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MOST IMPORTANT SCIENTIFIC DISCOVERIES
Probability Theory. History and essence of scientific discovery
Directory / The most important scientific discoveries “We can assume,” writes V.A. Nikiforovsky, “that probability theory is not a science, but a collection of empirical observations, information has existed for a long time, as long as there is a game of dice. Indeed, an experienced player knew and probably took into account in the game that different rolls of the number of points have different frequencies of occurrence.When throwing three dice, for example, three points can fall in only one way (one point on each die), and four points - in three ways: 2+1+1, 1+2+ 1, 1 + 1 + 2. The elementary concepts of the theory of probability arose, as already mentioned, in connection with the tasks of gambling, processing the results of astronomical observations, tasks of statistics, the practice of insurance companies. Insurance became widespread along with the development of navigation and maritime trade " . Back in the sixteenth century, the eminent mathematicians Tartaglia and Cardano turned to the problems of probability theory in connection with the game of dice and counted the various options for dropping points. Cardano, in his work "On Gambling," gave calculations very close to those obtained later, when the theory of probability had already established itself as a science. The same Cardano was able to calculate in how many ways throwing two or three dice will give one or another number of points. He determined the total number of possible fallouts. In other words, Cardano calculated the probabilities of certain occurrences. However, all the tables and calculations of Tartaglia and Cardano became only material for future science. "The calculus of probabilities, entirely built on exact conclusions, we find for the first time only in Pascal и Farm", says Zeiten. Fermat and Pascal really became the founders of the mathematical theory of probability. Blaise Pascal (1623–1662) was born in Clermont. The entire Pascal family was distinguished by outstanding abilities. As for Blaise himself, from early childhood he showed signs of extraordinary mental development. In 1631, when little Pascal was eight years old, his father moved with all the children to Paris, selling his office, according to the then custom, and investing a large part of his small capital in the Hotel de Ville. Having a lot of free time, Etienne Pascal almost exclusively engaged in the mental education of his son. He himself did a lot of mathematics and liked to gather mathematicians in his house. But, having drawn up a plan for his son's studies, he put aside mathematics until his son improved in Latin. What was the surprise of the father when he saw his son, who independently tried to prove the properties of the triangle. The meetings held at Father Pascal's and with some of his friends took on the character of genuine scholarly meetings. From the age of sixteen, young Pascal also began to take an active part in the classes of the circle. He was already so strong in mathematics that he mastered almost all the methods known at that time, and among the members who made new reports most often, he was one of the first. At the age of sixteen, Pascal wrote a very remarkable treatise on conic sections. However, intensive studies soon undermined Pascal's already poor health. At the age of eighteen, he already constantly complained of a headache, which initially did not pay much attention. But Pascal's health was finally upset during excessive work on the arithmetic machine he invented. The machine invented by Pascal was quite complex in design, and calculation with its help required considerable skill. This explains why it remained a mechanical curiosity that aroused the surprise of contemporaries, but did not enter into practical use. Since the invention of the arithmetic machine by Pascal, his name has become known not only in France, but also abroad. In 1643, Torricelli undertook experiments to lift various liquids in pipes and pumps. Torricelli deduced that the reason for the rise of both water and mercury is the weight of the air column pressing on the open surface of the liquid. These experiments interested Pascal. Knowing that air has weight, he decides to explain the phenomena observed in pumps and pipes by the action of this weight. The main difficulty, however, was to explain the mode of transmission of air pressure. Blaise reasoned as follows: if air pressure is indeed the cause of the phenomena in question, then it follows that the smaller or lower, all other things being equal, the column of air pressing on mercury, the lower the column of mercury in the barometric tube. As a result of the experiment, Pascal showed that the pressure of a liquid spreads uniformly in all directions and that almost all of their other mechanical properties follow from this property of liquids. Further, the scientist found that the pressure of air, in terms of its distribution, is completely similar to the pressure of water. In the field of mathematics, Pascal is primarily known for his contributions to probability theory. As Poisson put it, "the problem of gambling, set before the hard-nosed Jansenist layman, was the origin of the theory of probability." This secular man was the Chevalier de Mere, and the "severe Jansenist" was Pascal. It is believed that de Mere was a gambler. In fact, he was seriously interested in science. Be that as it may, de Mere asked Pascal the following question: how to divide the stark between the players if the game was not over? The solution of this problem did not lend itself to all the mathematical methods known up to that time. Here the question had to be decided, not knowing which of the players could win if the game continued? It is clear that this was a problem that had to be solved on the basis of the degree of probability of winning or losing one or another player. But until then, no mathematician had ever thought of calculating only probable events. It seemed that the problem allowed only a conjectural solution, that is, that it was necessary to divide the bet completely at random, for example, by throwing lots, which determines who should have the final win. It took the genius of Pascal and Fermat to understand that such problems admit of quite definite solutions, and that "probability" is a measurable quantity. Let's say we want to find out what is the probability of drawing a white ball from an urn containing two white balls and one black one. There are three balls in all, and there are twice as many white balls as black ones. It is clear that it is more plausible to assume, when drawn at random, that a white ball will be drawn than a black one. It may just happen that we take out a black ball; but still we can say that the probability of this event is less than the probability of drawing white. By increasing the number of white balls and leaving one black one, it is easy to see that the probability of taking out a black ball will decrease. So, if there were a thousand white balls, and one black ball, and if someone were offered to bet that a black ball would be drawn, and not a white one, then only a crazy or gambler would dare to stake a significant amount in favor of black ball. Having understood the concept of the measurement of probability, it is easy to understand how Pascal solved the problem proposed by de Mere. Obviously, to calculate the probability, you need to know the ratio between the number of cases of favorable events and the number of all possible cases (both favorable and unfavorable). The resulting ratio is the desired probability. So, if there are a hundred white balls, and let's say ten black balls, then there will be one hundred and ten "cases" in all, ten of them in favor of the black balls. Therefore, the probability of drawing a black ball is 10 to 110, or 1 to 11. The two tasks proposed by the Chevalier de Méré are as follows. First: how to find out how many times you need to throw two dice in the hope of getting the highest number of points, that is, twelve; the other is how to distribute the winnings between two players in case of an unfinished game. The first task is comparatively easy: it is necessary to determine how many different combinations of points there can be; only one of these combinations is favorable to the event, all the rest are unfavorable, and the probability is calculated very simply. The second task is much more difficult. Both were solved simultaneously in Toulouse by the mathematician Fermat and in Paris by Pascal. On this occasion, in 1654, a correspondence began between Pascal and Fermat, and, not being personally acquainted, they became best friends. Fermat solved both problems by means of the theory of combinations invented by him. Pascal's solution was much simpler: he proceeded from purely arithmetical considerations. Far from envying Fermat, Pascal, on the contrary, rejoiced at the coincidence of the results and wrote: “From now on, I would like to open my soul to you, I am so glad that our thoughts met. I see that the truth is one and the same in Toulouse and in Paris". Here is Pascal's concise solution. Suppose, says Pascal, that two players are playing and that the payoff is final after one of them wins three games. Suppose that each player's bet is 32 chervonets and that the first has already won two games (he is missing one), and the second has won one (he is missing two). They have one more game to play. If the first one wins it, he will receive the entire amount, that is, 64 chervonets; if the second, each will have two wins, the chances of both will become equal, and in case of termination of the game, each should obviously be given equally. So, if the first one wins, he will receive 64 chervonets. If the second wins, then the first will receive only 32. Therefore, if both agree not to play the upcoming game, then the first has the right to say: I will get 32 chervonets in any case, even if I lose the upcoming game, which we agreed to recognize as the last one. So, 32 chervonets are mine. As for the other 32 - maybe I will win them, maybe you too; so let's split this dubious amount in half. So, if the players disperse without playing the last game, then the first one must be given 48 chervonets, or s, the whole amount, the second 16 chervonets, or, from which it can be seen that the chances of the first of them to win are three times greater than the second (and not doubled, as one might think on a superficial basis). A little later than Pascal and Fermat turned to the theory of probability Heingens Christian Huygens (1629–1695). He was informed of their progress in the new field of mathematics. Huygens writes the work "On the calculations in gambling". It first appeared as an appendix to the "Mathematical Etudes" of his teacher Schooten in 1657. Until the beginning of the eighteenth century, "Etudes ..." remained the only guide to the theory of probability and had a great influence on many mathematicians. In a letter to Schooten, Huygens remarked: "I believe that upon careful study of the subject, the reader will notice that he is dealing not only with a game, but that the foundations of a very interesting and deep theory are being laid here." Such a statement suggests that Huygens deeply understood the essence of the subject under consideration. It was Huygens who introduced the concept of mathematical expectation and applied it to solving the problem of splitting the bet with a different number of players and a different number of missing games and to problems related to throwing dice. Mathematical expectation became the first major probabilistic concept. In the XNUMXth century, the first works on statistics appeared. They are mainly devoted to calculating the distribution of births of boys and girls, the mortality of people of different ages, the required number of people of different professions, the amount of taxes, national wealth, and income. At the same time, methods related to the theory of probability were used. Such work contributed to its development. Halley, when compiling a table of mortality in 1694, averaged observational data by age groups. In his opinion, the existing deviations are "apparently due to chance" that the data would not have sharp deviations with a "much larger" number of years of observation. Probability theory has a wide range of applications. By means of it, astronomers, for example, determine the probable errors of observations, and artillerymen calculate the probable number of shells that could fall in a certain area, and insurance companies - the amount of premiums and interest paid on life and property insurance. And in the second half of the nineteenth century, the so-called "statistical physics" was born, which is a branch of physics that specifically studies the huge collections of atoms and molecules that make up any substance, from the point of view of probabilities. Author: Samin D.K.
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