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Calculation Machine

Hypergeometric Distribution Calculator

Draws without replacement — cards, lotteries, quality samples.

Reviewed by the Calculation Machine team · How we check our maths

Your values

ⓘ Live results

Results
P(exactly k)
27.428%
P(at most k)
90.7233%
P(at least k)
36.7047%
Expected successes
1.25
Standard deviation
0.93

Quick answer

The hypergeometric distribution models sampling without replacement: P(X = k) = C(K,k)·C(N−K,n−k) ÷ C(N,n).

The formula

P(X = k) = C(K,k)·C(N−K,n−k) ÷ C(N,n)

Worked example

Worked example

Using the values the calculator loads with.

Inputs

  • Population size N52
  • Successes in the population K13
  • Sample drawn n5
  • Successes wanted k2

Result

  • P(exactly k)27.428%
  • P(at most k)90.7233%
  • P(at least k)36.7047%
  • Expected successes1.25
  • Standard deviation0.93

How to use the Hypergeometric Distribution Calculator

  1. 1Enter the total population and how many of them count as successes.
  2. 2Enter how many items you draw.
  3. 3Enter the number of successes you want in the draw.

Frequently asked questions

When should I use binomial instead?

When you replace each item, or when the sample is under about 5% of the population — then the two answers barely differ.

Can it handle lottery odds?

Yes. Set N to the ball count, K to the numbers on your ticket, n to the balls drawn and k to the matches.

About this calculator