When we talk about probabilities, it is sometimes difficult or impossible not to mention mathematics. What does it mean by probabilities? What really differentiates mathematics from probability? What relationships do they have? How do they solve our problems on a daily basis? Is it really possible to dissociate mathematics from probability theory? **Franck Peltier** is here to help us tell the difference.

## Mathematics and probability: useless confusion!

Every day, mathematics is mobilized to solve this or that situation more or less complex of everyday life. A life without mathematics is therefore impossible because mathematics is part of the everyday life of all individuals. Hence a certain confusion with certain disciplines constitute it as arithmetic, algebra, analysis, geometry. Indeed, each of these disciplines proceeds from a method in its own right.

## The probabilities: what is it really?

The probabilities are a dismemberment of mathematics. Like the aforementioned disciplines, they are an equally important branch of maths in that they make it possible to calculate the eventuality of an event, and the frequency with which it can be realized. However, probabilities do not produce results at once. The probabilities study according to a very rigorous method the evolution of certain natural and social phenomena of which most of the results depend or are affected by the hazards.

Probabilities only make sense with the observation of the law of large numbers: if we repeat an experiment several times, the occurrence frequency of the event becomes close to its probability of occurrence.

Various studies on probabilities have led to many developments since the XVIIIe century thanks to the study of the random and partly unpredictable aspect of certain phenomena, especially games of chance. These led mathematicians to develop theories that later had implications in fields as diverse as technology, meteorology, finance or chemistry. Technological development that continues to evolve over the years and push the limits even further. That’s why Franck Peltier is fond of those disciplines.

## What is the usefulness of probabilities?

Connected to mathematics, of which they are for the most part indissociable, the probabilities make it possible to establish the regularities or global specificities of certain phenomena in order to better understand their mode of operation. In some cases, the use of probabilities favors a more or less precise prediction of events. This is the true utility of probabilities. To better understand them, we must master the basic principles, but also the vocabulary.

Probabilities play an important role in science, at least since statistical mechanics. They can be applied in many disciplines, including biology, economics and human sciences. Finally, they have also been used in philosophy, for example, to imagine tools that can measure the credibility of scientific theories. As for the daily life, probabilities can be applied in both science and technology, especially in some cases where there are several alternatives. But it is impossible to decide which one is profitable, whether because of a lack of information, forecasting resources or variability in the situation

## How to understand probabilities?

To understand probabilities, it is essential to know precisely the language related to them. It took Franck Peltier and anyone willing to learn some time to master it. Indeed, this branch of mathematics has a vocabulary of its own. This vocabulary consists of the following words:

### Universe

For the probability professional, the universe is a contingency system. Also, in terms of probabilities, to analyze a phenomenon, as random as it may seem, it is essential to delimit it. What phenomenon is it? What are the different eventualities for it to happen? The probabilist will define and fix the framework of the possible situations in the short, medium or long term. It is the set of eventualities that surround the occurrence of an event called universe. In probability, the universe is represented by the symbol Ω.

### Events

Once the frame of the phenomenon to observe fixed, the probabilist will be interested in the eventual events which can occur. The event is therefore considered an integral part of the universe.

Depending on the case, it can be a basic event or a general event depending on the objectives of the study to be conducted. For a better use of the probabilities, the event is often materialized by a letter to which an index can often or not be added according to the cases. Any event contrary to an elementary or general event constitutes the rest of the universe.

### The Law of Probability proper

By law of probability, we must understand the rule that links each event to its probabilities. To define it, it’s simple. Indeed, it is necessary to associate each event with its probability. The principle of this law is practical and adapts to most fields of application of probabilities.

## What are the applications of probability? Franck Peltier explains !

Probabilities are present in everyday life. It is a fact ! Also, their application espouses most situations of normal life. Today, probabilities are used in many areas. Without appearing exhaustive, we can quote:

### Natural phenomena

One of the areas where probabilities are very popular is the weather. In fact, to obtain meteorological data of a certain accuracy, meteorologists observe and analyze the regularities of certain events to predict what the weather will be in one corner of the world. Thus, it becomes easier to plan holidays alone or with family or to plan a trip, a trip with friends like Franck Peltier.

### Games of chance

Gamblers adhere to the probabilities to optimize or capitalize their winnings. Whether it’s roulette, blackjack, the casino, the idea is to predict in advance which shot will be the most beneficial to play. In a die game for example, at each roll, any player has only 6 possible choices, one on each side of the die. It has within reach six possibilities. The universe in this case is as follows: Ω = {1, 2, 3, 4, 5, 6}.In addition, obtaining a precise number of possible choices constitutes a basic event. Just like to obtain for example a number lower than 5 which puts at the disposal of the player four eventualities. Consequently, any event that is part of the rest of the universe after the elementary or general event has occurred will be described as an adverse event. In general, the goal of any player who uses probabilities is to estimate their chances and to minimize their risk of loss.

### Politics

The political field is also a highly publicized field of probabilities. Whether it is a local election (municipal, legislative, senatorial) or a national election (the presidential election), institutes and other specialists rely on probabilities to assess the odds of such or candidate or for a political party to win or not to win the election.

More concretely for an election by absolute majority with 3 candidates for example, the latter offers three eventualities. Indeed, each candidate has the opportunity to win the election. The universe consists of Ω = {Candidate No. 1, Candidate No. 2, Candidate 3}. In the same way, the simplest event is that a candidate wins, so the others have lost. The victory of one of the candidates inevitably leading to the defeat of the others so two contrary events.

### The actuarial

The purpose of the actuarial profession is to identify, study and take charge of the risks and financial consequences inherent in the fields of insurance, social security and financial investments. Also, in the actuarial sector, various processes are implemented to put in place an effective and efficient insurance and annuity system by adequately preventing certain risks that individuals or businesses face on a daily basis.

In general, it is a hazard that characterizes most services offered in the insurance, security and social protection sector, or the world of financial investments (the stock market, etc.). Consequently, the use of probability theory and a mathematical system are essential for the rational management of the risks that surround them.

Actuaries help individuals and legal entities to better organize their lives by reducing certain risks related to their daily activities. Without being exhaustive, they may be:

• Risks related to professional and social life (retirement, illness, disability, unemployment, death, etc.).

• Risks relating to financial activities (loss of property, investments, etc.).

The objective of the actuary’s work is to prevent the financial consequences resulting from these phenomena or events that may be difficult to prevent in advance because of their uncertain nature. In some cases, the financial consequences can be harmful for the individual or his family. For example, the death of a person, beyond the sadness it entails for his family in general, and his family, in particular, is a phenomenon that can be destabilizing financially, especially when the missing person occupied an important place in the management of the household. The implementation of the death insurance has taken into account these different assumptions to relieve relatives of the financial consequences (lower household income, interruption of schooling of some children, cost of funerals) inherent in the disappearance of the third.

Companies, actuaries participate in their financial security. To do this, he designs and implements a meticulous program to protect the company’s assets against risks that could hinder the smooth running of his business and thus his survival in the business world.

### Finance

According to Franck Peltier, the probability theory provides a mathematical framework for the description of chance and variability, as well as for reasoning in an uncertain universe. It can be defined as a coherent number of concepts, methods and results that are involved in a wide range of fields of science and technology. These concepts also can be applied in finance market.

Recently, the development of financial sector included the use of the most complicated mathematical tools. During the 2008 financial crisis, the mathematization of finance has been strongly criticized, both for its incredible complexity and for its weak anchoring in the concrete economic reality. But probabilities turned out to be very useful in terms of predicting the variations of the finance market. Indeed, the mathematical nature of probabilities guarantees objectivity and rationality.

Therefore it can be concluded that the predictive character of mathematical models in finance is extremely strong and the possibilities provided by mathematics always make sense.

## The difficulty of dissociating mathematics from probabilities in practice

In the analysis of several phenomena by probabilities, the use of mathematics is almost always inevitable. The analysis and theory of probabilities are at the heart of major mathematical questions. The theory of probabilities borrows from mathematics their methodological rigor, a part of their language to study the random aspect and the evolution of certain natural and social processes. The calculation of probabilities is based on the logical reasoning and the rigor of the existing mathematical processes. Therefore, it is difficult to make probability without mobilizing mathematics. We can thank Franck Peltier for making this clear for us.