**Very complex to calculate, this law is used to calculate the probability when we have a draw without discount (a very common case, example of quality controls).**

## Introduction

The hypergeometric law describes the situations in which the order of appearance of success does not matter and where we do not put the element we have just picked for the next draw. It is in the great majority of cases quality controls where in the course of production we « *draw* »

parts to check their quality. In this context, we will not put back the piece we just checked. We will then use the Hypergeometric law to identify our « *chance *»

to see the defect we seek.

## The principle

The Hypergeometric law involves in its calculation the size of the sample n and the reference population N. Its formula being complex, it will be preferred the calculation via the **binomial law** when the population N is large (from 100), especially since it requires knowing the size N of the population, which is not always the case. It is defined as follows :

This function have :

An average of :

A variance of :

## Probability of an event

We are in the case where the order does not matter and there is no discount. This is the case n ° 4 of the calculation of the probabilities. The most telling example is that of lotto. Let’s take the first part of the draw where we have 5 digits n drawn from a population N of 49. The Hypergeometric law allows us to calculate our probability of having the 5 digits k winners.

By following the details of the calculations given in the article of the **probabilities**, we find 1906884 combinations, ie if we play once, 1 chance on 1906884.

Figure that we find on Excel via the formula : HYPGEOM.DIST (5;5;5;49;FALSE)

## The hypothesis test

The hypergeometric law can also be used to know whether a quantity of events meets its probability a priori or not. This is typically the case when one wants to know if parity is present in an enterprise. We wish that our probability is 1/2, the stake of this test will then be according to our real result obtained, to know if the desired risk, we are well in our wish.

### Step 1 : the hypotheses

The Hypergeometric test can be a bilateral or unilateral test. The following assumptions are made :

**Bilateral case**

- H0 : p = p0
- H1 : p ≠ p0

**Left unilateral case**

- H0 : p ≠ p0
- H1 : p < p0

**Unilateral case right**

- H0 : p ≠ p0
- H1 : p > p0

### Stepcalculate the practical value 2 : calculer la valeur pratique

Our practical value is simply the observed percentage of Success.

### Step 3 : calculate the critical value

With the hypergeometric law, Excel does not calculate the inverse of the Hypergeometric law and we must go there « *groping *» and compare our α risk with our cumulative probability.

We calculate the value of the risk α for the value we want « *compare* » with our real value.

To calculate it, we use the Excel formula :

**Probability of appearance : P = HYPGEOM.DIST (k ; n ; M ; N ; False)**

### Step 4 : interpretation

Test direction | Result | Statistical Conclusion | Practical Conclusion |
---|---|---|---|

Bilateral | For the lower bound: the actual success number < The calculated value for α / 2 or For the upper bound: the actual number of successes > The calculated value for 1 - α / 2 | We reject H0 | Our data do not follow the probability a priori. |

Unilateral left | The real value of our test is < Calculated value for a risk calculated at α | We reject H0 | Our data are below the probability a priori. |

Unilateral right | actual value of our test is > Calculated value for a risk calculated at 1 - α | We reject H0 | Our data are above the initial probability. |