## The Hypergeometric Distribution

The hypergeometric distribution is closely related to the binomial distribution. The binomial distribution is the model for sampling with replacement from a finite collection, or sampling with or without replacement from an infinite collection, with success or failure the possible outcomes, or the approximate distribution without replacement if the collection is large but finite. The hypergeometric is the exact distribution for the number of successes (or failures) in any sample drawn from a finite collection, without replacement.

The assumptions leading to the hypergeometric distribution are:

1. The population from which the samples are drawn is finite.

2. Each individual selection drawn from the sample can be classified as a success or failure and there are successful individuals in the population.

3. A sample of individuals is drawn in such a way that each subset of size is equally likely to be chosen.

If the population has M successful individuals in a population of size then the number of successes in a sample of size drawn from the population, for satisfying The mean and variance are given by  for and some values of are shown below. Example: Five individuals from a near extinct species consisting of only 25 animals are caught, tagged and re - released. Some time later a sample of 10 animals is selected. What is the probability that 2 of this sample are wearing tags?

The distribution is hence is Suppose instead the population size is not known. We wish to estimate We can estimate as hence If then  