Hypergeometric Calculator 2026

Your Free Online Probability Tool

Need to calculate exact probabilities for sampling without replacement? Our Hypergeometric Calculator delivers instant, accurate results for real-world scenarios like quality control, genetics research, and card games. Built with precision math and a modern interface, this tool helps students, researchers, and professionals make data-driven decisions fast.

Why trust this calculator? We use log-gamma approximations for reliable results even with large populations (up to 10,000), verified against standard statistical libraries. No ads interrupting your work—just clean, professional calculations.

What is the Hypergeometric Distribution?

Imagine drawing 5 cards from a deck of 52, wanting exactly 2 aces. Unlike coin flips (independent events), cards get removed each draw, making outcomes dependent. The hypergeometric distribution models this “without replacement” sampling perfectly.

Real-world examples:

  • Factory: 500 parts, 25 defective. Test 50—probability of finding exactly 2 defects?
  • Biology: 200 fish, 50 tagged. Catch 40—what’s the chance of 10 tagged?
  • Elections: 1,000 voters, 400 supporters. Poll 100—likely support range?

The formula is:
P(X=k)=(Kk)(N−Kn−k)(Nn)P(X=k)=(nN)(kK)(n−kN−K)

Where:

  • NN: Total population
  • KK: Success items in population
  • nn: Sample drawn
  • kk: Successes observed

HyperGeo

Probability Calculator
110000
052
152
05
Invalid value
Click value to copy
Exactly k P(X = 2)
0.000000
k or fewerP(X ≤ 2)
0.00
k or moreP(X ≥ 2)
0.00

Mean (μ)

0.00

Variance (σ²)

0.00

X-Axis: Number of Successes (x) | Y-Axis: Probability P(X=x)

How to Use This Hypergeometric Calculator

1

 Enter your values

  • Population (N): Total items (1-10,000). Slider or type.
  • Successes (K): Items of interest in population (0-N).
  • Sample (n): Items you draw/test (1-N).
  • Observed (k): Successes you see (auto-limits for validity).

Live updates: Change any input—results refresh instantly. Invalid combos auto-correct.

2

Pick a preset (optional)

Click the dropdown for ready-made scenarios:

  • Poker hands (N=52 aces=4)
  • Quality control testing
  • Lottery draws
  • Clinical trials
  • Fish tagging studies

3

Read your results

  • Highlighted: Exact probability P(X = k)—your main answer.
  • Cumulative: P(X ≤ k) or P(X ≥ k) for ranges.
  • Stats: Mean and variance for distribution shape.
  • Graph: Visual bars show full probability curve (hover for details, click to set k).

Pro tip: Click any result to copy it. Perfect for reports or spreadsheets.

Example: N=52, K=13 (hearts), n=5, k=2 → P(X=2) = 0.2744 (27.44% chance of exactly 2 hearts).

When to Use Hypergeometric (vs. Binomial)

ScenarioUse HypergeometricUse Binomial
Finite population, no replacement✅ Factory lots, card decks, urns❌
Large population (replacement or infinite)❌ (approximates to binomial)✅ Coin flips, large surveys
Sample >5% of population✅ Exact needed❌ (correction required)

Quick rule: If your sample depletes >5% of the group, hypergeometric gives precise answers binomial would overestimate.

Common Applications Across Industries

Manufacturing & Quality Control

Testing batches where defects matter: “From 1,000 widgets (20 faulty), inspect 100—risk of ≥5 faults?”

Research & Academia

Genetics (allele frequencies), ecology (species counts), psychology (response rates in small groups).

Gaming & Gambling

Card probabilities, lottery analysis, board game odds—exact math beats simulation.

Business & Marketing

Survey finite customer lists: “100 leads, 30 qualified. Sample 20—expected conversions?”

Ready to calculate? Scroll up and try the tool now. For custom needs or API access, contact us.

Found a bug or want a feature? Drop feedback below—we update based on user input.

Last updated: January 2026. Math verified against R’s dhyper() function.