Schoolboy about the theory of probability - Lyutikas V.S. Features of studying the foundations of probability theory in the school course of mathematics

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MINISTRY OF EDUCATION OF THE REPUBLIC OF BELARUS

ESTABLISHMENT OF EDUCATION "BELARUSIAN STATE PEDAGOGICAL

UNIVERSITY NAMED AFTER M. TANK "

FACULTY OF PHYSICAL AND MATHEMATICAL

DEPARTMENT OF METHODS OF TEACHING MATHEMATICS

THE THEORY OF PROBABILITY IN THE SCHOOL COURSE OF MATHEMATICS

Minsk, 2016

INTRODUCTION

The question of improving mathematical education in Russian schools was raised in the early 60s of the XX century by outstanding mathematicians B.V. Gnedenko, A.N. Kolmogorov, I.I. Kikoin, A.I. Markushevich, A. Ya. Khinchin. B.V. Gnedenko wrote: “The question of introducing elements of probabilistic and statistical knowledge into the school curriculum of mathematics has long ripened and does not tolerate further delay. The laws of rigid determination, the study of which our school education is wholly focused on, only unilaterally reveal the essence of the world around us. The random nature of many phenomena of reality is beyond the attention of our students. As a result, their ideas about the nature of many natural and social processes are one-sided and inadequate to modern science. It is necessary to acquaint them with statistical laws that reveal the multifaceted connections of the existence of objects and phenomena. " IN AND. Levin wrote: “… The statistical culture necessary for… activity must be brought up from an early age. It is no coincidence that much attention is paid to this in developed countries: students get acquainted with the elements of probability theory and statistics from the very first school years and throughout their education they acquire probabilistic and statistical approaches to the analysis of common situations encountered in everyday life. " With the reform of the 1980s, elements of the theory of probability and statistics were included in the curricula of specialized classes, in particular, in physics and mathematics and natural sciences, as well as in an optional course in the study of mathematics. Taking into account the urgent need for the development of individual qualities of students' thinking, the author's developments of optional courses in probability theory appear. An example of this can be the course of N.N. Avdeeva on statistics for grades 7 and 9 and a course on elements of mathematical statistics for grade 10 of secondary school. In grade 10, tests were carried out, the results of which, as well as observations of teachers and a survey of students, showed that the proposed material was quite accessible to students, aroused great interest in them, showing the specific application of mathematics to solving practical problems of science and technology. The process of introducing elements of the theory of probability into the compulsory course of school mathematics turned out to be a very difficult matter. There is an opinion that for the assimilation of the principles of the theory of probability, a preliminary stock of ideas, perceptions, habits is necessary, which are fundamentally different from those that are developed in schoolchildren during traditional teaching within the framework of familiarization with the patterns of strictly conditioning phenomena. Therefore, according to a number of educators - mathematicians, the theory of probability should enter school mathematics as an independent section, which would ensure the formation, systematization and development of ideas about the probabilistic nature of the phenomena of the world around us. Since the study of probability theory in the school course was introduced recently, there are currently problems with the implementation of this material in school textbooks. Also, due to the specificity of this course, the amount of methodological literature is also still small. According to the approaches set forth in the overwhelming majority of literature, it is believed that the main thing in the study of this topic should be the practical experience of students, therefore it is advisable to start learning with questions in which it is required to find a solution to the problem posed against the background of a real situation. In the learning process, one should not prove all the theorems, since a large amount of time is spent on this, while the task of the course is to develop useful skills, and the ability to prove theorems does not belong to such skills. The origin of the theory of probability occurred in search of an answer to the question: how often does this or that event occur in a larger series of trials with random outcomes that occur under the same conditions? When assessing the possibility of an event occurring, we often say: "It is very possible", "It will certainly happen", "It is unlikely", "It will never happen." By purchasing a lottery ticket, you can win, but you can not win; tomorrow in math class you may or may not be called to the blackboard; in the next elections, the ruling party may or may not win. Let's take a look at a simple example. How many people do you think there should be in a certain group for at least two of them to have the same birthdays with a probability of 100% (meaning the day and month without taking into account the year of birth)? This does not mean a leap year, i.e. a year in which there are 365 days. The answer is obvious - there should be 366 people in the group. Now another question: how many people must be in order to find a couple with the same birthday with a probability of 99.9%? At first glance, everything is simple - 364 people. In fact, 68 people are enough! In order to carry out such interesting calculations and make unusual discoveries for ourselves, we will study such a section of mathematics "Probability Theory".

CHAPTER I. PROBABILITY-STATISTICAL LINE IN BASIC SCHOOL COURSE OF MATHEMATICS

1.1 Statistical thinking and school mathematics education

Each era has its own requirements for mathematics and mathematics education. At the present time, the voices of methodologists are becoming more and more loud, who advocate strengthening the probabilistic - statistical line in the school mathematics course, starting from the lower grades of secondary school. But many teachers of mathematics for a long time have not faced questions of combinatorics, probability theory, statistics, that is, with everything that is included in the probabilistic-statistical direction of mathematics. They need to expand their knowledge of advanced issues. The most authoritarian researcher in our country in the field of probability theory and mathematical statistics was Boris Vladimirovich Gnedenko (1912-1995). He was the author of many articles in the journal Mathematics in School.

What and how to teach in school, apparently, will always belong to the number of eternal problems that constantly arise even after they have been given a solution that is better than the previous one. And this is inevitable, because our scientific knowledge and approaches to explaining the phenomena around us are constantly growing. Undoubtedly, the content of school teaching should change with the progress of science, somewhat lagging behind it and enabling new scientific ideas and concepts to take forms that are psychologically and methodologically acceptable.

However, it would be a gross mistake to believe that the content and nature of the school course of a particular science should be completely determined by the state of the corresponding scientific branch of knowledge and the prevailing ideas about its central concepts. The overwhelming majority of schoolchildren will not become specialists in this field of science. They will include representatives of other scientific interests and practical areas of activity, as well as representatives of the liberal professions - writers, artists, artists. That is why it is necessary for all students to obtain information at school about established scientific concepts and acquire a solid foundation of scientific knowledge, as well as the ability to reason logically and clearly express their thoughts. The school must give the idea that science and its concept are closely related to practice, from which it draws statements of its problems, ideas, and then returns to practice new opportunities for solving its main problems, creates new methods for it. Without this, education will be incomplete, divorced from life and create numerous difficulties for the pupils of the school. That is why the content of school education must be influenced by the broadly understood requirements of the practice of our day and the foreseeable future.

Elections and referendums, bank loans and insurance policies, employment tables and opinion poll charts have dominated our lives. Society begins to study itself deeper and deeper and strives to make predictions about itself and about natural phenomena that require ideas about probability. Even the weather reports in the newspapers report that "there is a 40% chance of rain tomorrow."

A full-fledged existence of a citizen in a complex, variable and multi-structured society is directly related to the right to receive information, with its availability and reliability, with the right to an informed choice, which cannot be carried out without the ability to make choices and forecasts based on the analysis and processing of often incomplete and contradictory information.

We must teach children to live in a probabilistic situation. And this means extracting, analyzing and processing information, making informed decisions in a variety of situations with random outcomes. The orientation towards democratic principles of thinking, the multivariance of the possible development of real situations and events, the formation of personality, the ability to live and work in a complex, constantly changing world, inevitably requires the development of probabilistic and statistical thinking in the younger generation. This problem can be solved in a school mathematics course on the basis of a set of questions related to descriptive statistics and elements of mathematical statistics, with the formation of combinatorial and probabilistic thinking (12). However, not only the socio - economic situation dictates the need for the formation of probabilistic thinking in the new generation. Probability laws are universal. They became the basis for describing the scientific picture of the world. Modern physics, chemistry, biology, demography, sociology, linguistics, philosophy, the whole complex of socio - economic sciences are built and develop on a probabilistic - statistical basis. A teenager is not separated from this world by a blank wall, and in his life he is constantly faced with probabilistic situations. Play and excitement are an essential part of a child's life. The range of issues related to the relationship between the concepts of "probability" and "reliability", the problem of choosing the best solution among several options, assessing the degree of risk and chances of success, the idea of ​​fairness and injustice in games and in real life conflicts - all this is undoubtedly in the sphere of real interests of a teenager. The preparation for solving such problems should be taken over by the course of school mathematics.

Today, in science, the concept of randomness has acquired fundamental importance and is confidently making its way to finding optimal solutions. There is a particularly urgent need to introduce the concept of the random into school teaching, and this is caused not only by the requirements of a scientific and practical order, but also by purely methodological considerations. At the same time, the classical system of Russian education is based primarily on clearly deterministic principles and approaches in mathematics and other subjects. If you do not remove, then at least weaken the contradiction between the deterministic picture of the world formed within the walls of the school and modern scientific concepts based on probabilistic and statistical laws, it is impossible without introducing the foundations of statistics and the theory of probability into compulsory school education. The modern concept of school mathematics education is focused primarily on taking into account the individuality of the child, his interests and inclinations. This determines the criteria for the selection of content, the development and implementation of new, interactive teaching methods, changes in the requirements for the mathematical preparation of the student. At the same time, the very acquaintance of schoolchildren with a very peculiar area of ​​mathematics, where between black and white there is a whole spectrum of colors and shades, possibilities and options, and between the unequivocal "yes" and "no" there is also "maybe" (and this is "maybe" lends itself to a strict quantitative assessment!), helps to eliminate the deep-rooted feeling that what is happening in a mathematics lesson has nothing to do with the world around us, with everyday life.

According to the data of physiologists and psychologists, as well as according to the numerous observations of mathematics teachers, the decline in interest in the learning process in general and in mathematics in particular. In mathematics lessons in basic school, in the fifth or ninth grades, conducted according to the usual scheme and on traditional material, the student often has a feeling of an impenetrable wall between the abstract-formal objects being described and the world around him. It is the probabilistic-statistical line, or, as it has come to be called lately, the stochastic line, the study of which is impossible without relying on the processes observed in the world around, on the child's real life experience, which is able to contribute to the return of interest in the very subject of "mathematics". promoting its importance and universality. Finally, the concept of an open society, the processes of European and world integration are inextricably linked with the mutual rapprochement of countries and peoples, including in the field of education. Russia, having one of the most powerful and recognized traditions of school mathematics education in the world, at the same time remains hardly the only developed country where in the main school course in mathematics there are no foundations of statistics and the theory of probability. The trends of economic transformations that have emerged in our country suggest that in the very near future, society will be in demand for organizers and participants in a new type of production, which many school graduates will have to become. The stochastic culture, which is so necessary for their activities, must be brought up from an early age. It is no coincidence that much attention is paid to this in developed countries: students get acquainted with the elements of probability theory and statistics from the very first school years and throughout their education they acquire probabilistic - statistical approaches to the analysis of common situations encountered in everyday life.

The number of examples of approaches to the study of probabilistic - statistical material in secondary school could be given a lot, since over the past two decades, almost every country has introduced this material into the school curriculum and has proposed one or several approaches to its study. Interesting works appeared in Poland, Sweden, Israel, France. The problems associated with the creation of a system for the study of probabilistic - statistical material in secondary school are not sufficiently covered in our country. An analysis of the approaches we know to studying the elements of the theory of probability and statistics in secondary schools in different countries allows us to draw the following conclusions:

In the vast majority of countries, this material begins to be studied in primary school;

Throughout all the years of study, students get acquainted with probabilistic and statistical approaches to the analysis of empirical data, and an important role in this is played by tasks of an applied nature, analysis of real situations;

In the learning process, a large role is assigned to tasks that require students to work in small groups, collect data on their own, summarize the results of group work, conduct independent research, practical work, set up experiments, conduct small laboratory work, prepare long-term coursework - all this is dictated by the originality probabilistic - statistical material, its close connection with practical activities;

The study of stochastics, as it were, breaks down into probabilistic and statistical components, closely related to each other, in many countries they are supplemented with a small fragment of combinatorics.

In our country, unsuccessful attempts have already been made to introduce the concept of the probability of an event into the school course of mathematics. Due to its isolation and foreignness in relation to the traditional school course, this material was soon removed from programs and textbooks.

Some experience of teaching the elements of probability theory has been accumulated in schools with advanced study of mathematics, but it only confirms the fact that attempts to solve the problem by introducing a new isolated section into the traditional course of mathematics are doomed to failure. The study of the elements of the theory of probability as a closed section of the program related to "pure", theoretical mathematics, completely discredited itself in the eyes of teachers and led to the fact that some of them generally express doubts that it can and should be studied in secondary school. At the same time, teachers of physics, chemistry, biology feel an urgent need to express the basic laws of these sciences in the language of probabilistic concepts. After all, the current state of human knowledge about the world allows us to consider that the random nature is inherent in the basic (basic) phenomena of the microworld.

The appearance in the school curriculum of a probabilistic - statistical line, focused on acquainting students with the probabilistic nature of most of the phenomena of the surrounding reality, will contribute to the strengthening of its general cultural potential, the emergence of new, deeply grounded intersubject connections, and the humanization of school mathematics education.

When selecting material for a new line of the school course, it is necessary to take into account the general educational significance and worldview potential of the proposed topics. It is important to correctly assess what knowledge a modern person needs in everyday life and activities, what of them a student will need to study other school subjects, to continue education, what contribution this knowledge can make to the formation of various aspects of a student's intellect. It is also necessary to take care that the proposed content provides the possibility of organic conjugation of the new educational material with the traditional one, and promotes the development of intra-subject connections.

And in our country today there is an inevitable process of stochastics entering as an equal component in compulsory school mathematics education.

All state educational documents of recent years contain a probabilistic-statistical line in the mathematics course of basic school, along with such familiar lines as "Numbers", "Functions", "Equations and Inequalities", "Geometric Shapes", etc.

1.2 Psychological and pedagogical aspects of studying probability theory in secondary school

probability theory school middle

The research of psychologists (J. Piaget, E. Fishbein) shows that a person is initially poorly adapted to probabilistic assessment, to understanding and correct interpretation of probabilistic and statistical information. Experiments conducted by E. A. Bunimovich (Moscow, one of the authors of textbooks containing elements of stochastics) on the basis of Moscow gymnasium 710, Yaroslavl gymnasium 20, and Kaluga gymnasium 2 indicate the same. classes who have embarked on an advanced course in mathematics, but have not yet studied the probabilistic sections. The research results unequivocally indicate that even a good knowledge and understanding of other branches of mathematics in itself does not provide the development of probabilistic thinking and does not get rid of even trivial probabilistic prejudices and delusions (7).

Let's give one example. Students were asked the question:

"One sport lotto card (6 out of 49) has numbers crossed out

1, 2, 3, 4, 5 and 6,

and on the other

5, 12, 17, 23, 35 and 41.

Which set of numbers do you think is more likely to win? "

Of all the participants in the experiment, 22% of high school students answered that the second card was more likely. Interesting is the almost identical answer of two schoolchildren from different schools (Moscow and Yaroslavl): "Generally, both cases are equally probable, but the second case is more probable," which expresses an obvious contradiction between everyday and scientific ideas of schoolchildren.

It is curious that specialized chemical and biological economic classes, where the mathematics course is much deeper than the basic one, but there is no probabilistic and statistical material, give almost the same result (up to 30% of answers - “winning the second set is more likely”). The results of answers to a similar question in the test offered in 1998 to mathematics teachers at refresher courses in Moscow do not differ much from the given data.

Let us note by the way that the well-known lover of mathematical games and paradoxes Martin Gardner wrote on a similar occasion that in fact it is more profitable to cross out combinations 1, 2, 3, 4, 5 and 6 or another "regular" combination. The chances of winning are the same, but the amount in case of winning may turn out to be significantly higher, since it is unlikely that anyone will cross out numbers of the order from 1 to 6, and therefore, in case of success, you will not have to share the prize fund with anyone.

At the age of elementary grades, much in the students' ideas about the world is still insufficiently formed, and there is not enough mathematical apparatus (first of all, simple fractions) to explain the concepts of probability. At the same time, the basics of descriptive statistics, tables and bar charts, as well as the basics of combinatorics, a systematic enumeration of possible options on a small set of subjects is possible and even necessary to be introduced into the primary school course.

It is ineffective to start outlining the basics of probability theory in high school. The desire to quickly formalize knowledge developed by this age, formed by the traditional course of mathematics, the desire to learn in the lesson, first of all, a certain set of rules, algorithms and calculation methods actually replaces the formation of probabilistic representations by formal learning of formulas of combinatorics and calculating probability according to the classical Laplace model.

Elements of statistical thinking should be introduced at school in a number of subjects, not just in mathematics. It is necessary to make sure that in the lessons of botany and zoology, astronomy and physics, the Russian language and history, from time to time, in the right place, reasonable remarks are made about the randomness of the phenomena studied by this scientific discipline. Naturally, mathematics at the same time cannot stand aside. Children get the very first ideas about the random world from observing them in the surrounding life. At the same time, the important characteristic features of the observed phenomena are clarified in the course of collecting statistical information and their visual presentation. The ability to register statistical information and present them in the form of the simplest tables and diagrams already in itself characterizes the presence of some statistical experience in the student. It reflects the very first, albeit not yet fully realized, ideas about the ambiguity and variability of real phenomena, about random, reliable and impossible results of observations, about specific types of statistical aggregate, their features and general properties. These skills make it possible to form a correct idea not only about phenomena with a pronounced randomness, but also about such phenomena, the random nature of which is not obvious, and is obscured by many factors that complicate perception.

In everyday life and at work, a high school graduate is constantly faced with the need to obtain and formalize some information. In physics, chemistry, biology lessons, when performing laboratory and practical work, the student must be able to draw up the results of observation and experiments; in the lessons of geography, history, social studies, he needs to use tables and reference books, to perceive information presented in graphic form. These skills are necessary for every person, because with statistical material presented in various forms, it is constantly found in all sources of information designed for a mass audience - in newspapers, magazines, books, on television, etc.

Understanding the nature of the studied stochastic phenomenon is associated with the ability to highlight the main thing, to see the features and trends when examining tables, diagrams and graphs. The simplest skills when "reading" tables and graphs allow one to notice some patterns of the observed phenomena, to see specific properties of phenomena with their inherent features and causal relationships behind the forms of presentation of statistical data.

Typical features of the studied phenomena, their general tendencies can be identified using average statistical characteristics. The ability to use them characterizes the student's presence of ideas associated with central tendencies in the world of randomness. Understanding the meaning of the simplest averages, such as the arithmetic mean, is necessary for every student.

The stochastic nature of the surrounding phenomena cannot be disclosed without understanding the degree of variability. Therefore, there is a need for a quantitative assessment of the scatter of statistical data, which contributes to a deeper understanding of the essence of phenomena and processes, makes it possible to compare statistical populations by the degree of their variation.

One of the most important components of stochastic thinking is the understanding of randomness that is stable in the world, the ordering of random facts. Students should not be allowed to perceive the individual aspects of random phenomena spontaneously in life without any interconnections. The central place is occupied here by concepts associated with various experimental representations of the law of large numbers. The simplest and most accessible way is to form the concept of probability as a "theoretically expected" frequency value with an increase in the number of observations. At the same time, understanding the relationship between probability and its empirical prototype - frequency, leads to an awareness of the statistical stability of frequency. At the same time, an important role is played by the understanding that a quantitative assessment of the possibility of the occurrence of a certain event can be carried out before the experiment, based on some theoretical considerations. Thus, we come to the calculation of probabilities in the classical scheme.

In the case when probabilistic intuition does not develop in teaching mathematics, instead of correct ideas and concepts, students acquire false views, they express erroneous judgments.

One of the important goals of studying probabilistic - statistical material in school is the development of probabilistic intuition, the formation of adequate ideas about the properties of random phenomena. Indeed, in life, very often it is necessary to assess the chances, put forward hypotheses and proposals, predict the development of the situation, talk about the possibilities of confirming one or another hypothesis, etc., the idea of ​​probability, which is assimilated in the process of organized, systematic study, differs from the ordinary, everyday precisely by the fact that it is the bearer of ideas about stability, regularity in the world of the random, it allows the most complete and correct drawing of conclusions from the available information.

Note that both early formalization and the other extreme, which is now reflected in some experimental programs - endless discussions about probability outside the course of mathematics, outside the construction of probabilistic models, are equally ineffective and even dangerous.

CHAPTER 2. BASIC CONCEPTS

2.1 Elements of combinatorics

The study of the course should begin with the study of the basics of combinatorics, and in parallel, the theory of probability should be studied, since combinatorics is used to calculate probabilities. Combinatorial methods are widely used in physics, chemistry, biology, economics, and other fields of knowledge. In science and practice, problems are often encountered, solving which one has to make various combinations of a finite number of elements and count the number of combinations. Such problems are called combinatorial problems, and the branch of mathematics that deals with these problems is called combinatorics. Combinatorics studies ways of counting the number of elements in finite sets. Combinatorial formulas are used to calculate probabilities. Consider some set X consisting of n elements. We will choose from this set various ordered subsets of Y of k elements. An arrangement of n elements of a set X by k elements is any ordered set of elements of a set X. If the selection of elements of a set Y from X occurs with a return, i.e. each element of the set X can be selected several times, then the number of placements from n to k is found by the formula (placements with repetitions). If the choice is made without returning, i.e. each element of the set X can be selected only once, then the number of placements from n to k is denoted and determined by equality (placements without repetitions). The special case of placement for n = k is called a permutation of n elements. The number of all permutations of n elements is equal. Let now an unordered subset Y be selected from the set X (the order of the elements in the subset does not matter). Combinations of n elements by k are called subsets of k elements that differ from each other by at least one element. The total number of all combinations from n to k is denoted and is equal. The equalities are valid:, When solving problems, combinatorics use the following rules: The rule of sum. If some object A can be selected from a collection of objects in m ways, and another object B can be selected in n ways, then either A or B can be selected in m + n ways. Product rule. If object A can be selected from a set of objects in m ways, and after each such choice, object B can be selected in n ways, then a pair of objects (A, B) in the specified order can be selected in m * n ways.

2.2 Probability theory

In everyday life, in practical and scientific activities, we often observe certain phenomena, conduct certain experiments. An event that may or may not occur in the course of observation or experiment is called a random event. For example, there is a light bulb hanging from the ceiling - no one knows when it will burn out. Each random event is a consequence of the action of very many random variables (the force with which the coin is thrown, the shape of the coin, and much more). It is impossible to take into account the influence on the result of all these reasons, since their number is large and the laws of action are unknown. The laws of random events are studied by a special branch of mathematics called probability theory. Probability theory does not set itself the task of predicting whether a single event will happen or not - it simply cannot do it. If we are talking about mass homogeneous random events, then they obey certain laws, namely, probabilistic laws. First, let's look at the classification of events. Distinguish between joint and inconsistent events. Events are called joint events if the occurrence of one of them does not exclude the occurrence of the other. Otherwise, the events are said to be inconsistent. For example, two dice are tossed. Event A - three points on the first die, event B - three points on the second die. A and B are joint events. Let the store receive a batch of shoes of the same style and size, but in a different color. Event A - the box taken at random will end up with black shoes, event B - the box will end up with brown shoes, A and B are incompatible events. An event is called reliable if it will necessarily happen in the conditions of a given experience. An event is called impossible if it cannot happen under the conditions of a given experience. For example, the event that a standard part is taken from a batch of standard parts is reliable, but a non-standard part is impossible. An event is called possible, or accidental, if as a result of experience it can appear, but it may not appear. An example of a random event is the identification of product defects during the control of a batch of finished products, a discrepancy between the size of the processed product and a given one, failure of one of the links of the automated control system. Events are called equally possible if, according to the test conditions, none of these events is objectively more possible than others. For example, suppose a store is supplied with light bulbs (and in equal quantities) from several manufacturers. The events of buying a light bulb from any of these factories are equally possible. An important concept is the complete group of events. Several events in a given experience form a complete group, if at least one of them necessarily appears as a result of the experience. For example, there are ten balls in an urn, six of them are red, four are white, and five balls are numbered. A - the appearance of a red ball with one extraction, B - the appearance of a white ball, C - the appearance of a numbered ball. Events A, B, C form a complete group of joint events. The event can be opposite or complementary. An opposite event is understood as an event that must necessarily occur if some event A has not occurred. Opposite events are incompatible and are the only possible. They form a complete group of events. For example, if a batch of manufactured products consists of good and defective ones, then when one product is removed, it may turn out to be either good - event A, or defective - event. Let's look at an example. A dice is thrown (i.e. a small cube, on the edges of which points 1, 2, 3, 4, 5, 6 are knocked out). When throwing a dice on its top edge, one point, two points, three points, etc. can fall out. Each of these outcomes is random. Have carried out such a test. The dice were thrown 100 times and observed how many times the event "6 points fell on the die." It turned out that in this series of experiments, the "six" fell out 9 times. The number 9, which shows how many times the event in question occurred in this test, is called the frequency of this event, and the ratio of the frequency to the total number of tests, equal to, is called the relative frequency of this event. In general, let a certain test be carried out repeatedly under the same conditions and each time it is fixed whether the event of interest to us A has occurred or not. The probability of an event is denoted by the capital letter P. Then the probability of event A will be denoted by P (A). The classical definition of probability: The probability of an event A is equal to the ratio of the number of cases m, favorable to it, out of the total number n of the only possible, equally possible and inconsistent cases to the number n, that is, Consequently, to find the probability of an event it is necessary to: consider various test outcomes; find the set of the only possible, equally possible and incompatible cases, calculate their total number n, the number of cases m, favorable to this event; perform the calculation using the formula. It follows from the formula that the probability of an event is a non-negative number and can vary from zero to one, depending on what proportion is the favorable number of cases from the total number of cases: Consider another example. The box contains 10 balls. 3 of them are red, 2 are green, the rest are white. Find the probability that a ball drawn at random will be red, green, or white. The appearance of the red, green and white balls make up a complete group of events. Let's designate the appearance of the red ball - event A, the appearance of the green - event B, the appearance of the white - event C. Then, in accordance with the above formulas, we get:; ; Note that the probability of the occurrence of one of the two pairwise incompatible events is equal to the sum of the probabilities of these events. The relative frequency of event A is the ratio of the number of experiments that resulted in event A to the total number of experiments. The difference between the relative frequency and the probability is that the probability is calculated without a direct product of experiments, and the relative frequency - after the experiment. So in the example considered above, if 5 balls are taken at random from the box and 2 of them turned out to be red, then the relative frequency of the appearance of the red ball is: As you can see, this value does not coincide with the found probability. With a sufficiently large number of experiments performed, the relative frequency changes little, fluctuating around one number. This number can be taken as the likelihood of an event. Geometric probability. The classical definition of probability assumes that the number of elementary outcomes is finite, which also limits its application in practice. In the case when a test with an infinite number of outcomes takes place, the definition of geometric probability is used - the point hitting the area. When determining the geometric probability, it is assumed that there is a region N and in it a smaller region M. A point is dropped at random on the region N (this means that all points of the region N are "equal" in relation to the hit there of a point thrown at random). Event A - "hit of a dropped point on area M". Area M is called favorable to event A. The probability of hitting any part of area N is proportional to the measure of this part and does not depend on its location and shape. The area covered by the geometric probability can be: a segment (the measure is the length) a geometric figure on a plane (the measure is the area) a geometric body in space (the measure is the volume) Let us define the geometric probability for the case of a flat figure. Let the area M be a part of the area N. The event A consists in the hit of a point randomly thrown on the area N into the area M. The geometric probability of the event A is the ratio of the area of ​​the area M to the area of ​​the area N: In this case, the probability of a randomly thrown point on the border of the area is assumed to be zero ... Consider an example: A twelve-hour mechanical watch broke and stopped walking. Find the probability that the hour hand froze at 5 but did not reach 8 o'clock. Decision. The number of outcomes is infinite, we apply the definition of geometric probability. The sector between 5 and 8 o'clock is part of the area of ​​the entire dial, therefore,. Operations on events: Events A and B are called equal if the implementation of event A entails the implementation of event B and vice versa. A combination or sum of events is called event A, which means the occurrence of at least one of the events. A = The intersection or product of events is the event A, which consists in the implementation of all events. A =? The difference between events A and B is called event C, which means that event A occurs, but event B does not occur. C = A \ B Example: A + B - “2 dropped out; four; 6 or 3 points ”A B -“ 6 points were drawn ”A - B -“ 2 and 4 points were drawn ”An additional event to event A is an event that means that event A does not occur. Elementary outcomes of experience are those results of experience that mutually exclude each other and as a result of the experience one of these events occurs, also whatever event A is, by the elementary outcome that has occurred, one can judge whether this event occurs or does not occur. The totality of all elementary outcomes of experience is called the space of elementary events. Properties of probabilities: Property 1. If all cases are favorable for a given event A, then this event will surely occur. Consequently, the event under consideration is reliable, and the probability of its occurrence, since in this case Property 2. If there is not a single case favorable to this event A, then this event cannot occur as a result of experience. Consequently, the event under consideration is impossible, and the probability of its occurrence, since in this case m = 0: Property 3. The probability of occurrence of events forming a complete group is equal to one. Property 4. The probability of occurrence of the opposite event is determined in the same way as the probability of occurrence of event A: where (n-m) is the number of cases favorable for the occurrence of the opposite event. Hence, the probability of occurrence of the opposite event is equal to the difference between unity and the probability of occurrence of event A: Addition and multiplication of probabilities. Event A is called a special case of event B if, when A occurs, B also occurs. The fact that A is a special case of B, we write A? B. Events A and B are called equal if each of them is a special case of the other. The equality of events A and B is written A = B. The sum of events A and B is called an event A + B, which occurs if and only if at least one of the events occurs: A or B. Theorem on the addition of probabilities 1. The probability of occurrence of one of two incompatible events is equal to the sum of the probabilities of these events. P = P + P Note that the theorem formulated is valid for any number of incompatible events: If random events form a complete group of incompatible events, then the equality P + P + ... + P = 1 is the product of events A and B, which occurs then and only when both events occur: A and B simultaneously. Random events A and B are called joint if both of these events can occur during a given test. The theorem on the addition of probabilities 2. The probability of the sum of joint events is calculated by the formula P = P + P-P Examples of problems for the addition theorem. In the geometry exam, the student gets one question from the list of exam questions. The probability that this is an inscribed circle question is 0.2. The probability that this is a Parallelogram question is 0.15. There are no questions that simultaneously relate to these two topics. Find the probability that a student will get a question on one of these two topics on the exam. Decision. The probability of the sum of two incompatible events is equal to the sum of the probabilities of these events: 0.2 + 0.15 = 0.35. Answer: 0.35. In the mall, two identical vending machines sell coffee. The probability that the machine will run out of coffee by the end of the day is 0.3. The probability that both machines will run out of coffee is 0.12. Find the probability that coffee will remain in both machines by the end of the day. Decision. Let us consider events A - “coffee ends in the first machine”, B - “coffee ends in the second machine”. Then A · B - "coffee will run out in both vending machines", A + B - "coffee will run out in at least one vending machine". By condition, P (A) = P (B) = 0.3; P (A B) = 0.12. Events A and B are joint, the probability of the sum of two joint events is equal to the sum of the probabilities of these events without the probability of their product: P (A + B) = P (A) + P (B)? P (AB) = 0.3 + 0.3? 0.12 = 0.48. Therefore, the probability of the opposite event, that coffee remains in both machines, is equal to 1? 0.48 = 0.52. Answer: 0.52. Events of events A and B are called independent if the appearance of one of them does not change the probability of the appearance of the other. Event A is said to be dependent on event B if the probability of event A changes depending on whether event B has occurred or not. The conditional probability P (A | B) of event A is the probability calculated under the condition that event B has occurred. Similarly, P (B | A) denotes the conditional probability of event B, provided that A has occurred. For independent events, by definition, P (A | B) = P (A); P (B | A) = P (B) Multiplication theorem for dependent events The probability of producing dependent events is equal to the product ve0.01 = 0.0198 + 0.0098 = 0.0296. Answer: 0.0296.

In 2003, it was decided to include elements of the theory of probability in the school mathematics course of a comprehensive school (instruction letter No. 03-93in / 13-03 of 23.09.2003 of the Ministry of Education of the Russian Federation "On the introduction of elements of combinatorics, statistics and probability theory in the content of mathematical education basic school "," Mathematics in school ", No. 9 for 2003). By this time, elements of the theory of probability for more than ten years were present in various forms in well-known school textbooks of algebra for different classes (for example, I.F. "Algebra: Textbooks for grades 7-9 of educational institutions" edited by G.V. Dorofeev; " Algebra and the beginning of analysis: Textbooks for 10-11 grades of educational institutions "GV Dorofeev, LV Kuznetsova, EA Sedova"), and in the form of separate textbooks. However, the presentation of material on the theory of probability in them, as a rule, was not systematic, and teachers, most often, did not refer to these sections, did not include them in the curriculum. The document, adopted by the Ministry of Education in 2003, provided for the gradual, phased inclusion of these sections in school courses, allowing the teaching community to prepare for the corresponding changes. In 2004-2008. a number of textbooks are published to supplement existing algebra textbooks. These are the editions of Tyurin Yu.N., Makarov A.A., Vysotsky I.R., Yashchenko I.V. "Probability theory and statistics", Tyurin Yu.N., Makarov A.A., Vysotsky I.R., Yashchenko I.V. "Probability theory and statistics: Methodological guide for teachers", Makarychev Yu.N., Mindyuk N.G. “Algebra: elements of statistics and probability theory: textbook. A manual for students of 7-9 grades. general education. institutions ", Tkacheva MV, Fedorova N.Ye. “Elements of statistics and probability: Textbook. Allowance for 7-9 grades. general education. institutions ". Methodological manuals were also published to help teachers. For a number of years, all these teaching aids have been tested in schools. In conditions when the transitional period of introduction into school curricula has ended, and the sections of statistics and probability theory have taken their place in the curricula of grades 7-9, an analysis and understanding of the consistency of the basic definitions and notations used in these textbooks is required. All these textbooks were created in the absence of traditions of teaching these branches of mathematics at school. This absence, willingly or unwillingly, provoked the authors of textbooks to compare with the existing textbooks for universities. The latter, depending on the prevailing traditions in individual specializations of higher education, often allowed significant terminological inconsistencies and differences in the designations of basic concepts and the recording of formulas. An analysis of the content of the above school textbooks shows that today they have inherited these features from the textbooks of higher education. With a greater degree of accuracy, it can be argued that the choice of specific educational material for new to school branches of mathematics, concerning the concept of "random", occurs at the moment in the most random way, right down to the names and designations. Therefore, the teams of authors of leading school textbooks on probability theory and statistics decided to unite their efforts under the auspices of the Moscow Institute of Open Education to develop agreed positions on the unification of the basic definitions and notations used in textbooks for the school on probability theory and statistics. Let's analyze the introduction of the topic "Probability Theory" in school textbooks. General characteristics: The content of training on the topic "Elements of the theory of probability", highlighted in the "Program for educational institutions. Mathematics", ensures the further development of students' mathematical abilities, orientation to professions that are significantly related to mathematics, preparation for university studies. The specificity of the mathematical content of the topic under consideration allows us to concretize the selected main task of in-depth study of mathematics as follows. 1. To continue the disclosure of the content of mathematics as a deductive system of knowledge. - to build a system of definitions of basic concepts; - to reveal additional properties of the introduced concepts; - to establish connections between introduced and previously studied concepts. 2. To systematize some probabilistic ways of solving problems; to reveal the operational composition of the search for solutions to problems of certain types. 3. To create conditions for understanding and understanding by students of the main idea of ​​the practical significance of the theory of probability by analyzing the basic theoretical facts. To reveal the practical applications of the theory studied in this topic. The achievement of the set educational goals will be facilitated by the solution of the following tasks: 1. To form an idea of ​​various ways to determine the probability of an event (statistical, classical, geometric, axiomatic) 2. To form knowledge of the basic operations on events and the ability to apply them to describe some events through others. 3. To reveal the essence of the theory of addition and multiplication of probabilities; define the limits of the use of these theorems. Show their applications for deriving total probability formulas. 4. Identify algorithms for finding the probabilities of events a) according to the classical definition of probability; b) on the theory of addition and multiplication; c) according to the formula 0.99 + 0.98P (A | Bn) Consider an example: An automatic line produces batteries. The probability that a finished battery is defective is 0.02. Before packing, each battery goes through a control system. The probability that the system will reject a faulty battery is 0.99. The probability that the system mistakenly rejects a good battery is 0.01. Find the probability that a battery randomly selected from the package will be rejected. Decision. A situation in which the battery will be rejected may arise as a result of the following events: A - “the battery is really faulty and rightly rejected” or B - “the battery is working properly, but was rejected by mistake”. These are inconsistent events, the probability of their sum is equal to the sum of the probabilities of these events. We have: P (A + B) = P (A) + P (B) = 0.02P (A | B3) + ... + P (Bn) P (A | B2) + P (B3) P (A | B1 ) + Р (В2) the probability of one of them on the conditional probability of the other, provided that the first happened: P (AB) = P (A) P (B | A) P (AB) = P (B) P (A | B) (depending on which event happened first). Consequences of the theorem: Multiplication theorem for independent events. The probability of the product of independent events is equal to the product of their probabilities: P (A B) = P (A) P (B) If A and B are independent, then the pairs are also independent: (;), (; B), (A;). Examples of problems on the multiplication theorem: If Grandmaster A. plays White, then he wins against Grandmaster B. with probability 0.52. If A. plays black, then A. wins against B. with a probability of 0.3. Grandmasters A. and B. play two games, and in the second game they change the color of the pieces. Find the probability that A. wins both times. Decision. The chances of winning the first and second games are independent of each other. The probability of the product of independent events is equal to the product of their probabilities: 0.52 · 0.3 = 0.156. Answer: 0.156. There are two payment machines in the store. Each of them can be faulty with a probability of 0.05, regardless of the other machine. Find the probability that at least one machine is operational. Decision. Let us find the probability that both automata are faulty. These events are independent, the probability of their product is equal to the product of the probabilities of these events: 0.05 · 0.05 = 0.0025. The event that at least one machine is operational is the opposite. Therefore, its probability is 1? 0.0025 = 0.9975. Answer: 0.9975. The formula of total probability A consequence of the theorems of addition and multiplication of probabilities is the formula of total probability: The probability P (A) of an event A, which can occur only if one of the events (hypotheses) B1, B2, B3 ... Bn appears, forming a complete group of pairwise incompatible events, is equal to the sum of the products of the probabilities of each of the events (hypotheses) B1, B2, B3, ..., Bn by the corresponding conditional probabilities of the event A: P (A) = P (B1) of the total probability. 5. Formulate a prescription that allows you to rationally choose one of the algorithms when solving a specific problem. The selected educational goals for the study of the elements of the theory of probability are supplemented by the setting of developmental and educational goals. Developing goals: to form students' steady interest in the subject, to identify and develop mathematical abilities; in the learning process, develop speech, thinking, emotional-volitional and concrete-motivational areas; students' independent finding of new ways to solve problems and problems; application of knowledge in new situations and circumstances; develop the ability to explain facts, connections between phenomena, transform material from one form of presentation to another (verbal, sign-symbolic, graphic); to teach to demonstrate the correct application of methods, to see the logic of reasoning, the similarity and difference of phenomena. Educational goals: to form in schoolchildren moral and aesthetic ideas, a system of views on the world, the ability to follow the norms of behavior in society; to form the needs of the individual, the motives of social behavior, activities, values ​​and value orientations; educate a person capable of self-education and self-education. Let's analyze the textbook on algebra for the 9th grade "Algebra: elements of statistics and probability theory" Makarychev Yu.N. This textbook is intended for students in grades 7-9, it supplements the textbooks: Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. "Algebra 7", "Algebra 8", "Algebra 9", edited by S. Telyakovsky. The book consists of four sections. Each paragraph contains theoretical information and related exercises. Exercises for review are given at the end of the paragraph. For each paragraph, additional exercises of a higher level of complexity are given in comparison with the main exercises. According to the "Program for educational institutions" for the study of the topic "Probability theory and statistics" in the school algebra course is given 15 hours. Material on this topic falls on the 9th grade and is presented in the following paragraphs: §3 "Elements of combinatorics" contains 4 items: Examples of combinatorial problems. Solving combinatorial problems by enumerating possible options is demonstrated using simple examples. This technique is illustrated by building a tree of possibilities. The rule of multiplication is considered. Permutations. The concept itself and the formula for calculating permutations are introduced. Accommodation. The concept is introduced with a specific example. The formula for the number of placements is displayed. Combinations. The concept and formula for the number of combinations. The purpose of this section is to provide students with different ways of describing all possible elementary events in different types of random experiences. §4 "Initial information from the theory of probability". The presentation of the material begins with an examination of the experiment, after which the concepts of "random event" and "relative frequency of a random event" are introduced. The statistical and classical definition of probability is introduced. The paragraph ends with the addition and multiplication of probabilities. The theorems of addition and multiplication of probabilities are considered, the related concepts of incompatible, opposite, independent events are introduced. This material is designed for students with an interest and aptitude for mathematics and can be used for individual work or in extracurricular activities with students. Methodological recommendations for this textbook are given in a number of articles by Makarychev and Mindyuk ("Elements of combinatorics in the school course of algebra", "Initial information from the theory of probability in the school course of algebra"). And also some critical remarks on this tutorial are contained in the article by Studenetskaya and Fadeeva, which will help prevent mistakes when working with this tutorial. Purpose: transition from a qualitative description of events to a mathematical description. The topic "Probability Theory" in the textbooks of Mordkovich A.G., Semenov P.V. for grades 9-11. At the moment, one of the current textbooks at the school is the textbook of A.G. Mordkovich, P.V. Semenov. "Events, probabilities, statistical processing of data", there are also additional chapters for grades 7-9. Let's analyze it. According to the "Work Program on Algebra" 20 hours are allocated for the study of the topic "Elements of combinatorics, statistics and probability theory". Material on the topic "Probability theory" is disclosed in the following paragraphs: § 1. The simplest combinatorial problems. The rule of multiplication and the tree of variants. Permutations. It starts with an examination of simple combinatorial problems, a table of possible options is considered, which shows the principle of the multiplication rule. Then candidate trees and permutations are considered. After the theoretical material, there are exercises for each of the sub-items. § 2. Selection of several elements. Combinations. First, the formula is displayed for 2 elements, then for three, and then the general formula for n elements. § 3. Random events and their probabilities. The classical definition of probability is introduced. The advantage of this tutorial is that it is one of the few that contains items that deal with tables and trees of options. These points are necessary because it is the tables and variant trees that teach students the presentation and initial analysis of the data. Also, in this tutorial, a combination formula is successfully introduced, first for two elements, then for three, and generalizes for n elements. In terms of combinatorics, the material is presented just as well. Each paragraph contains exercises that allow you to consolidate the material. For notes on this tutorial, see Studenetskaya and Fadeeva's article. In grade 10, three sections are devoted to this topic. In the first of them “The rule of multiplication. Permutations and factorials ”, in addition to the rule of multiplication itself, the main emphasis was made on the derivation from this rule of two basic combinatorial identities: for the number of permutations and for the number of all possible subsets of a set consisting of n elements. Moreover, factorials were introduced as a convenient way to shorten the answer in many specific combinatorial problems before the very concept of "permutation". In the second paragraph of the 10th class “Selection of several elements. Binomial coefficients ”considered the classical combinatorial problems associated with the simultaneous (or sequential) selection of several elements from a given finite set. The most significant and really new for the Russian general education school was the final paragraph "Random events and their probabilities." It considered the classical probabilistic scheme, analyzed the formulas P (A + B) + P (AB) = P (A) + P (B), P () = 1-P (A), P (A) = 1- P () and how to use them. The paragraph ended with a transition to independent repetitions of the trial with two outcomes. This is the most important from a practical point of view probabilistic model (Bernoulli tests), which has a significant number of applications. The latter material formed the transition between the content of the educational material in the 10th and 11th grades. In the 11th grade, the topic "Elements of the theory of probability" is devoted to two sections of the textbook and the book of problems. Section 22 deals with geometric probabilities, and Section 23 repeats and expands knowledge about independent repetitions of trials with two outcomes.

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In everyday life, in practical and scientific activities, we often observe certain phenomena, conduct certain experiments. An event that may or may not occur in the course of observation or experiment is called a random event. For example, there is a light bulb hanging from the ceiling - no one knows when it will burn out. Each random event is a consequence of the action of very many random variables (the force with which the coin is thrown, the shape of the coin, and much more). It is impossible to take into account the influence on the result of all these reasons, since their number is large and the laws of action are unknown. The laws of random events are studied by a special branch of mathematics called probability theory. Probability theory does not set itself the task of predicting whether a single event will happen or not - it simply cannot do it. If we are talking about mass homogeneous random events, then they obey certain laws, namely, probabilistic laws. First, let's look at the classification of events. Distinguish between joint and inconsistent events. Events are called joint events if the occurrence of one of them does not exclude the occurrence of the other. Otherwise, the events are said to be inconsistent. For example, two dice are tossed. Event A - three points on the first die, event B - three points on the second die. A and B are joint events. Let the store receive a batch of shoes of the same style and size, but in a different color. Event A - the box taken at random will end up with black shoes, event B - the box will end up with brown shoes, A and B are incompatible events. An event is called reliable if it will necessarily happen in the conditions of a given experience. An event is called impossible if it cannot happen under the conditions of a given experience. For example, the event that a standard part is taken from a batch of standard parts is reliable, but a non-standard part is impossible. An event is called possible, or accidental, if as a result of experience it can appear, but it may not appear. An example of a random event is the identification of product defects during the control of a batch of finished products, a discrepancy between the size of the processed product and a given one, failure of one of the links of the automated control system. Events are called equally possible if, according to the test conditions, none of these events is objectively more possible than others. For example, suppose a store is supplied with light bulbs (and in equal quantities) from several manufacturers. The events of buying a light bulb from any of these factories are equally possible. An important concept is the complete group of events. Several events in a given experience form a complete group, if at least one of them necessarily appears as a result of the experience. For example, there are ten balls in an urn, six of them are red, four are white, and five balls are numbered. A - the appearance of a red ball with one extraction, B - the appearance of a white ball, C - the appearance of a numbered ball. Events A, B, C form a complete group of joint events. The event can be opposite or complementary. An opposite event is understood as an event that must necessarily occur if some event A has not occurred. Opposite events are incompatible and are the only possible. They form a complete group of events. For example, if a batch of manufactured products consists of good and defective ones, then when one product is removed, it may turn out to be either good - event A, or defective - event. Let's look at an example. A dice is thrown (i.e. a small cube, on the edges of which points 1, 2, 3, 4, 5, 6 are knocked out). When throwing a dice on its top edge, one point, two points, three points, etc. can fall out. Each of these outcomes is random. Have carried out such a test. The dice were thrown 100 times and observed how many times the event "6 points fell on the die." It turned out that in this series of experiments, the "six" fell out 9 times. The number 9, which shows how many times the event in question occurred in this test, is called the frequency of this event, and the ratio of the frequency to the total number of tests, equal to, is called the relative frequency of this event. In general, let a certain test be carried out repeatedly under the same conditions and each time it is fixed whether the event of interest to us A has occurred or not. The probability of an event is denoted by the capital letter P. Then the probability of event A will be denoted by P (A). The classical definition of probability: The probability of an event A is equal to the ratio of the number of cases m, favorable to it, out of the total number n of the only possible, equally possible and inconsistent cases to the number n, that is, Consequently, to find the probability of an event it is necessary to: consider various test outcomes; find the set of the only possible, equally possible and incompatible cases, calculate their total number n, the number of cases m, favorable to this event; perform the calculation using the formula. It follows from the formula that the probability of an event is a non-negative number and can vary from zero to one, depending on what proportion is the favorable number of cases from the total number of cases: Consider another example. The box contains 10 balls. 3 of them are red, 2 are green, the rest are white. Find the probability that a ball drawn at random will be red, green, or white. The appearance of the red, green and white balls make up a complete group of events. Let's designate the appearance of the red ball - event A, the appearance of the green - event B, the appearance of the white - event C. Then, in accordance with the above formulas, we get:; ; Note that the probability of the occurrence of one of the two pairwise incompatible events is equal to the sum of the probabilities of these events. The relative frequency of event A is the ratio of the number of experiments that resulted in event A to the total number of experiments. The difference between the relative frequency and the probability is that the probability is calculated without a direct product of experiments, and the relative frequency - after the experiment. So in the example considered above, if 5 balls are taken at random from the box and 2 of them turned out to be red, then the relative frequency of the appearance of the red ball is: As you can see, this value does not coincide with the found probability. With a sufficiently large number of experiments performed, the relative frequency changes little, fluctuating around one number. This number can be taken as the likelihood of an event. Geometric probability. The classical definition of probability assumes that the number of elementary outcomes is finite, which also limits its application in practice. In the case when a test with an infinite number of outcomes takes place, the definition of geometric probability is used - the point hitting the area. When determining the geometric probability, it is assumed that there is a region N and in it a smaller region M. A point is dropped at random on the region N (this means that all points of the region N are "equal" in relation to the hit there of a point thrown at random). Event A - "hit of a dropped point on area M". Area M is called favorable to event A. The probability of hitting any part of area N is proportional to the measure of this part and does not depend on its location and shape. The area covered by the geometric probability can be: a segment (the measure is the length) a geometric figure on a plane (the measure is the area) a geometric body in space (the measure is the volume) Let us define the geometric probability for the case of a flat figure. Let the area M be a part of the area N. The event A consists in the hit of a point randomly thrown on the area N into the area M. The geometric probability of the event A is the ratio of the area of ​​the area M to the area of ​​the area N: In this case, the probability of a randomly thrown point on the border of the area is assumed to be zero ... Consider an example: A twelve-hour mechanical watch broke and stopped walking. Find the probability that the hour hand froze at 5 but did not reach 8 o'clock. Decision. The number of outcomes is infinite, we apply the definition of geometric probability. The sector between 5 and 8 o'clock is part of the area of ​​the entire dial, therefore,. Operations on events: Events A and B are called equal if the implementation of event A entails the implementation of event B and vice versa. A combination or sum of events is called event A, which means the occurrence of at least one of the events. A = The intersection or product of events is the event A, which consists in the implementation of all events. A =? The difference between events A and B is called event C, which means that event A occurs, but event B does not occur. C = AB Example: A + B - “2 dropped out; four; 6 or 3 points ”A B -“ 6 points were drawn ”A - B -“ 2 and 4 points were drawn ”An additional event to event A is an event that means that event A does not occur. Elementary outcomes of experience are those results of experience that mutually exclude each other and as a result of the experience one of these events occurs, also whatever event A is, by the elementary outcome that has occurred, one can judge whether this event occurs or does not occur. The totality of all elementary outcomes of experience is called the space of elementary events. Properties of probabilities: Property 1. If all cases are favorable for a given event A, then this event will surely occur. Consequently, the event under consideration is reliable, and the probability of its occurrence, since in this case Property 2. If there is not a single case favorable to this event A, then this event cannot occur as a result of experience. Consequently, the event under consideration is impossible, and the probability of its occurrence, since in this case m = 0: Property 3. The probability of occurrence of events forming a complete group is equal to one. Property 4. The probability of occurrence of the opposite event is determined in the same way as the probability of occurrence of event A: where (n-m) is the number of cases favorable for the occurrence of the opposite event. Hence, the probability of occurrence of the opposite event is equal to the difference between unity and the probability of occurrence of event A: Addition and multiplication of probabilities. Event A is called a special case of event B if, when A occurs, B also occurs. The fact that A is a special case of B, we write A? B. Events A and B are called equal if each of them is a special case of the other. The equality of events A and B is written A = B. The sum of events A and B is called an event A + B, which occurs if and only if at least one of the events occurs: A or B. Theorem on the addition of probabilities 1. The probability of occurrence of one of two incompatible events is equal to the sum of the probabilities of these events. P = P + P Note that the theorem formulated is valid for any number of incompatible events: If random events form a complete group of incompatible events, then the equality P + P + ... + P = 1 is the product of events A and B, which occurs then and only when both events occur: A and B simultaneously. Random events A and B are called joint if both of these events can occur during a given test. The theorem on the addition of probabilities 2. The probability of the sum of joint events is calculated by the formula P = P + P-P Examples of problems for the addition theorem. In the geometry exam, the student gets one question from the list of exam questions. The probability that this is an inscribed circle question is 0.2. The probability that this is a Parallelogram question is 0.15. There are no questions that simultaneously relate to these two topics. Find the probability that a student will get a question on one of these two topics on the exam. Decision. The probability of the sum of two incompatible events is equal to the sum of the probabilities of these events: 0.2 + 0.15 = 0.35. Answer: 0.35. In the mall, two identical vending machines sell coffee. The probability that the machine will run out of coffee by the end of the day is 0.3. The probability that both machines will run out of coffee is 0.12. Find the probability that coffee will remain in both machines by the end of the day. Decision. Let us consider events A - “coffee ends in the first machine”, B - “coffee ends in the second machine”. Then A · B - "coffee will run out in both vending machines", A + B - "coffee will run out in at least one vending machine". By condition, P (A) = P (B) = 0.3; P (A B) = 0.12. Events A and B are joint, the probability of the sum of two joint events is equal to the sum of the probabilities of these events without the probability of their product: P (A + B) = P (A) + P (B)? P (AB) = 0.3 + 0.3? 0.12 = 0.48. Therefore, the probability of the opposite event, that coffee remains in both machines, is equal to 1? 0.48 = 0.52. Answer: 0.52. Events of events A and B are called independent if the appearance of one of them does not change the probability of the appearance of the other. Event A is said to be dependent on event B if the probability of event A changes depending on whether event B has occurred or not. The conditional probability P (A | B) of event A is the probability calculated under the condition that event B has occurred. Similarly, P (B | A) denotes the conditional probability of event B, provided that A has occurred. For independent events, by definition, P (A | B) = P (A); P (B | A) = P (B) Multiplication theorem for dependent events The probability of producing dependent events is equal to the product ve0.01 = 0.0198 + 0.0098 = 0.0296. Answer: 0.0296.

In 2003, it was decided to include elements of the theory of probability in the school mathematics course of a comprehensive school (instruction letter No. 03-93in / 13-03 of 23.09.2003 of the Ministry of Education of the Russian Federation "On the introduction of elements of combinatorics, statistics and probability theory in the content of mathematical education basic school "," Mathematics in school ", No. 9 for 2003). By this time, elements of the theory of probability for more than ten years were present in various forms in well-known school textbooks of algebra for different classes (for example, I.F. "Algebra: Textbooks for grades 7-9 of educational institutions" edited by G.V. Dorofeev; " Algebra and the beginning of analysis: Textbooks for 10-11 grades of educational institutions "GV Dorofeev, LV Kuznetsova, EA Sedova"), and in the form of separate textbooks. However, the presentation of material on the theory of probability in them, as a rule, was not systematic, and teachers, most often, did not refer to these sections, did not include them in the curriculum. The document, adopted by the Ministry of Education in 2003, provided for the gradual, phased inclusion of these sections in school courses, allowing the teaching community to prepare for the corresponding changes. In 2004-2008. a number of textbooks are published to supplement existing algebra textbooks. These are the editions of Tyurin Yu.N., Makarov A.A., Vysotsky I.R., Yashchenko I.V. "Probability theory and statistics", Tyurin Yu.N., Makarov A.A., Vysotsky I.R., Yashchenko I.V. "Probability theory and statistics: Methodological guide for teachers", Makarychev Yu.N., Mindyuk N.G. “Algebra: elements of statistics and probability theory: textbook. A manual for students of 7-9 grades. general education. institutions ", Tkacheva MV, Fedorova N.Ye. “Elements of statistics and probability: Textbook. Allowance for 7-9 grades. general education. institutions ". Methodological manuals were also published to help teachers. For a number of years, all these teaching aids have been tested in schools. In conditions when the transitional period of introduction into school curricula has ended, and the sections of statistics and probability theory have taken their place in the curricula of grades 7-9, an analysis and understanding of the consistency of the basic definitions and notations used in these textbooks is required. All these textbooks were created in the absence of traditions of teaching these branches of mathematics at school. This absence, willingly or unwillingly, provoked the authors of textbooks to compare with the existing textbooks for universities. The latter, depending on the prevailing traditions in individual specializations of higher education, often allowed significant terminological inconsistencies and differences in the designations of basic concepts and the recording of formulas. An analysis of the content of the above school textbooks shows that today they have inherited these features from the textbooks of higher education. With a greater degree of accuracy, it can be argued that the choice of specific educational material for new to school branches of mathematics, concerning the concept of "random", occurs at the moment in the most random way, right down to the names and designations. Therefore, the teams of authors of leading school textbooks on probability theory and statistics decided to unite their efforts under the auspices of the Moscow Institute of Open Education to develop agreed positions on the unification of the basic definitions and notations used in textbooks for the school on probability theory and statistics. Let's analyze the introduction of the topic "Probability Theory" in school textbooks. General characteristics: The content of training on the topic "Elements of the theory of probability", highlighted in the "Program for educational institutions. Mathematics", ensures the further development of students' mathematical abilities, orientation to professions that are significantly related to mathematics, preparation for university studies. The specificity of the mathematical content of the topic under consideration allows us to concretize the selected main task of in-depth study of mathematics as follows. 1. To continue the disclosure of the content of mathematics as a deductive system of knowledge. - to build a system of definitions of basic concepts; - to reveal additional properties of the introduced concepts; - to establish connections between introduced and previously studied concepts. 2. To systematize some probabilistic ways of solving problems; to reveal the operational composition of the search for solutions to problems of certain types. 3. To create conditions for understanding and understanding by students of the main idea of ​​the practical significance of the theory of probability by analyzing the basic theoretical facts. To reveal the practical applications of the theory studied in this topic. The achievement of the set educational goals will be facilitated by the solution of the following tasks: 1. To form an idea of ​​various ways to determine the probability of an event (statistical, classical, geometric, axiomatic) 2. To form knowledge of the basic operations on events and the ability to apply them to describe some events through others. 3. To reveal the essence of the theory of addition and multiplication of probabilities; define the limits of the use of these theorems. Show their applications for deriving total probability formulas. 4. Identify algorithms for finding the probabilities of events a) according to the classical definition of probability; b) on the theory of addition and multiplication; c) according to the formula 0.99 + 0.98P (A | Bn) Consider an example: An automatic line produces batteries. The probability that a finished battery is defective is 0.02. Before packing, each battery goes through a control system. The probability that the system will reject a faulty battery is 0.99. The probability that the system mistakenly rejects a good battery is 0.01. Find the probability that a battery randomly selected from the package will be rejected. Decision. A situation in which the battery will be rejected may arise as a result of the following events: A - “the battery is really faulty and rightly rejected” or B - “the battery is working properly, but was rejected by mistake”. These are inconsistent events, the probability of their sum is equal to the sum of the probabilities of these events. We have: P (A + B) = P (A) + P (B) = 0.02P (A | B3) + ... + P (Bn) P (A | B2) + P (B3) P (A | B1 ) + Р (В2) the probability of one of them on the conditional probability of the other, provided that the first happened: P (AB) = P (A) P (B | A) P (AB) = P (B) P (A | B) (depending on which event happened first). Consequences of the theorem: Multiplication theorem for independent events. The probability of the product of independent events is equal to the product of their probabilities: P (A B) = P (A) P (B) If A and B are independent, then the pairs are also independent: (;), (; B), (A;). Examples of problems on the multiplication theorem: If Grandmaster A. plays White, then he wins against Grandmaster B. with probability 0.52. If A. plays black, then A. wins against B. with a probability of 0.3. Grandmasters A. and B. play two games, and in the second game they change the color of the pieces. Find the probability that A. wins both times. Decision. The chances of winning the first and second games are independent of each other. The probability of the product of independent events is equal to the product of their probabilities: 0.52 · 0.3 = 0.156. Answer: 0.156. There are two payment machines in the store. Each of them can be faulty with a probability of 0.05, regardless of the other machine. Find the probability that at least one machine is operational. Decision. Let us find the probability that both automata are faulty. These events are independent, the probability of their product is equal to the product of the probabilities of these events: 0.05 · 0.05 = 0.0025. The event that at least one machine is operational is the opposite. Therefore, its probability is 1? 0.0025 = 0.9975. Answer: 0.9975. The formula of total probability A consequence of the theorems of addition and multiplication of probabilities is the formula of total probability: The probability P (A) of an event A, which can occur only if one of the events (hypotheses) B1, B2, B3 ... Bn appears, forming a complete group of pairwise incompatible events, is equal to the sum of the products of the probabilities of each of the events (hypotheses) B1, B2, B3, ..., Bn by the corresponding conditional probabilities of the event A: P (A) = P (B1) of the total probability. 5. Formulate a prescription that allows you to rationally choose one of the algorithms when solving a specific problem. The selected educational goals for the study of the elements of the theory of probability are supplemented by the setting of developmental and educational goals. Developing goals: to form students' steady interest in the subject, to identify and develop mathematical abilities; in the learning process, develop speech, thinking, emotional-volitional and concrete-motivational areas; students' independent finding of new ways to solve problems and problems; application of knowledge in new situations and circumstances; develop the ability to explain facts, connections between phenomena, transform material from one form of presentation to another (verbal, sign-symbolic, graphic); to teach to demonstrate the correct application of methods, to see the logic of reasoning, the similarity and difference of phenomena. Educational goals: to form in schoolchildren moral and aesthetic ideas, a system of views on the world, the ability to follow the norms of behavior in society; to form the needs of the individual, the motives of social behavior, activities, values ​​and value orientations; educate a person capable of self-education and self-education. Let's analyze the textbook on algebra for the 9th grade "Algebra: elements of statistics and probability theory" Makarychev Yu.N. This textbook is intended for students in grades 7-9, it supplements the textbooks: Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. "Algebra 7", "Algebra 8", "Algebra 9", edited by S. Telyakovsky. The book consists of four sections. Each paragraph contains theoretical information and related exercises. Exercises for review are given at the end of the paragraph. For each paragraph, additional exercises of a higher level of complexity are given in comparison with the main exercises. According to the "Program for educational institutions" for the study of the topic "Probability theory and statistics" in the school algebra course is given 15 hours. Material on this topic falls on the 9th grade and is presented in the following paragraphs: §3 "Elements of combinatorics" contains 4 items: Examples of combinatorial problems. Solving combinatorial problems by enumerating possible options is demonstrated using simple examples. This technique is illustrated by building a tree of possibilities. The rule of multiplication is considered. Permutations. The concept itself and the formula for calculating permutations are introduced. Accommodation. The concept is introduced with a specific example. The formula for the number of placements is displayed. Combinations. The concept and formula for the number of combinations. The purpose of this section is to provide students with different ways of describing all possible elementary events in different types of random experiences. §4 "Initial information from the theory of probability". The presentation of the material begins with an examination of the experiment, after which the concepts of "random event" and "relative frequency of a random event" are introduced. The statistical and classical definition of probability is introduced. The paragraph ends with the addition and multiplication of probabilities. The theorems of addition and multiplication of probabilities are considered, the related concepts of incompatible, opposite, independent events are introduced. This material is designed for students with an interest and aptitude for mathematics and can be used for individual work or in extracurricular activities with students. Methodological recommendations for this textbook are given in a number of articles by Makarychev and Mindyuk ("Elements of combinatorics in the school course of algebra", "Initial information from the theory of probability in the school course of algebra"). And also some critical remarks on this tutorial are contained in the article by Studenetskaya and Fadeeva, which will help prevent mistakes when working with this tutorial. Purpose: transition from a qualitative description of events to a mathematical description. The topic "Probability Theory" in the textbooks of Mordkovich A.G., Semenov P.V. for grades 9-11. At the moment, one of the current textbooks at the school is the textbook of A.G. Mordkovich, P.V. Semenov. "Events, probabilities, statistical processing of data", there are also additional chapters for grades 7-9. Let's analyze it. According to the "Work Program on Algebra" 20 hours are allocated for the study of the topic "Elements of combinatorics, statistics and probability theory". Material on the topic "Probability theory" is disclosed in the following paragraphs: § 1. The simplest combinatorial problems. The rule of multiplication and the tree of variants. Permutations. It starts with an examination of simple combinatorial problems, a table of possible options is considered, which shows the principle of the multiplication rule. Then candidate trees and permutations are considered. After the theoretical material, there are exercises for each of the sub-items. § 2. Selection of several elements. Combinations. First, the formula is displayed for 2 elements, then for three, and then the general formula for n elements. § 3. Random events and their probabilities. The classical definition of probability is introduced. The advantage of this tutorial is that it is one of the few that contains items that deal with tables and trees of options. These points are necessary because it is the tables and variant trees that teach students the presentation and initial analysis of the data. Also, in this tutorial, a combination formula is successfully introduced, first for two elements, then for three, and generalizes for n elements. In terms of combinatorics, the material is presented just as well. Each paragraph contains exercises that allow you to consolidate the material. For notes on this tutorial, see Studenetskaya and Fadeeva's article. In grade 10, three sections are devoted to this topic. In the first of them “The rule of multiplication. Permutations and factorials ”, in addition to the rule of multiplication itself, the main emphasis was made on the derivation from this rule of two basic combinatorial identities: for the number of permutations and for the number of all possible subsets of a set consisting of n elements. Moreover, factorials were introduced as a convenient way to shorten the answer in many specific combinatorial problems before the very concept of "permutation". In the second paragraph of the 10th class “Selection of several elements. Binomial coefficients ”considered the classical combinatorial problems associated with the simultaneous (or sequential) selection of several elements from a given finite set. The most significant and really new for the Russian general education school was the final paragraph "Random events and their probabilities." It considered the classical probabilistic scheme, analyzed the formulas P (A + B) + P (AB) = P (A) + P (B), P () = 1-P (A), P (A) = 1- P () and how to use them. The paragraph ended with a transition to independent repetitions of the trial with two outcomes. This is the most important from a practical point of view probabilistic model (Bernoulli tests), which has a significant number of applications. The latter material formed the transition between the content of the educational material in the 10th and 11th grades. In the 11th grade, the topic "Elements of the theory of probability" is devoted to two sections of the textbook and the book of problems. Section 22 deals with geometric probabilities, and Section 23 repeats and expands knowledge about independent repetitions of trials with two outcomes.

Tarasevich Alena Konstantinovna, Student of Smolensk State University, Smolensk [email protected];

Morozova Elena Valentinovna, Degree of Candidate of Pedagogical Sciences, position of Associate Professor of the Department of Information and Educational Technologies, Smolensk State University, city of Smolensk [email protected]

Features of studying the foundations of probability theory in the school course of mathematics

Annotation. The article is devoted to the peculiarities of studying the foundations of the theory of probability in the school course of mathematics. Special attention is paid to the goals of teaching, characteristics and periods, as well as examples of studying this discipline using specially created programs.

Key words: methods of studying probability theory at school, methods of studying basic concepts, methods of teaching mathematics.

The study of the basics of probability theory in the school course of mathematics has some peculiarities. On the one hand, this is a rather capacious and difficult process, which is difficult to assimilate sometimes already at a more conscious age, not to mention school age, however, at the moment, no one doubts the need to include this discipline in the pre-university course, since it helps to develop a number of skills in the child. , which will be useful to him not only in further education, but also in life in general. It is necessary to teach schoolchildren to think, taking into account all kinds of probabilities. That is, you need to teach them to receive, analyze and process information, make weighted, deliberate actions in various situations with unexpected outcomes. Schoolchildren in their lives are faced with such situations every day. Play and courage occupy a definite, significant place in our life. All these questions related to comparing the concepts of "probability" and "reliability", the difficulty of choosing the best of several options for action, assessing the likelihood of success and fiasco, the idea of ​​good and evil in games and in real life situations - all this, of course, is in a circle true and necessary hobbies of a teenager. The mathematical activity of schoolchildren must go beyond the framework of ready-made probabilistic models. The performance of tasks by schoolchildren, which then help to make decisions in real life situations, plays a huge role and requires correct and experienced teaching of the material by the teacher. Knowledge of stochastics is one of the most important factors in the future activity of a mathematics teacher. We need a multilateral view of stochastics, including as a special methodology that includes probabilistic and statistical inference and their relationship. The teacher must thoroughly know and understand the causes of the risk of making wrong decisions during the analysis of events occurring in the mind of the case. Deceptive understanding, for example, can arise from little statistical information. Teachers are developing unusual approaches to teaching. A teacher, determining the level of knowledge by schoolchildren of any kind of stochastic skills, may encounter some difficulties, for example, when solving problems, schoolchildren often have, so to speak, to think sensibly, and not act strictly according to the algorithm, rules, so their answers to the same questions may In this case, the task of the teacher will be to assess the student's right to error, since it is of a possible nature. It should be borne in mind that the most developed children quickly begin to do things related to conducting experiments and studies of interest to us, take, so to speak, custody of their comrades.

Therefore, it is not a little important to differentiate the level of skills and abilities individually and without the help of outsiders to draw conclusions about what has been learned. When starting to teach stochastics to students, the teacher is obliged to understand why it became necessary to introduce a new program into the curriculum. The teacher's correct understanding of the goals of teaching stochastics at school, a clear understanding of their relationship with mathematics and the place of stochastics among other topics, knowledge of the final requirements for this training of students is the main basis of a mathematics teacher for the implementation of a new line. on the mental development of adolescents, because it endows them with the skills of correct logical thinking, based solely on the correct and necessary concepts. All of the above in full applies to teaching probability theory, but teaching "the law of the case" is much more important, going beyond the ordinary. Studying the course in probability theory, the student begins to understand how to apply the techniques of logical thinking when faced with uncertainty (and there are a lot of such cases in practice).

All of the above can be defined as the goals of studying this discipline, but what exactly does it teach us in the school course, what do the students study and what basic concepts are found there?

If you approach in detail and step by step, then it is better to start the school course of probability theory in grade 5, where the basic definitions of probability theory will be introduced using specific, "live", understandable examples. The beginning of probability theory is combinatorics, where problems will be solved by an enumeration method, that is, students explore all possible solutions. Of course, it is necessary to consider the solution of combinatorial problems using a tree of possible options.

The next stage of learning is the consideration of events: random, reliable, impossible, equally possible, equally probable events, which are illustrated by everyday examples. It is also necessary to consider the rule of multiplication, which is a new means of solving combinatorial problems, which sounds like this: "if the first element of a pair can be chosen in m ways and for each of these ways the second element can be selected in n ways, then this pair can be selected in m * n ways. " It is necessary to illustrate the possibilities of this rule with specific examples.

In a separate chapter, it is necessary to consider the basic statistical characteristics: the arithmetic mean (the arithmetic mean of a series of numbers is the quotient of dividing the sum of these numbers by their number), fashion (the fashion is the number of the series that occurs most often in this series), the range (the range is the difference between the largest and the smallest values ​​of the data series), the median (the median is the number that divides the data series into two parts with the same number of members), which should be illustrated by many examples from life. The most important thing in learning is to consider examples that relate to practice, various life examples are described, which will be useful and interesting for children.

Having analyzed the above, we can formulate the classical definition of probability theory, which was first given in the works of the French mathematician Laplace, and also consider the elements of combinatorics: placement and combination. The classic definition can be illustrated using the table: Table 1 Problem solving using the classic definition

Already in high school, statistical research is studied, a definition of statistics is introduced (the science that studies, processes and analyzes quantitative data on a wide variety of mass phenomena in life), new concepts of sampling, representativeness, general population, ranking, sample size are considered. A new way of graphical presentation of polygons results is introduced. New concepts of sample variance and standard deviation are studied.

The study of the latter requires not only an understanding of the fundamentals given earlier, but also a more detailed and attentive attitude, because in mathematics, as in life, the further, the more difficult.

Of course, as in all disciplines, the school course for studying probability theory has its own special method of studying theorems, the main of which are the theorem of addition of probabilities and their consequences and the theorem of multiplication of probabilities. The study of theorems must be demonstrated with specific examples illustrating their application, but we will leave this to the school teachers, and we will simply announce the content of these theorems, and so, the probability addition theorem sounds like this: "the probability of the sum of two incompatible events is equal to the sum of the probabilities of these events", and , respectively, the formula to this theorem is P (A + B) = P (A) + P (B). Probability multiplication theorem “The probability of the product of two events is equal to the product of the probability of one event by the conditional probability of another, provided that the first event happened”, the formula for it looks like this P (AB) = P (A) * P (B / A). Along with these theorems, set theory is also studied in the course of mathematics, a branch of mathematics in which the general properties of sets-collections of elements of arbitrary nature that have some general property are studied. If students have knowledge of set theory, they will be able to see the connection between operations on events and operations on sets ... Thanks to this, students will be able to conclude that objects and relationships in probability theory are similar to objects and relationships in set theory. The difference lies in the names of the terms used. At first, it is necessary to create a summary table that reflects the basic information. Experiment Number n of possible outcomes of the experiment Event A Number of outcomes, favorable for this event Probability of the occurrence of event A: P (A) = m / n Throw a coin2 Heads fell 11/2 Draw out an examination ticket 24 Drawn out an unlucky ticket 11/24 Roll a die6 On a die, the number of points drawn33/6 = 1/2 Play the lottery / 250 = 250

In the process of studying operations on events, it is necessary to use as many examples as possible, which reflect not only the essence of these operations, but also the differences in them. Students can easily find the sum and the product of events using the definition. The difficulty lies in developing students' understanding and awareness of the essence of operations on events. To do this, you can use various tasks for working with operations on events. The problem that you may encounter in explaining this topic is the difficulty of isolating simple events. The solution is obvious, the whole point is in experience, the more problems are solved, the more understanding and a minimum of erroneous judgments. The study of this topic will lead students to a much detailed understanding and comprehension of concepts such as "elementary events", "incompatible events", "reliable events", "impossible events "," opposite events ", since all these concepts can be defined on the basis of an operation on events. Of course, any system has its drawbacks and remarks. One of the flaws of the generally accepted definition of probability is its limited use, since it is suitable only for classical experiments that are not so common in modern practice. the number of approaches to the interpretation of the concept of probability. One of the most important approaches from a practical point of view is the statistical approach to defining the concept of "probability". Its implementation is considered as the next stage in the formation of probabilistic theory in students. Mastering the statistical definition of the concept of "probability" is important for its subsequent application in the sections of mathematical statistics to assess the statistical characteristics of a wide class of phenomena of a different nature. Practice has shown that the study of probability theory is a very laborious and difficult process for students at school, and it is just as difficult for teachers , in terms of its transmission to students. Therefore, it does not simplify any mistakes and shortcomings, which, say, can be made in fine arts and music lessons, primarily because it is consistent, structural, and every particle of its structure complements each other.

References to sources 1. Morozova E.V. Ways of development of logical thinking and logical reflection of students in the context of modernization of school education // Modern problems of science and education. –2014. –№ 5; URL: http://www.scienceeducation.ru/ru/article/view?id=14962 (date of access: 10.02.2016). Dorofeev, I.F. Sharygin, S.B. Suvorova. Tutorial: Algebra. Grade 7: textbook for general education institutions / –M .: Education 2014 –288 p.3.G. V. Dorofeev, S. B. Suvorova, E. A. Bunimovich and others Algebra. Grade 8: study, for general education. institutions / A45; ed. G. V. Dorofeeva; Grew up. acad. Sciences, Ros. acad. education, publishing house "Enlightenment". - 5th ed. -M. : Enlightenment, 2010.-288 p. 4 See: G.V. Dorofeev, I.F. Sharygin, S.B. Suvorova. Tutorial: Algebra. Grade 7: textbook for general education institutions / –M .: Education 2014 –288 p. 5.

N.L.Stefanov, N. S. Podkhodov. Methods and technology of teaching mathematics. Course of lectures: a manual for universities /. -M. : Bustard, 2005.-416 p. 6.

See: N. L. Stefanov, N. S. Podkhodov. Methods and technology of teaching mathematics. Course of lectures: a manual for universities /. -M. : Bustard, 2005. -416 p.