Bordeaux is a port city on the Garonne in the Gironde department in Southwestern France. The municipality of Bordeaux proper has a population of 252,040. Together with its suburbs and satellite towns, Bordeaux is the centre of the Bordeaux Métropole. With 1,195,335 in the metropolitan area, it is the sixth-largest in France, after Paris, Lyon and Lille, it is the capital of the Nouvelle-Aquitaine region, as well as the prefecture of the Gironde department. Its inhabitants are called "Bordelais" or "Bordelaises"; the term "Bordelais" may refer to the city and its surrounding region. Being at the center of a major wine-growing and wine-producing region, Bordeaux remains a prominent powerhouse and exercises significant influence on the world wine industry although no wine production is conducted within the city limits, it is home to the world's main wine fair and the wine economy in the metro area takes in 14.5 billion euros each year. Bordeaux wine has been produced in the region since the 8th century.
The historic part of the city is on the UNESCO World Heritage List as "an outstanding urban and architectural ensemble" of the 18th century. After Paris, Bordeaux has the highest number of preserved historical buildings of any city in France. In historical times, around 567 BC it was the settlement of a Celtic tribe, the Bituriges Vivisci, who named the town Burdigala of Aquitanian origin; the name Bourde is still the name of a river south of the city. In 107 BC, the Battle of Burdigala was fought by the Romans who were defending the Allobroges, a Gallic tribe allied to Rome, the Tigurini led by Divico; the Romans were defeated and their commander, the consul Lucius Cassius Longinus, was killed in the action. The city fell under Roman rule around its importance lying in the commerce of tin and lead, it became capital of Roman Aquitaine, flourishing during the Severan dynasty. In 276 it was sacked by the Vandals. Further ravage was brought by the same Vandals in 409, the Visigoths in 414, the Franks in 498, beginning a period of obscurity for the city.
In the late 6th century, the city re-emerged as the seat of a county and an archdiocese within the Merovingian kingdom of the Franks, but royal Frankish power was never strong. The city started to play a regional role as a major urban center on the fringes of the newly founded Frankish Duchy of Vasconia. Around 585, Gallactorius is fighting the Basque people; the city was plundered by the troops of Abd er Rahman in 732 after they stormed the fortified city and overwhelmed the Aquitanian garrison. Duke Eudes mustered a force ready to engage the Umayyads outside Bordeaux taking them on in the Battle of the River Garonne somewhere near the river Dordogne; the battle had a high death toll. Although Eudes was defeated here, he saved part of his troops and kept his grip on Aquitaine after the Battle of Poitiers. In 735, the Aquitanian duke Hunald led a rebellion after his father Eudes's death, at which Charles responded by sending an expedition that captured and plundered Bordeaux again, but did not retain it for long.
The following year, the Frankish commander descended again to Aquitaine, but clashed in battle with the Aquitanians and left to take on hostile Burgundian authorities and magnates. In 745, Aquitaine faced yet another expedition by Charles's sons Pepin and Carloman, against Hunald, the Aquitanian princeps strong in Bordeaux. Hunald was defeated, his son Waifer replaced him, confirmed Bordeaux as the capital city. During the last stage of the war against Aquitaine, it was one of Waifer's last important strongholds to fall to King Pepin the Short's troops. Next to Bordeaux, Charlemagne built the fortress of Fronsac on a hill across the border with the Basques, where Basque commanders came over to vow loyalty to him. In 778, Seguin was appointed count of Bordeaux undermining the power of the Duke Lupo, leading to the Battle of Roncevaux Pass that year. In 814, Seguin was made Duke of Vasconia, but he was deposed in 816 for failing to suppress or sympathise with a Basque rebellion. Under the Carolingians, sometimes the Counts of Bordeaux held the title concomitantly with that of Duke of Vasconia.
They were meant to keep the Basques in check and defend the mouth of the Garonne from the Vikings when the latter appeared c. 844 in the region of Bordeaux. In Autumn 845, count Seguin II marched on the Vikings, who were assaulting Bordeaux and Saintes, but he was captured and executed. No bishops were mentioned during part of the 9th in Bordeaux. From the 12th to the 15th century, Bordeaux regained importance following the marriage of Duchess Eléonore of Aquitaine with the French-speaking Count Henri Plantagenet, born in Le Mans, who became, within months of their wedding, King Henry II of England; the city flourished due to the wine trade, the cathedral of St. André was built, it was the capital of an independent state under Edward, the Black Prince, but in the end, after the Battle of Castillon, it was annexed by France which extended its territory. The Château Trompette and the Fort du Hâ, built by Charles VII of France, were the symbols of the new domination, which however deprived the city of its wealth by halting the wine commerce with England.
In 1462, Bordeaux obtained a parliament, but regained importance only in the 16th century when it became the centre of the distribution of sugar and slaves from the West Indies along with the traditional wine. Bordeaux adhered to the Fronde
Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to
Probability is the measure of the likelihood that an event will occur. See glossary of probability and statistics. Probability quantifies as a number between 0 and 1, loosely speaking, 0 indicates impossibility and 1 indicates certainty; the higher the probability of an event, the more it is that the event will occur. A simple example is the tossing of a fair coin. Since the coin is fair, the two outcomes are both probable; these concepts have been given an axiomatic mathematical formalization in probability theory, used in such areas of study as mathematics, finance, science, artificial intelligence/machine learning, computer science, game theory, philosophy to, for example, draw inferences about the expected frequency of events. Probability theory is used to describe the underlying mechanics and regularities of complex systems; when dealing with experiments that are random and well-defined in a purely theoretical setting, probabilities can be numerically described by the number of desired outcomes divided by the total number of all outcomes.
For example, tossing a fair coin twice will yield "head-head", "head-tail", "tail-head", "tail-tail" outcomes. The probability of getting an outcome of "head-head" is 1 out of 4 outcomes, or, in numerical terms, 1/4, 0.25 or 25%. However, when it comes to practical application, there are two major competing categories of probability interpretations, whose adherents possess different views about the fundamental nature of probability: Objectivists assign numbers to describe some objective or physical state of affairs; the most popular version of objective probability is frequentist probability, which claims that the probability of a random event denotes the relative frequency of occurrence of an experiment's outcome, when repeating the experiment. This interpretation considers probability to be the relative frequency "in the long run" of outcomes. A modification of this is propensity probability, which interprets probability as the tendency of some experiment to yield a certain outcome if it is performed only once.
Subjectivists assign numbers per subjective probability. The degree of belief has been interpreted as, "the price at which you would buy or sell a bet that pays 1 unit of utility if E, 0 if not E." The most popular version of subjective probability is Bayesian probability, which includes expert knowledge as well as experimental data to produce probabilities. The expert knowledge is represented by some prior probability distribution; these data are incorporated in a likelihood function. The product of the prior and the likelihood, results in a posterior probability distribution that incorporates all the information known to date. By Aumann's agreement theorem, Bayesian agents whose prior beliefs are similar will end up with similar posterior beliefs. However, sufficiently different priors can lead to different conclusions regardless of how much information the agents share; the word probability derives from the Latin probabilitas, which can mean "probity", a measure of the authority of a witness in a legal case in Europe, correlated with the witness's nobility.
In a sense, this differs much from the modern meaning of probability, which, in contrast, is a measure of the weight of empirical evidence, is arrived at from inductive reasoning and statistical inference. The scientific study of probability is a modern development of mathematics. Gambling shows that there has been an interest in quantifying the ideas of probability for millennia, but exact mathematical descriptions arose much later. There are reasons for the slow development of the mathematics of probability. Whereas games of chance provided the impetus for the mathematical study of probability, fundamental issues are still obscured by the superstitions of gamblers. According to Richard Jeffrey, "Before the middle of the seventeenth century, the term'probable' meant approvable, was applied in that sense, unequivocally, to opinion and to action. A probable action or opinion was one such as sensible people would undertake or hold, in the circumstances." However, in legal contexts especially,'probable' could apply to propositions for which there was good evidence.
The sixteenth century Italian polymath Gerolamo Cardano demonstrated the efficacy of defining odds as the ratio of favourable to unfavourable outcomes. Aside from the elementary work by Cardano, the doctrine of probabilities dates to the correspondence of Pierre de Fermat and Blaise Pascal. Christiaan Huygens gave the earliest known scientific treatment of the subject. Jakob Bernoulli's Ars Conjectandi and Abraham de Moivre's Doctrine of Chances treated the subject as a branch of mathematics. See Ian Hacking's The Emergence of Probability and James Franklin's The Science of Conjecture for histories of the early development of the concept of mathematical probability; the theory of errors may be traced back to Roger Cotes's Opera Miscellanea, but a memoir prepared by Thomas Simpson in 1755 first applied the theory to the discussion of errors of observation. The reprint of this memoir lays down the axioms that positive and negative errors are probable, that certain assignable limits define the range of all errors.
Simpson discusses c
Maxima and minima
In mathematical analysis, the maxima and minima of a function, known collectively as extrema, are the largest and smallest value of the function, either within a given range or on the entire domain of a function. Pierre de Fermat was one of the first mathematicians to propose a general technique, for finding the maxima and minima of functions; as defined in set theory, the maximum and minimum of a set are the greatest and least elements in the set, respectively. Unbounded infinite sets, such as the set of real numbers, have no maximum. A real-valued function f defined on a domain X has a global maximum point at x∗ if f ≥ f for all x in X. Similarly, the function has a global minimum point at x∗ if f ≤ f for all x in X; the value of the function at a maximum point is called the maximum value of the function and the value of the function at a minimum point is called the minimum value of the function. Symbolically, this can be written as follows: x 0 ∈ X is a global maximum point of function f: X → R if f ≥ f.
For global minimum point. If the domain X is a metric space f is said to have a local maximum point at the point x∗ if there exists some ε > 0 such that f ≥ f for all x in X within distance ε of x∗. The function has a local minimum point at x∗ if f ≤ f for all x in X within distance ε of x∗. A similar definition can be used when X is a topological space, since the definition just given can be rephrased in terms of neighbourhoods. Mathematically, the given definition is written as follows: Let be a metric space and function f: X → R. X 0 ∈ X is a local maximum point of function f if such that d X < ε ⟹ f ≥ f. For a local minimum point. In both the global and local cases, the concept of a strict extremum can be defined. For example, x∗ is a strict global maximum point if, for all x in X with x ≠ x∗, we have f > f, x∗ is a strict local maximum point if there exists some ε > 0 such that, for all x in X within distance ε of x∗ with x ≠ x∗, we have f > f. Note that a point is a strict global maximum point if and only if it is the unique global maximum point, for minimum points.
A continuous real-valued function with a compact domain always has a maximum point and a minimum point. An important example is a function. Finding global maxima and minima is the goal of mathematical optimization. If a function is continuous on a closed interval by the extreme value theorem global maxima and minima exist. Furthermore, a global maximum either must be a local maximum in the interior of the domain, or must lie on the boundary of the domain. So a method of finding a global maximum is to look at all the local maxima in the interior, look at the maxima of the points on the boundary, take the largest one; the most important, yet quite obvious, feature of continuous real-valued functions of a real variable is that they decrease before local minima and increase afterwards for maxima. A direct consequence of this is the Fermat's theorem, which states that local extrema must occur at critical points. One can distinguish whether a critical point is a local maximum or local minimum by using the first derivative test, second derivative test, or higher-order derivative test, given sufficient differentiability.
For any function, defined piecewise, one finds a maximum by finding the maximum of each piece separately, seeing which one is largest. The function x2 has a unique global minimum at x = 0; the function x3 has maxima. Although the first derivative is 0 at x = 0, this is an inflection point; the function x. The function x−x has a unique global maximum over the positive real numbers at x = 1/e; the function x3/3 − x has first derivative x2 − second derivative 2x. Setting the first derivative to 0 and solving for x gives stationary points at −1 and +1. From the sign o
François Viète, Seigneur de la Bigotière, was a French mathematician whose work on new algebra was an important step towards modern algebra, due to its innovative use of letters as parameters in equations. He was a lawyer by trade, served as a privy councillor to both Henry III and Henry IV of France. Viète was born at Fontenay-le-Comte in present-day Vendée, his grandfather was a merchant from La Rochelle. His father, Etienne Viète, was a notary in Le Busseau, his mother was the aunt of Barnabé Brisson, a magistrate and the first president of parliament during the ascendancy of the Catholic League of France. Viète went to a Franciscan school and in 1558 studied law at Poitiers, graduating as a Bachelor of Laws in 1559. A year he began his career as an attorney in his native town. From the outset, he was entrusted with some major cases, including the settlement of rent in Poitou for the widow of King Francis I of France and looking after the interests of Mary, Queen of Scots. In 1564, Viète entered the service of Antoinette d’Aubeterre, Lady Soubise, wife of Jean V de Parthenay-Soubise, one of the main Huguenot military leaders and accompanied him to Lyon to collect documents about his heroic defence of that city against the troops of Jacques of Savoy, 2nd Duke of Nemours just the year before.
The same year, at Parc-Soubise, in the commune of Mouchamps in present-day Vendée, Viète became the tutor of Catherine de Parthenay, Soubise's twelve-year-old daughter. He taught her science and mathematics and wrote for her numerous treatises on astronomy and trigonometry, some of which have survived. In these treatises, Viète used decimal numbers and he noted the elliptic orbit of the planets, forty years before Kepler and twenty years before Giordano Bruno's death. John V de Parthenay presented him to King Charles IX of France. Viète wrote a genealogy of the Parthenay family and following the death of Jean V de Parthenay-Soubise in 1566, his biography. In 1568, Lady Soubise, married her daughter Catherine to Baron Charles de Quellenec and Viète went with Lady Soubise to La Rochelle, where he mixed with the highest Calvinist aristocracy, leaders like Coligny and Condé and Queen Jeanne d’Albret of Navarre and her son, Henry of Navarre, the future Henry IV of France. In 1570, he refused to represent the Soubise ladies in their infamous lawsuit against the Baron De Quellenec, where they claimed the Baron was unable to provide an heir.
In 1571, he enrolled as an attorney in Paris, continued to visit his student Catherine. He lived in Fontenay-le-Comte, where he took on some municipal functions, he began publishing his Universalium inspectionum ad canonem mathematicum liber singularis and wrote new mathematical research by night or during periods of leisure. He was known to dwell on any one question for up to three days, his elbow on the desk, feeding himself without changing position. In 1572, Viète was in Paris during the St. Bartholomew's Day massacre; that night, Baron De Quellenec was killed after having tried to save Admiral Coligny the previous night. The same year, Viète met Françoise de Rohan, Lady of Garnache, became her adviser against Jacques, Duke of Nemours. In 1573, he became a councillor of the Parliament of Brittany, at Rennes, two years he obtained the agreement of Antoinette d'Aubeterre for the marriage of Catherine of Parthenay to Duke René de Rohan, Françoise's brother. In 1576, duc de Rohan took him under his special protection, recommending him in 1580 as "maître des requêtes".
In 1579, Viète printed his canonem mathematicum. A year he was appointed maître des requêtes to the parliament of Paris, committed to serving the king; that same year, his success in the trial between the Duke of Nemours and Françoise de Rohan, to the benefit of the latter, earned him the resentment of the tenacious Catholic League. Between 1583 and 1585, the League persuaded Henry III to release Viète, Viète having been accused of sympathy with the Protestant cause. Henry of Navarre, at Rohan's instigation, addressed two letters to King Henry III of France on March 3 and April 26, 1585, in an attempt to obtain Viète's restoration to his former office, but he failed. Vieta retired with François de Rohan, he spent four years devoted to mathematics. In 1589, Henry III took refuge in Blois, he commanded the royal officials to be at Tours before 15 April 1589. Viète was one of the first, he deciphered other enemies of the king. He had arguments with the classical scholar Joseph Juste Scaliger. Viète triumphed against him in 1590.
After the death of Henry III, Vieta became a Privy Councillor to Henry of Navarre, now Henry IV. He was appreciated by the king. Viète was given the position of councillor of the parlement at Tours. In 1590, Viète discovered the key to a Spanish cipher, consisting of more than 500 characters, this meant that all dispatches in that language which fell into the hands of the French could be read. Henry IV published a letter from Commander Moreo to the king of Spain; the contents of this letter, read by Viète, revealed that the head of the League in France, the Duke of Mayenne, planned to become king in place of Henry IV. This publication led to the settlement of the Wars of Religion; the king of Spain accused Viète of having used magical powers. In 1593, Viète published his arguments against Scaliger. Beginning in 1594, he was appointed deciphering the enemy's secret codes. In 1582, Pope Gregory X
Castres is a commune, arrondissement capital in the Tarn department and Occitanie region in southern France. It lies in the former French province of Languedoc. Castres is the fourth largest industrial centre of the predominantly rural Midi-Pyrénées région and the largest in that part of Languedoc lying between Toulouse and Montpellier. Castres is noted for being the birthplace of the famous socialist leader Jean Jaurès and home to the important Goya Museum of Spanish painting. In 1831, the population of Castres was 12,032. One of the few industrial towns in the region of Albigeois, the population of the commune proper grew to 19,483 in 1901, 34,126 by 1954. However, with the decline of its industries, population growth diminished. Albi surpassed Castres as the most populous metropolitan area of Tarn; the population of Castres is now stagnating: after small growth in the 1970s and 1980s, it registered zero growth in the 1990s. Castres is located at an altitude of 172 metres above sea level, it is located 45 km south-southeast of Albi, the préfecture of Tarn, 79 km east of Toulouse, the capital of Midi-Pyrénées.
Castres is intersected from north to south by the Durenque rivers. The Thoré forms most of the commune's south-eastern border flows into the Agout, which forms part of its western border. Between 1790 and 1797 Castres was the prefecture of Tarn. Since 2001, the mayor of Castres has been Pascal Bugis, who defeated the socialist mayor in the 2001 election after a campaign focused on the bad records of the socialist mayor on fighting crime, the high level of insecurity in the town. Castres has teamed up with the nearby town of Mazamet and the independent suburbs and villages in between to create the Greater Castres-Mazamet Council, established in January 2000; the Greater Castres-Mazamet Council groups 16 independent communes, with a total population of 79,988 inhabitants, 54% of these living in the commune of Castres proper, 13% in the commune of Mazamet, the rest in the communes in between. The Greater Castres-Mazamet Council was created in order to better coordinate transport, infrastructure and economic policies between the communes of the area.
The current president of the Greater Castres-Mazamet Council is Jacques Limouzy, former mayor of Castres before 1995, who became president in 2001. The name of the town comes from Latin castrum, means "fortified place". Castres grew up round the Benedictine abbey of Saint Benoît, believed to have been founded in AD 647 on the site of an old Roman fort. Castres became an important stop on the international pilgrimage routes to Santiago de Compostela in Spain because its abbey-church, built in the 9th century, was keeping the relics of Saint Vincent, the renowned martyr of Spain, it was a place of some importance as early as the 12th century, ranked as the second town of the Albigeois behind Albi. Despite the decline of its abbey, which in 1074 came under the authority of Saint Victor abbey in Marseille, Castres was granted a liberal charter in the 12th century by the famous Trencavel family, viscounts of Albi. Resulting from the charter, Castres was ruled by a college of consuls. During the Albigensian Crusade it surrendered of its own accord to Simon de Montfort, thus entered into the kingdom of France in 1229.
In 1317, Pope John XXII established the bishopric of Castres. In 1356, the town of Castres was raised to a countship by King John II of France. However, the town suffered from the Black Plague in 1347-1348 from the Black Prince of England and the Free Companies who laid waste the country during the Hundred Years' War. By the late 14th century Castres entered a period of sharp decline. In 1375, there were only 4,000 inhabitants left in only half the figure from a century before. Following the confiscation of the possessions of Jacques d'Armagnac, duke of Nemours, to which the countship of Castres had passed, it was bestowed in 1476 by King Louis XI on Boffille de Juge, an Italian nobleman and adventurer serving as a diplomat for Louis XI, but the appointment led to so much disagreement that the countship was united to the crown by King Francis I in 1519. Around 1560, the majority of the population of Castres converted to Protestantism. In the wars of the latter part of the 16th century the inhabitants sided with the Protestant party, fortified the town, established an independent republic.
Castres was one of the largest Protestant strongholds in southern France, along with Montauban and La Rochelle. Henry of Navarre, leader of the Protestant party, who became King Henry IV of France, stayed in Castres in 1585; the Protestants of Castres were brought to terms, however, by King Louis XIII in 1629, Richelieu came himself to Castres to have its fortifications dismantled. Nonetheless, after these religious wars, the town, now in peace, enjoyed a period of rapid expansion. Business and traditional commercial activities revived, in particular fur and leather-dressing and above all wool trade. Culture flourished anew, with the founding of the Academy of Castres in 1648. Castres was turned by the Cath
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms; these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of these outcomes is called an event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, stochastic processes, which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion. Although it is not possible to predict random events, much can be said about their behavior. Two major results in probability theory describing such behaviour are the law of large numbers and the central limit theorem.
As a mathematical foundation for statistics, probability theory is essential to many human activities that involve quantitative analysis of data. Methods of probability theory apply to descriptions of complex systems given only partial knowledge of their state, as in statistical mechanics. A great discovery of twentieth-century physics was the probabilistic nature of physical phenomena at atomic scales, described in quantum mechanics; the mathematical theory of probability has its roots in attempts to analyze games of chance by Gerolamo Cardano in the sixteenth century, by Pierre de Fermat and Blaise Pascal in the seventeenth century. Christiaan Huygens published a book on the subject in 1657 and in the 19th century, Pierre Laplace completed what is today considered the classic interpretation. Probability theory considered discrete events, its methods were combinatorial. Analytical considerations compelled the incorporation of continuous variables into the theory; this culminated on foundations laid by Andrey Nikolaevich Kolmogorov.
Kolmogorov combined the notion of sample space, introduced by Richard von Mises, measure theory and presented his axiom system for probability theory in 1933. This became the undisputed axiomatic basis for modern probability theory. Most introductions to probability theory treat discrete probability distributions and continuous probability distributions separately; the measure theory-based treatment of probability covers the discrete, continuous, a mix of the two, more. Consider an experiment that can produce a number of outcomes; the set of all outcomes is called the sample space of the experiment. The power set of the sample space is formed by considering all different collections of possible results. For example, rolling an honest die produces one of six possible results. One collection of possible results corresponds to getting an odd number. Thus, the subset is an element of the power set of the sample space of die rolls; these collections are called events. In this case, is the event that the die falls on some odd number.
If the results that occur fall in a given event, that event is said to have occurred. Probability is a way of assigning every "event" a value between zero and one, with the requirement that the event made up of all possible results be assigned a value of one. To qualify as a probability distribution, the assignment of values must satisfy the requirement that if you look at a collection of mutually exclusive events, the probability that any of these events occurs is given by the sum of the probabilities of the events; the probability that any one of the events, or will occur is 5/6. This is the same as saying that the probability of event is 5/6; this event encompasses the possibility of any number except five being rolled. The mutually exclusive event has a probability of 1/6, the event has a probability of 1, that is, absolute certainty; when doing calculations using the outcomes of an experiment, it is necessary that all those elementary events have a number assigned to them. This is done using a random variable.
A random variable is a function that assigns to each elementary event in the sample space a real number. This function is denoted by a capital letter. In the case of a die, the assignment of a number to a certain elementary events can be done using the identity function; this does not always work. For example, when flipping a coin the two possible outcomes are "heads" and "tails". In this example, the random variable X could assign to the outcome "heads" the number "0" and to the outcome "tails" the number "1". Discrete probability theory deals with events. Examples: Throwing dice, experiments with decks of cards, random walk, tossing coins Classical definition: Initially the probability of an event to occur was defined as the number of cases favorable for the event, over the number of total outcomes possible in an equiprobable sample space: see Classical definition of probability. For example, if the event is "occurrence of an number when a die is