Hypergeometric Distribution

The Hypergeometric Distribution is like the binomial distribution since there are TWO outcomes. The difference is the trials are done WITHOUT replacement.

For example when flipping a coin each outcome (head or tail) has the same probability each time. Both heads and tails are outcomes every time on each trial. Therefore the binomial distribution formulas apply in that case.

The following assumptions and rules apply to use the Hypergeometric Distribution:

  • Discrete distribution.
  • Population, N, is finite and a known value.
  • Two outcomes - call them SUCCESS (S) and FAILURE (F).
  • Number of successes in the population is known, S.
  • Used when sample size, n, is greater than or equal to 5% of N.
  • Trials are done without replacement, dependent.

A random variable (x) follows this distribution if its probability mass function is given by the formula shown below:

Hypergeometric Distribution Probability Formula


A sample of 5 parts are drawn without replacement from a total population of 30 parts. Determine the probability of getting exactly 2 defective parts. The population is known to have 14 defective parts.


There are two outcomes and n/N = 5/30 = 16.6% which satisfies assumptions.

n = 5
N = 30
s = 2
S = 14

Substitute the values above into the probability formula above:

Hypergeometric Distribution Example

The probability of getting exactly 2 defective parts is 0.3576 or 35.76%

The denominator represents the total amount of combinations of selecting 5 parts from 30 parts, which is 142,506 for this example.

Keep in mind, the this says "defective" parts. Each part may have one or more "defects" that cause the part to be appraised as a "defective" part. There is a difference between and defective part and a defect on a part. 

Using Excel

Reminder, most statistical software can solve for probabilities when given the inputs.


Return to other Discrete Distributions

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