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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 :

Avec :

• 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
Called binomial coefficient « 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 :
$\frac{n!}{k!×\left(n-k\right)!}$

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.

## Interpretation practical value / critical value

test directionResultStatistical conclusionPractical conclusion
BilateralPractical value > + Critical value or Practical value < - Critical valueWe reject H0Our data do not follow the Binomial Law
Unilateral rightPractical value > Critical valueWe reject H0Our data is above the Binomiale law
Unilateral leftPractical value < Critical valueWe reject H0Our data is below the Binomial Law

## p-value interpretation

ResultStatistical ConclusionPractical Conclusion
p-value > αWe retain H0Our data follow the Binomial Law
p-Value ≤ αWe reject H0Our data do not follow the Binomial law.