = 2)
27.428 %
Expected Successes in Draw (n·K/N)
1.25 successes
Finite-Population Standard Deviation
0.93 SD
Total Possible Combinations C(N, n)
2,598,960 ways

Hypergeometric Probability P(X = 2)

Graph 1 • Hypergeometric Probability • Domain Architecture & Live Telemetry Maps Population Size N, Success States In Pop K, Sample Draws Count Small N through Hypergeometric Probability Formula.
Hypergeometric Probability P(X: 27.428 % Hypergeometric Probability • Invariants & Metrics Hypergeometric Probability P 27.428 % Expected Successes in Draw ( 1.25 successes Finite-Population Standard D 0.93 SD Total Possible Combinations 2,598,960 ways
Graph 2 • Hypergeometric Probability P(X = 2) Across Population Size N Sweep (40% → 160%) Evaluates P(X = k) = [ C(K, k) × C(N − K, n − k) ] / C(N, n); Mean = n × (K / N) across 7 operating points (current input highlighted).
0 % 20.60 % 41.21 % 21.46 20.80N 37.46 31.20N 33.51 41.60N 27.43 52N 22.21 62.40N 17.77 72.80N 14.71 83.20N
Graph 3 • Joint Sensitivity: Population Size N × Success States In Pop K Compares Hypergeometric Probability P(X = 2) when Success States In Pop K is shifted +25% (solid) vs. −25% (dashed).
0 % 19.96 % 39.92 % --- Success States In Pop K @ 75% 36.29 -30% 36.20 -15% 32.97 +0% 29.10 +15% 25.44 +30%
Summary: Drawing n=5 from N=52 (with K=13 successes) gives P(X = 2) = 27.428%, expected successes E[X] = 1.25, and SD = 0.93.
Summary: Drawing n=5 from N=52 (with K=13 successes) gives P(X = 2) = 27.428%, expected successes E[X] = 1.25, and SD = 0.93.
⚡ Send result to: Scientific →Statistics & SD →Probability & P-Value →
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Formula & Mathematical Methodology: Hypergeometric Probability Formula

P(X = k) = [ C(K, k) × C(N − K, n − k) ] / C(N, n)
Mean = n × (K / N)
P(X = k) = [ C(K, k) × C(N − K, n − k) ] / C(N, n); Mean = n × (K / N)
  • populationSizeN: Population Size N (N)
  • successStatesInPopK: Success States In Pop K (K)
  • sampleDrawsCountSmallN: Sample Draws Count Small N (n draws)
  • observedSuccessesSmallK: Observed Successes Small K (k)

Rounding: Internal floating-point calculations run in 64-bit IEEE-754 double precision; display values round to 2–4 decimal places.

Worked Example: Worked Mathematical Example: Hypergeometric Probability

Compute exact probability P(X=k) when drawing n items without replacement from population N.

  1. Specify the function parameters, matrix entries, or confusion matrix counts.
  2. Apply the closed-form or iterative formula: P(X = k) = [ C(K, k) × C(N − K, n − k) ] / C(N, n); Mean = n × (K / N).
  3. Examine the primary result, error bounds, and 3 interactive SVG plots.
Result: Computes Hypergeometric Probability via P(X = k) = [ C(K, k) × C(N − K, n − k) ] / C(N, n); Mean = n × (K / N).

Frequently Asked Questions

How does the Hypergeometric Distribution (Sampling Without Replacement / Lottery & Audit) Calculator compute the solution?

It evaluates P(X = k) = [ C(K, k) × C(N − K, n − k) ] / C(N, n); Mean = n × (K / N) directly in your browser and updates three custom SVG visualizations in real time.

Can I solve backwards for an input parameter using Goal Seek?

Yes. Click "🎯 Goal Seek" in the results toolbar to solve backwards for any numeric input field.

Does this calculator run locally offline?

Yes. All calculations run 100% locally in your browser and work offline via the CalculatorGraph PWA.

Mathematics & Statistics · Probability Distributions & Survival Analysis

Hypergeometric Distribution (Sampling Without Replacement / Lottery & Audit) Calculator: inputs, method, and how to read the result

Compute exact probability P(X=k) when drawing n items without replacement from population N.

Choose inputs for the case you are checking

The defaults are illustrative. Replace them with values for your case and keep the units shown beside each field.

  • Population Size N: Total finite population size N (e.g., 52 cards in a deck).
  • Success States In Pop K: Total success items K in the population (e.g., 13 hearts).
  • Sample Draws Count Small N: Number of items drawn without replacement (n).
  • Observed Successes Small K: Target number of successes k in the sample.

What the calculation includes

With the default values shown, Hypergeometric Probability P(X = 2) is 27.428 %. The calculation follows Hypergeometric Probability Formula; the formula and variable definitions above show how the inputs combine.

One-input check: With the other defaults held constant, changing Population Size N from 52 N to 57 N changes Hypergeometric Probability P(X = 2) from 27.428 % to 24.672 %. This is a comparison inside the calculator's stated model, not a prediction.

Where to use caution

  • Uses exact analytical or standard numerical analysis formulations from NIST DLMF.
  • Double-precision floating-point arithmetic has ~15–17 significant decimal digits of precision.

Scope: Global Mathematical & Statistical Standard Check the assumptions and limitations against your situation before relying on the estimate.

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