The Hypergeometric Random Variable

The hypergeometric distribution is a discrete probability distribution that describes the probability of $k$ successes in $n$ draws without replacement from a finite population of size $N$ containing a maximum of $K$ successes. This is in contrast to the binomial distribution, which describes the probability of $k$ successes in $n$ draws with replacement.

The hypergeometric distribution applies to sampling without replacement from a finite population whose elements can be classified into two mutually exclusive categories like pass/fail, male/female or employed/unemployed. As random selections are made from the population, each subsequent draw decreases the population causing the probability of success to change with each draw. The following conditions characterize the hypergeometric distribution:

• The result of each draw can be classified into one or two categories.
• The probability of a success changes on each draw.

A random variable follows the hypergeometric distribution if its probability mass function is given by:

$\displaystyle P(X=k) = \frac{{{K}\choose{k}}{{N-K}\choose{n-k}}}{{{N}\choose{n}}}$

Where:

• $N$ is the population size,
• $K$ is the number of success states in the population,
• $n$ is the number of draws,
• $k$ is the number of successes, and
• $\displaystyle {{a}\choose{b}}$ is a binomial coefficient.

A hypergeometric probability distribution is the outcome resulting from a hypergeometric experiment. The characteristics of a hypergeometric experiment are:

1. You take samples from 2 groups.
2. You are concerned with a group of interest, called the first group.
3. You sample without replacement from the combined groups. For example, you want to choose a softball team from a combined group of 11 men and 13 women. The team consists of 10 players.
4. Each pick is not independent, since sampling is without replacement. In the softball example, the probability of picking a women first is $\frac{13}{24}$. The probability of picking a man second is $\frac{11}{23}$, if a woman was picked first. It is $\frac{10}{23}$ if a man was picked first. The probability of the second pick depends on what happened in the first pick.
5. You are not dealing with Bernoulli Trials.