The hypergeometric law is a discrete probability distribution that models the number of successes in a fixed-size sample drawn without replacement from a finite population containing a specified number of success and failure objects. It is often used to model situations where selection is not independent, as each draw affects the composition of the population.

Here are the key features of the hypergeometric law:

**Finite population**The total population contains a fixed number of objects, e.g. cards, marbles, coins, etc.

**Objects of success and failure**The population is divided into two groups: success objects (e.g. marked cards, winning marbles) and failure objects (e.g. unmarked cards, losing marbles).

**Pulling without replacement**The sample is drawn without putting the objects back into the population after each draw. This means that the probability of drawing an object changes after each draw.

The probability mass function (PMF) of the hypergeometric law is given by the formula :

𝑃(𝑋=𝑘)=𝐶𝐾𝑘∗𝐶𝑁-𝐾𝑛-𝑘𝐶𝑁𝑛PX=k=CkK∗Cn-kN-KCnN

where :

- N is the total population size,
- K is the number of successful objects in the population,
- n is the sample size,
- k is the number of successes in the sample,

The mean (expectation) of the hypergeometric distribution is :

\mu=n*\frac{K}{N}

The variance of a binomial distribution is :

\sigma=n*\frac{K*(N-K)*(N-n)}{N^{2}(N-1)}

The hypergeometric law is used in fields such as statistics, genetics, economics and others where samples are drawn without replacement from a finite population. It differs from the binomial law in that it takes into account the evolution of the probability of success with each draw.