**Queen of chance, this law is the basis of probability calculations.**

## Introduction

The Binomial law test is the test of chance par excellence which is at the origin of the calculations of **probabilities**. She was born in 1713 with the publication of “* Ars conjectandi *“, the posthumous work of the Swiss Bâlois Jacob Bernoulli.

## The principle

Each draw has the probability p of being a success and the probability q = 1 – p of being a failure. If the distribution follows the binomial Theoretical law, the ** probability of success is written in the following manner :**

**n :**the number of draws in total**k :**The number of successes we want to see among our n draws**P :**the probability of success**q :**the probability of failure

*k among n*» (norm ISO 31-11), it indicates the number of paths (p-Suites) possible to obtain the k successes among the n draws. It is calculated as follows :

His average = n * p

his variance = n * P * (1 – p)

## Probability of an event

The Binomial Law is used to calculate the probability of occurrence of an event in the event that :

- the order does not matter.
- With each draw, we put back the element that we just drew.

So we are in case 2 of the calculation of **probabilities** “*without order, with put back*“.

The most typical example is the roll of the die. To get at least 3 times the number 6 in 4 throws, our probability this calculation as follows :

That is, 1.54% chance of having our success. Result that can be found in Excel in use the binomial function : BINOM.DIST (k;n;1/6;0).

It should be noted that the larger our population (from N = 100), at most Binomial Law gets closer to the **Loi Hypergeométric**, hence the fact that often the approximation is made, the Binomial law being easier to calculate both by hand and by spreadsheets.

## The hypothesis test

The binomial law can also be used to know if a quantity of events matches its probability to “* priori *” or not. This is typically the case when one wants to know if a die is *load *. We know that we have a probability of 1/6, the stake of this test will then be according to our real result obtained, to know if the die is well * to the standard* ».

**Step 1 : the hypothesis**

The binomial test can be a two-sided or one-sided test. If we put p the theoretical value of p, we have the following hypotheses :

**bilateral**

- H0 : p = p0
- H1 : p ≠ p0

**Unilateral left test**

- H0 : p ≠ p0
- H1 : p < p0

**Unilateral right**

- H0 : p ≠ p0
- H1 : p > p0

### Step 2: Calculate the practical value

The practical value is simply our observed percentage of Success.

### Step 3: Calculate the critical value

The critical value will depend on the direction of the test. We must therefore calculate in the following way for taking into account the risk :

- bilateral : α/2 et 1 – α/2
- Unilateral left : α
- Unilateral right : 1 – α

The function in Excel is BINOM.INV.

### Step 4 : Calculate the p-value

For p-Value, we use the approximation of the binomial distribution, which will be the same whatever the meaning of the test. In Excel, the function is BINOM.DIST.

### Step 5: Interpretation

## Interpretation practical value / critical value

test direction | Result | Statistical conclusion | Practical conclusion |
---|---|---|---|

Bilateral | Practical value > + Critical value or Practical value < - Critical value | We reject H0 | Our data do not follow the Binomial Law |

Unilateral right | Practical value > Critical value | We reject H0 | Our data is above the Binomiale law |

Unilateral left | Practical value < Critical value | We reject H0 | Our data is below the Binomial Law |

## p-value interpretation

Result | Statistical Conclusion | Practical Conclusion |
---|---|---|

p-value > α | We retain H0 | Our data follow the Binomial Law |

p-Value ≤ α | We reject H0 | Our data do not follow the Binomial law. |