Hypergeometric Distribution

Modeling the probability of successes in a sample drawn without replacement.

The "Drawing from a Deck" Distribution

The Hypergeometric distribution is used when you are sampling from a finite population without replacement. This is a key difference from the Binomial distribution, where each trial is independent.

The classic example is drawing cards from a deck. If you draw a 5-card hand, what's the probability of getting exactly 2 spades? In finance, this can model credit risk in a portfolio of bonds: if you have a portfolio of 100 bonds and know that 5 will default, what is the probability that if you randomly select 10 bonds, exactly 1 of them will be a defaulter?

The Formula

The probability of getting

k

successes in a sample of size

n

is:

P(X=k) = \frac{\binom{K}{k} \binom{N-K}{n-k}}{\binom{N}{n}}

$N$ is the total population size.
$K$ is the total number of "success" items in the population.
$n$ is the size of the sample drawn.
$k$ is the number of "successes" in the sample.

Interactive Hypergeometric Distribution

Adjust the parameters of the population and sample to see how the probabilities change.

Population Size (N): 52

Population Successes (K): 13

Sample Size (n): 5