L a efrosinina educational reader.

In everyday life, in practical and scientific activity We often observe certain phenomena and conduct certain experiments. An event that may or may not occur during an 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. Every random event is a consequence of the action of 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 of all these reasons on the result, since their number is large and the laws of action are unknown. The patterns 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 occur or not - it simply cannot do this. If we're talking about about mass homogeneous random events, then they obey certain patterns, namely probabilistic patterns. First, let's look at the classification of events. A distinction is made between joint and non-joint events. Events are called joint if the occurrence of one of them does not exclude the occurrence of the other. Otherwise, the events are called incompatible. For example, two dice are tossed. Event A - getting three points on the first dice, event B - getting 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 different color. Event A - a box taken at random will contain black shoes, event B - the box will contain brown shoes, A and B are incompatible events. An event is called reliable if it is sure to occur under the conditions of a given experience. An event is called impossible if it cannot occur under the conditions of a given experience. For example, the event that a standard part will be taken from a batch of standard parts is reliable, but a non-standard part is impossible. An event is called possible, or random, if as a result of experience it may appear, but may not appear. An example of a random event could be the identification of product defects during inspection of a batch of finished products, a discrepancy between the size of the processed product and the specified one, or the failure of one of the links in the automated control system. Events are called equally possible if, according to the test conditions, none of these events is objectively more possible than the others. For example, let a store be supplied with light bulbs (and in equal quantities) several manufacturing plants. Events involving the purchase of a light bulb from any of these factories are equally possible. The important concept is the complete group of events. Several events in this experiment form full group, if at least one of them is sure to appear as a result of the experiment. For example, an urn contains ten balls, six of them are red, four are white, and five balls have numbers. A - the appearance of a red ball with one draw, B - the appearance of a white ball, C - the appearance of a numbered ball. Events A,B,C form a complete group joint events . The event may be the opposite, or additional. An opposite event is understood as an event that must necessarily occur if some event A does not occur. Opposite events are incompatible and are the only possible ones. They form a complete group of events. For example, if a batch of manufactured products consists of good and defective products, 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. They throw a dice (i.e. a small cube with points 1, 2, 3, 4, 5, 6 stamped on its sides). When throwing a die, one point, two points, three points, etc. may appear on its top face. Each of these outcomes is random. We carried out such a test. The dice were thrown 100 times and the number of times the event “the die scored 6” occurred was observed. It turned out that in this series of experiments the “six” fell 9 times. The number 9, which shows how many times the event in question occurred in this trial, is called the frequency of this event, and the ratio of the frequency to the total number of trials, equal, 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 recorded whether the event A of interest to us has occurred or not. The probability of the event is denoted by the capital letter P. Then the probability of the event A will be denoted by: P(A). Classic definition of probability: The probability of event A is equal to the ratio of the number of cases m, favorable to it, from the total number n of uniquely possible, equally possible and incompatible cases to the number n, i.e. Therefore, to find the probability of an event it is necessary: ​​to consider various test outcomes; find a set of uniquely possible, equally possible and incompatible cases, count their total number n, the number of cases m, favorable for a given 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 the favorable number of cases is from the total number of cases: Consider another example. There are 10 balls in the box. 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 red, green and white balls constitute a complete group of events. Let us denote the appearance of a red ball as event A, the appearance of a green ball as event B, and the appearance of a white ball as event C. Then, in accordance with the formulas written above, we obtain: ; ; Note that the probability of one of the two occurring in pairs 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 experiences as a result of which event A occurred to the total number of experiences. The difference between relative frequency and probability is that probability is calculated without direct experimentation, and relative frequency is calculated after experimentation. So, in the example discussed above, if 5 balls are drawn at random from the box and 2 of them turn out to be red, then the relative frequency of occurrence of the red ball is equal to: 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 probability of an event. Geometric probability. The classical definition of probability assumes that the number of elementary outcomes is finite, which also limits its use in practice. In the case where there is a test with an infinite number of outcomes, the definition of geometric probability is used - the point falling into the region. When determining the geometric probability, it is assumed that there is a region N and in it a smaller region M. A point is thrown at random onto the region N (this means that all points of the region N are “equal” in terms of whether a randomly thrown point falls there). Event A is “the thrown point hits area M.” Region M is called favorable for event A. The probability of falling into any part of region N is proportional to the measure of this part and does not depend on its location and shape. The area to which geometric probability extends can be: a segment (the measure is length) a geometric figure on the plane (the measure is area) a geometric body in space (the measure is volume) Let us give a definition of geometric probability for the case of a plane figure. Let region M be part of region N. Event A consists of a randomly thrown point on region N falling into region M. The geometric probability of event A is the ratio of the area of ​​region M to the area of ​​region N: In this case, the probability of a randomly thrown point hitting the boundary of the region is considered equal to zero . Let's look at an example: Mechanical watches with a twelve-hour dial broke and stopped working. Find the probability that hour hand froze, reaching the 5 o'clock mark, but did not reach the 8 o'clock mark. Solution. 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 said to be equal if the implementation of event A entails the implementation of event B and vice versa. The union or sum of events is the 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 is rolled; 4; 6 or 3 points" A B - "6 points rolled" A - B - "2 and 4 points rolled" Additional to event A is an event that means that event A does not occur. Elementary outcomes of experience are those results of experience that are mutually exclusive and as a result of the experience one of these events occurs, and whatever event A is, by the elementary outcome that occurs one can judge whether this event occurs or does not occur. The set of all elementary outcomes of experience is called the space of elementary events. Properties of probability: Property 1. If all cases are favorable for a given event A, then this event will definitely occur. Consequently, the event in question is reliable, and the probability of its occurrence, since in this case Property 2. If there is not a single case favorable to a given event A, then this event cannot occur as a result of experience. Consequently, the event in question is impossible, and the probability of its occurrence, since in this case m = 0: Property 3. The probability of the occurrence of events that form a complete group is equal to one. Property 4. The probability of the occurrence of an opposite event is determined in the same way as the probability of the occurrence of event A: where (n-m) is the number of cases favorable to the occurrence of the opposite event. Hence, the probability of the occurrence of the opposite event is equal to the difference between one and the probability of the 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 said to be equal if each of them is a special case of the other. If events A and B are equal, we write A = B. The sum of events A and B is the 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 the occurrence of one of two incompatible events is equal to the sum of the probabilities of these events. P=P+P Note that the formulated theorem 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 The product of events A and B is called the event AB, which occurs then and only when both events A and B occur simultaneously. Random events A and B are called joint if both of these events can occur during a given test. Theorem on the addition of probabilities 2. The probability of the sum of joint events is calculated using the formula P=P+P-P Examples of problems on the addition theorem. At 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 question on the topic “Parallelogram” 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 in the exam. Solution. 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 a shopping center, two identical 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 at the end of the day there will be coffee left in both machines. Solution. Let's consider events A - “coffee will run out in the first machine”, B - “coffee will run out in the second machine”. Then A·B - “coffee will run out in both machines”, A + B - “coffee will run out in at least one 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(A·B) = 0.3 + 0.3 ? 0.12 = 0.48. Therefore, the probability of the opposite event, that the coffee will remain in both machines, is equal to 1? 0.48 = 0.52. Answer: 0.52. Events A and B are called independent if the occurrence of one of them does not change the probability of the occurrence 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 occurs or not. The conditional probability P(A|B) of event A is the probability calculated given that event B occurred. Similarly, P(B|A) denotes the conditional probability of event B given that A occurs. For independent events, by definition, P(A|B) = P(A); P(B|A) = P(B) Multiplication theorem for dependent events The probability of the product of dependent events is equal to the product be0.01 = 0.0198 + 0.0098 = 0.0296. Answer: 0.0296.

In 2003, it was decided to include elements of probability theory in the school mathematics course secondary school(instruction letter No. 03-93in/13-03 dated 09/23/2003 of the Ministry of Education of the Russian Federation “On the introduction of elements of combinatorics, statistics and probability theory into the content of mathematics education in primary schools”, “Mathematics at School”, No. 9 for 2003). By this time, elements of probability theory had been present in various forms for more than ten years in well-known school algebra textbooks for different classes (for example, I.F. “Algebra: Textbooks for grades 7-9 of general education institutions,” edited by G.V. Dorofeev; “ Algebra and the beginnings of analysis: Textbooks for grades 10-11 of general education institutions "G.V. Dorofeev, L.V. Kuznetsova, E.A. Sedova"), and in the form of separate textbooks. However, the presentation of material on probability theory in them, as a rule, was not systematic, and teachers, most often, did not refer to these sections and did not include them in the curriculum. The document adopted by the Ministry of Education in 2003 provided for the gradual, stage-by-stage inclusion of these sections in school courses, giving the teaching community the opportunity to prepare for the corresponding changes. In 2004-2008 A number of textbooks are being published to supplement existing algebra textbooks. These are the publications 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: A manual for teachers”, Makarychev Yu.N., Mindyuk N.G. “Algebra: elements of statistics and probability theory: textbook. A manual for students of grades 7-9. general education institutions”, Tkacheva M.V., Fedorova N.E. “Elements of statistics and probability: Textbook. Manual for 7-9 grades. general education institutions." Methodological manuals were also published to help teachers. Over the course of a number of years, all these teaching aids have been tested in schools. In conditions where the transition period of implementation in school programs ended, and the sections of statistics and probability theory took 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 sections of mathematics at school. This absence, wittingly or unwittingly, provoked the authors of textbooks to compare them with existing textbooks for universities. The latter, depending on the established traditions in individual specializations high school often allowed significant terminological discrepancies 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 higher school textbooks. With a greater degree of accuracy, we can say that the choice of a specific educational material in new branches of mathematics for the school concerning the concept of “random”, occurs in currently in the most random way, right down to the names and designations. Therefore, teams of authors of leading school textbooks on probability theory and statistics decided to join forces under the auspices of the Moscow Institute of Open Education to develop agreed positions on unifying the basic definitions and notations used in school textbooks on probability theory and statistics. Let us analyze the introduction of the topic “Probability Theory” in school textbooks. General characteristics: The content of teaching the topic “Elements of Probability Theory”, highlighted in the “Program for General Education Institutions. Mathematics”, ensures the further development of students’ mathematical abilities, orientation towards professions significantly related to mathematics, and preparation for studying at a university. The specificity of the mathematical content of the topic under consideration allows us to specify the identified main task of in-depth study of mathematics as follows. 1. Continue revealing the content of mathematics as a deductive system of knowledge. - build a system of definitions of basic concepts; - identify additional properties of the introduced concepts; - establish connections between introduced and previously studied concepts. 2. Systematize some probabilistic methods for solving problems; reveal the operational structure of searching for solutions to certain types of problems. 3. Create conditions for students to understand and understand the main idea practical significance probability theory by analyzing basic theoretical facts. Reveal practical applications of the theory studied in this topic. Achieving the educational goals will be facilitated by solving the following tasks: 1. To form an idea of ​​different ways of determining the probability of an event (statistical, classical, geometric, axiomatic) 2. To develop knowledge of basic operations on events and the ability to use them to describe some events through others. 3. Reveal the essence of the theory of addition and multiplication of probabilities; determine the limits of 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 faulty is 0.02. Before packaging, 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 will mistakenly reject a working battery is 0.01. Find the probability that a battery randomly selected from the package will be rejected. Solution. 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 was correctly rejected” or B - “the battery is working, but was rejected by mistake.” These are incompatible 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)probability of one of them to the conditional probability of the other, provided that the first occurred: P(A B) = P(A) P(B|A) P(A B) = P(B) P(A| B) (depending on which event happened first). Corollaries of the theorem: Multiplication theorem for independent events. The probability of a 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 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 probability 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. Solution. The possibility of winning the first and second games does not depend on each other. The probability of a 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 probability 0.05, regardless of the other machine. Find the probability that at least one machine is working. Solution. Let's find the probability that both machines are faulty. These events are independent, the probability of their occurrence is equal to the product of the probabilities of these events: 0.05 · 0.05 = 0.0025. An event consisting in the fact that at least one machine is working, the opposite. Therefore, its probability is 1? 0.0025 = 0.9975. Answer: 0.9975. Formula of total probability A consequence of the theorems of addition and multiplication of probabilities is the formula of total probability: 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 event A: P(A) = P(B1) total probability. 5. Create a prescription that allows you to rationally choose one of the algorithms when solving a specific problem. We will supplement the identified educational goals for studying the elements of probability theory by setting developmental and educational goals. Developmental goals: to form in students a sustainable 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 discovery of new ways to solve problems and tasks; application of knowledge in new situations and circumstances; develop the ability to explain facts, connections between phenomena, transform material from one form of representation to another (verbal, sign-symbolic, graphic); learn to demonstrate the correct application of methods, see the logic of reasoning, the similarities and differences 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; form the needs of the individual, motives social behavior, activities, values ​​and value orientations; to educate a person capable of self-education and self-education. Let us analyze the algebra textbook for grade 9 “Algebra: elements of statistics and probability theory” Makarychev Yu.N. This textbook is intended for students in grades 7-9, it complements the textbooks: Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. “Algebra 7”, “Algebra 8”, “Algebra 9”, edited by Telyakovsky S.A. The book consists of four paragraphs. Each paragraph contains theoretical information and relevant exercises. At the end of the paragraph there are exercises for repetition. For each paragraph there are given additional exercises more high level difficulty compared to basic exercises. According to the “Program for General Education Institutions”, 15 hours are allocated for studying the topic “Probability Theory and Statistics” in the school algebra course. The material on this topic falls on grade 9 and is presented in the following paragraphs: §3 “Elements of combinatorics” contains 4 points: Examples of combinatorial problems. Simple examples demonstrate the solution of combinatorial problems by enumerating possible options. This method is illustrated by constructing a tree of possible options. The multiplication rule is considered. Rearrangements. The concept itself and the formula for calculating permutations are introduced. Placements. The concept is introduced on specific example. The formula for the number of placements is derived. Combinations. Concept and formula for the number of combinations. The purpose of this paragraph is to give students various ways of describing all possible elementary events in various types random experience. §4 “Initial information from probability theory.” The presentation of the material begins with a review 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 paragraph “adding and multiplying probabilities.” Theorems of addition and multiplication of probabilities are considered, and the associated 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 to individual work or in extracurricular activities with students. Guidelines To this textbook are given in a number of articles by Makarychev and Mindyuk (“Elements of combinatorics in a school algebra course”, “Initial information from probability theory in a school algebra course”). And also some critical comments on this textbook are contained in the article by Studenetskaya and Fadeeva, which will help to avoid mistakes when working with this textbook. Goal: transition from a qualitative description of events to a mathematical description. Topic “Theory of Probability” in the textbooks of Mordkovich A.G., Semenov P.V. for grades 9-11. On this moment One of the current textbooks at school is the textbook by Mordkovich A.G., Semenov P.V. “Events, probabilities, statistical data processing”, it also has additional chapters for grades 7-9. Let's analyze it. According to the “Work Program for Algebra,” 20 hours are allotted for studying the topic “Elements of Combinatorics, Statistics and Probability Theory.” The material on the topic “Probability Theory” is covered in the following paragraphs: § 1. The simplest combinatorial problems. Multiplication rule and tree of options. Rearrangements. It begins with a consideration of simple combinatorial problems, a table of possible options is considered, which shows the principle of the multiplication rule. Trees of possible options and permutations are then considered. After the theoretical material there are exercises for each of the subpoints. § 2. Selecting several elements. Combinations. First, the formula is displayed for 2 elements, then for three, and then the general one for n elements. § 3. Random events and their probabilities. The classical definition of probability is introduced. The advantage of this manual is that it is one of the few that contains paragraphs that discuss tables and option trees. These points are necessary, since it is tables and option trees that teach students the presentation and initial analysis of data. Also in this textbook, the combination formula is successfully introduced, first for two elements, then for three and generalized for n elements. On combinatorics, the material is presented just as successfully. Each paragraph contains exercises, which allows you to consolidate the material. Comments on this textbook are contained in the article by Studenetskaya and Fadeeva. In 10th grade this topic three paragraphs are given. In the first of them, “The Rule of Multiplication. Permutations and Factorials,” in addition to the multiplication rule itself, the main emphasis was on the derivation from this rule of two main combinatorial identities: for the number of permutations and for the number of all possible subsets of a set consisting of n elements. In this case, factorials are introduced as convenient way abbreviated notation of the answer in many specific combinatorial problems earlier than the very concept of “permutation”. In the second paragraph of grade 10 “Selecting several elements. Binomial coefficients" considered classical combinatorial problems associated with the simultaneous (or sequential) selection of several elements from a given finite set. The most significant and truly new for the Russian secondary school was the final paragraph “Random events and their probabilities.” It examined 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 test with two outcomes. This is the most important probabilistic model from a practical point of view (Bernoulli Tests), which has a significant number of applications. The latter material formed a transition between the content of educational material in grades 10 and 11. In the 11th grade, two paragraphs of the textbook and a problem book are devoted to the topic “Elements of Probability Theory”. § 22 deals with geometric probabilities; § 23 repeats and expands knowledge about independent repetitions of tests with two outcomes.

(from work experience)

mathematic teacher

gymnasium No. 8 named after L.M. Marasinova

Rybinsk, 2010

Introduction 3

1. Software-content design of a stochastic line in secondary school 4

3. Methodological notes: from work experience 10

4. Probability graph – a visual tool of probability theory 13

5. Module “Entropy and Information” - meta-subject of the school course Probability Theory 19

6.Organization of project and research activities of students when mastering the course theory of probability 24

Annex 1. Thematic site "Probability Theory". Abstract and multimedia manual 27

Appendix 2. Analysis of educational and methodological complexes for the effectiveness of introducing the stochastic line into school education 31

Appendix 3. Control test. Electronic control system 33

Appendix 4. Test No. 1 34

Appendix 5. Routing topics "Elements of probability theory" 36

Appendix 7. Presentation for the lesson “Subject of probability theory. Basic concepts" 53

Appendix 8. Technological map for constructing the lesson “Conditional probability. Total probability" 60

Appendix 9. Technological map for constructing the lesson “Random events and gambling” 63

Appendix 10. Methodological manual “Entropy and information. Solution logical problems" 36s. 66

Appendix 11. “Entropy and information” multimedia – complex. CD – disk, teaching aid. 12s. 67

Appendix 12. Booklet of the thematic module “Entropy and information” 68

Appendix 13. Technological map for constructing the lesson “Solving logical problems by calculating entropy and the amount of information” 69

Appendix 14. Thematic abstract “History of the formation of probability theory” 73

Appendix 16. Presentation of the launch of the project “Theory of Probability and Life” 78

Appendix 17. Booklet “From probability theory to the theory of gambling” within the framework of the project “Probability Theory and Life” 80

Appendix 18. Presentation “Children in the world of adult vices” within the framework of the project “Theory of Probability and Life” 81

Appendix 19. Abstract of the research work “Probabilistic games” for 8th grade students 83

Appendix 20. Presentation for the research work “Probability Games” 86

Introduction

The most effective way to develop these skills is in the course “Probability Theory and Mathematical Statistics”, the need for which to be studied in Russian schools by scientists has been debated over the last century. At different periods of formation Russian education approaches to the stochastic line have varied from its complete exclusion from mathematics education in high school to partial and complete study of basic concepts. One of the main aspects of the modernization of Russian school mathematics education in the 21st century is the inclusion of probabilistic theoretical knowledge in general education. The stochastic line (a combination of elements of probability theory and mathematical statistics) is intended to form an understanding of determinism and randomness, to help realize that many laws of nature and society are probabilistic in nature, real phenomena and processes are described by probabilistic models.

As a student at Yaroslavl State Pedagogical University named after K.D. Ushinsky, under the guidance of Professor V.V. Afanasyev, I was quite actively involved in this particular course, methods of solving problems and studying theoretical knowledge, searching for applied opportunities. The introduction of probability theory into the second generation standards increased the relevance of the formed body of knowledge, understanding of the importance of human probabilistic culture, and the need to search for methodological and didactic “highlights”.

The practical significance and novelty of the presented work experience lies in its author's exclusive use of systematic graphs in solving problems, in the methodological and didactic meta-subject of the formation of information culture. The program requirements of the standards are continued in the project and research activities of the teacher and students. The openness of the experience is confirmed by a working thematic website 1, that is, the possibility of repeated broadcast and interpretation.

On the pages of this work, the experience of programmatic and meaningful construction of the stochastic line of mathematics in general and probability theory in particular is presented, methodological advice is offered on the use of methodological and didactic techniques studying theory and application in practice. A feature of the author’s experience in mastering the course in probability theory is the presentation of the subject with the systematic use of graphs, which makes the material under consideration more visual and accessible. Options for using modern interactive learning and knowledge control tools are proposed: interactive whiteboard, electronic knowledge control systems. The appendices present specific results of the joint work of the teacher and students of gymnasium No. 8 named after L.M. Marasinova.

1. Software-content design of a stochastic line in high school

The problem of choosing an appropriate educational and methodological complex that most fully accompanies educational process, and the selection of those didactic techniques that will allow optimal implementation of the required tasks of stochastic education. A detailed substantive analysis of the teaching and learning systems in force at the time of 2007 is presented on the pages of the author’s thematic site 2 (Appendix 2).

An analysis of the approved educational and methodological complexes shows that the compulsory mastery of the stochastic line of mathematics in primary school and at the 3rd stage of education, only the textbook by G.V. Dorofeev and I.F. Sharygina suggests the following:

• 
  Grade 5 – in the topic “Natural Numbers” - “Data Analysis”
• 
  Grade 6 - Combinatorics (6 hours) and Probability of random events (9 hours)
• 
  Grade 7 - Frequency and probability (6 hours);
• 
  Grade 8 – Probability and Statistics (5 hours)
• 
  Grade 9 – Statistical research (9 hours)

• 
  Grades 8-9: Sets and elements of combinatorics.
• 
  10-11th grade – Elements of combinatorics and probability theory. Elements of probability theory and mathematical statistics.

To compensate for the lack of content in textbooks, the authors of some of them have developed additional paragraphs for the algebra course for grades 7-9, offering lesson planning: A.G. Mordkovich and P.V. Semenov; M.V. Tkachev and N.E. Fedorov “Elements of statistics and probability”

Such manuals have not yet been developed for other educational and methodological complexes. The way out for the teacher - practice from the current situation is the author's development of a work program, an elective course, taking into account all the contradictions that have arisen regarding the introduction of the stochastic line into the high school course and the proposed ways to resolve them.

Considering that no science should be mastered by students separately, in isolation from each other, I made an attempt to find a meaningful interpenetration of geometry, algebra, arithmetic, computer science and stochastics.

Funding of the mathematics section of the basic school

“Elements of logic, combinatorics, statistics and probability theory” (45 hours)

5
Arithmetic:

operations with natural numbers

Sets and combinatorics
Class
6
Probability of random events
Arithmetic:

operations with fractions;

average
Class

Statistical data, random variables

Computer science:

Working with diagrams (Excel)

7th grade

Proof

Geometry: Proving Theorems

8
Geometric probability

Geometry:

areas of figures;

Funding of the secondary school mathematics section

“Elements of combinatorics, statistics, probability theory”

20 hours – base, 25 hours – prof. humanitarian,
Combinatorics formulas

Solving combinatorial problems

Tabular and graphical presentation of data

Incompatible events

their probability

Elementary and complex events

Solving practical problems using probabilistic methods, graph method
20 hours – prof. mathematical

Grade 10

Thus, by creatively building a work program, the teacher has the opportunity to use the educational base of other sections or science, creating conditions for the meta-subjectivity of each issue. But the teacher’s creativity does not end there. Much greater opportunities for the manifestation of authorship and, accordingly, creativity of a mathematics teacher appear with the choice of didactic techniques for introducing and further applying the basic concepts of the stochastics course. Structurally author's vision of the spiral foundation of the concepts of probability theory in secondary school in conjunction with additional education as follows

1. 
   Basic concepts of probability theory

Any exact science studies not the phenomena themselves that occur in nature and in society, but their mathematical models, that is, the description of phenomena using a set of strictly defined symbols and operations on them. At the same time, to construct a mathematical model of a real phenomenon, in many cases it is enough to take into account only the main factors and patterns that make it possible to predict the result of an experiment (observation, experiment) based on its given initial conditions. However, there are many problems for which it is necessary to take into account random factors that add an element of uncertainty to the outcome of the experiment.

Probability theory- a mathematical science that studies the patterns inherent in mass random phenomena. In this case, the phenomena being studied are considered in an abstract form, regardless of their specific nature. That is, probability theory does not consider the real phenomena themselves, but their simplified schemes - mathematical models. The subject of probability theory is mathematical models of random phenomena (events). At the same time, under random occurrence understand a phenomenon whose outcome is impossible to predict (when the same experience is repeated repeatedly, it proceeds slightly differently each time). Examples of random phenomena: a coat of arms falling out when tossing a coin, winning a purchased lottery ticket, the result of measuring a quantity, the duration of the TV, etc. The purpose of probability theory is to make a forecast in the field of random phenomena, influence the course of these phenomena, control them, limit the scope of randomness. Currently, there is practically no field of science in which probabilistic methods are not used to one degree or another.

Random event(or simply: an event) is any outcome of an experience that may or may not happen. Events are usually designated by capital letters of the Latin alphabet: A, B, C, ....

If the occurrence of one event in a single trial excludes the occurrence of another, such events are called incompatible. If, when considering a group of events, only one of them can occur, then it is called the only possible. Most attention mathematicians have been attracted to equally possible events(one of the sides of the cube falls out).

Examples: a) when throwing a dice, the space of elementary events P consists of six points: P = (1,2,3,4,5,6); b) toss the coin twice in a row, then P = (GG, GR, RG, RR), where G is the “coat of arms”, P is the “lattice” and the total number of outcomes (power P) |P| = 4; c) toss a coin until the first appearance of the “coat of arms”, then P = (G, RG, RRG, RRRG,...). In this case, P is called a discrete space of elementary events.

One is usually interested not in what specific outcome occurs as a result of a trial, but in whether the outcome belongs to one or another subset of all outcomes. All those subsets of A for which, according to the experimental conditions, an answer of one of two types is possible: “the outcome belongs to A” or “the outcome does not belong to A”, we will call events. In example b), the set A = (GG, GR, RG) is the event that at least one “coat of arms” appears. Event A consists of three elementary outcomes of space P, therefore |A| = 3.

The sum of two events A and B is the event C=A+B, which consists of the execution of event A or event B. The product of events A and B is called an event D=A·B, consisting in the joint execution of event A and event B. Opposite to event A is an event consisting in the non-occurrence of A and, therefore, complementing it to P. If each occurrence of event A is accompanied by the appearance of B, then write A to B and say that A precedes B or A entails B.

Historically, the first definition of the concept of probability is the definition that is currently called classical, or classical probability: classical probability event A is called the ratio of the number of favorable outcomes (obligatory to occur) to the total number of incompatible only possible and equally possible outcomes: P(A) = m/n, where m is the number of outcomes favorable for event A; n is the total number of incompatible only possible and equally possible outcomes. In terms of the meaning of randomness, all events can be classified as follows:

Several events are called joint, if the occurrence of one of them in a single trial does not exclude the occurrence of other events in the same trial. Otherwise the events are called incompatible.

The two events are called dependent, if the probability of one event depends on the occurrence or non-occurrence of another. The two events are called independent, if the probability of one event does not depend on the occurrence or non-occurrence of another. Several events are called collectively independent if each of them and any combination of other events are independent events. Several events are called pairwise independent, if any two of these events are independent.

The requirement for joint independence is stronger than the requirement for pairwise independence. This means that several events can be independent in pairs, but they will not be independent in the aggregate. If several events are independent in the aggregate, then their pairwise independence follows. Due to the fact that in the future it will often be necessary to consider the probabilities of some events depending on the occurrence or non-occurrence of others, it is necessary to introduce another concept.

Conditional probability RA(B) is the probability of event B calculated given that event A has already occurred.

One of the most important concepts in probability theory (along with a random event and probability) is the concept random variable .

A random variable is understood as a quantity that, as a result of experiment, takes on one or another value, and it is not known in advance which one. Examples of a random variable include: 1) X - the number of points that appear when throwing a dice; 2) Y - the number of shots before the first hit on the target; 3) Z - device uptime, etc. A random variable that takes a finite or countable set of values ​​is called discrete. If the set of possible values ​​of a random variable is uncountable, then such a value is called continuous.

That is, a discrete random variable takes individual values ​​isolated from each other, and a continuous random variable can take any values ​​from a certain interval (for example, values ​​on a segment, on the entire number line, etc.). Random variables X and Y (examples 1) and 2)) are discrete. The random variable Z (example 3)) is continuous: its possible values ​​belong to the interval . Example. The experiment consists of tossing a coin 2 times. You can consider a random event - the appearance of a coat of arms and a random variable X - the number of appearances of the coat of arms.

The main characteristics of a random variable are position characteristics (mathematical expectation, mode, median) and dispersion characteristics (variance, standard deviation).

Expected value is calculated using the formula M[X]=Σxipi and characterizes the average value of a random variable.

Fashion (M 0 ) – this is the value of a random variable for which the corresponding probability value is maximum.

Median discrete random quantity (Me) is such a value x k in a series of possible values ​​of a random variable, which it takes with certain probability values, such that it is approximately equally likely that the process will end before x k or continue after it.

Variance(scattering) of a discrete random variable is the mathematical expectation of the squared deviation of a random variable from its mathematical expectation: D[X]=M(X-M[X]) 2 = M[X 2 ]-M 2 [X].

Standard deviation random variable X is called a positive value square root from the variance: σ[Х]=.

Problems related to the concepts of a random event and a random variable can be effectively considered through a graphical illustration using a probability graph, on the edges of which the corresponding probability values ​​are inscribed.

Let the probability of winning one game for the first player be equal to 0.3, and the probability of winning for the second player, respectively, equal to 0.7. How to split the bet in this case?

Answer: proportional to the probability of winning.

A distribution law can have a geometric illustration in the form of a distribution graph.

As you know, there are two in the community bookshelves, dedicated to literature in probability theory and mathematical statistics
Literature on probability theory and mathematical statistics (part 1)
Literature on probability theory and mathematical statistics (part 2)
However, it contains mainly books for university students and teachers.
This entry will be devoted to books on probability theory and mathematical statistics for schoolchildren and teachers.
Some of them have already been posted in the community.
The books are arranged in alphabetical order.

Books on probability theory and mathematical statistics for schoolchildren and teachers

Bunimovich B. A., Bulychev V. A. Probability and statistics. Grades 5-9: A manual for general education. textbook avedenia M.: Bustard, 2002. - 160 p.: ill. - (School course topics). ISBN 5-7107-4582-0 The manual contains the necessary theoretical and practical material for studying the probabilistic-statistical line, which today is becoming an integral part of the school mathematics course. The study of probability is expected within the framework basic course mathematics grades 5-9. To successfully master it, it is enough to master the basic theoretical material and solving problems of group A. The manual can be used in conjunction with any existing mathematics textbook. Provided by Robot Download (djvu/rar, 2.41 MB) ifolder.ru || onlinedisk

Varga T. Mathematics 1. Flowcharts, punched cards, probabilities: (Mathematical games and experiments) Per. with him. - M.: Pedagogy, 1978. - 112 p., ill. The book reveals effective ways to introduce such sections of modern mathematics into school teaching as introduction to probability theory, flowcharts and punched cards. The author focuses on learning by children aged 10-14 years mathematical concepts during entertaining games. Found on the web Download (djvu/rar, 1.74 MB) ifolder.ru || onlinedisk

Varga Tamash, Esther Nemeny-Chervenak, Maria Halmos Mathematics 2. Plane and space. Trees and graphs. Combinatorics and probability. (Mathematical games and experiments). Per. with him. E.Ya. Gabovich. M.: Pedagogika, 1978. – 112 p. with ill. The book reveals effective ways for the early introduction into school teaching of a number of concepts of the geometry of polygons and polyhedra, the introduction of such concepts of modern mathematics as the simplest graphs, trees and probability, as well as their simplest applications. Chapter 3 of the book is a continuation of Chapter 3 of the book "Mathematics 1", together forming the basis of an introduction to probability theory for middle and high school students school age. The author focuses on the acquisition of mathematical concepts by children aged 10–14 years through entertaining games. The publication is intended for methodologists and teachers organizing experimental work to determine effective methods teaching mathematics. For the book Thanks a lot Ak-sakal Download (djvu, 2.6 MB) ifolder.ru || onlinedisk

Vysotsky I. R., Yashchenko I. V. Unified State Examination 2012. Mathematics. Problem B10. Probability theory. Workbook M.: MTsNMO, 2012. -48 p. ISBN 978-5-94057-860-4 At various stages of training, the manual will help to provide a leveled approach to organizing repetition, to monitor and self-monitor knowledge on the topic “Probability Theory”. The workbook is focused on one academic year, however, if necessary, it will allow as soon as possible fill gaps in the graduate’s knowledge. The notebook is intended for students high school, mathematics teachers, parents. Book provided Robot Download (djvu/rar, 690 kb) rghost.ru || onlinedisk

Kolmogorov A. N., Zhurbenko I. G., Prokhorov A. V. Introduction to the theory of probability. - M., Nauka, 1982. - 160 p. - Bible book "Quantum". Issue 23 The book introduces the basic concepts of probability theory using simple examples. Along with the combinatorial definition of probability, we consider statistical definition. Analyzed in detail random walk on a straight line, describing the physical processes of one-dimensional Brownian motion of particles, as well as a number of other examples. For schoolchildren, students, teachers, people engaged in self-education. Found on the web Download (djvu, 1.9 MB) Rapida || rghost.ru

Kordemsky B. A. Mathematics studies randomness. A manual for students. M., “Enlightenment”, 1975. -223 pp. (World of Knowledge). The goal that the author of this book has set for himself is to help the reader independently master the initial concepts and methods of probability theory and the simplest apparatus of mathematical statistics. This is a book for educational reading with a pencil in hand and workbook on the table. In the initial part of the book, a free form of presentation prevails, not constrained by the framework of the program, using entertaining and game material; gradually the book “gets more serious”, but does not lose its accessibility for high school students and readers who have already graduated from high school. To self-test the effectiveness of acquired knowledge and “probabilistic thinking,” the penultimate chapter offers about fifty sketch problems. Some evidence, conclusions and theoretical comments are included in the final chapter, “Appendix”. Book provided Robot Download (djvu/rar, 4.17 MB, 600dpi+OCR) ifolder or onlinedisk

Makarychev Yu. N. Algebra: elements of statistics and probability theory: textbook. manual for students 7-9 grades. general education institutions / Yu. N. Makarychev, N. G. Mindyuk; edited by S. A. Telyakovsky. 3rd ed. - M.: Education, 2005. - 78 p. : ill. - ISBN 5-09-014164-9. This manual is intended for studying probabilistic and statistical material when working on the textbooks “Algebra, 7”, “Algebra, 8”, “Algebra, 9” by Yu. N. Makarycheva, N. G. Mindyuk, K. I. Peshkova, S. B. Suvorova, ed. S. A. Telyakovsky. Book provided by acub Download (djvu, 1.3 mb)mediafire.com || ifolder.ru

F. Mosteller Fifty entertaining probability problems with solutions. - Per. from English M.: Science. Ch. ed. physics and mathematics lit., 1975.- 112 p. The book actually contains 57 entertaining tasks(seven problems are discussed rather than solved). Most tasks are easy. Only a very few of them require knowledge of the analysis course, but even in these cases, an untrained reader will still be able to understand the problem statement and the answer. The book is addressed to a wide range of readers: high school students, teachers, students. Found on the web Download (djvu, 1.9 mb)alleng.ru || mediafire.com

Tyurin Yu. N. et al. Probability theory and statistics / Yu. N. Tyurin, A. A. Makarov, I. R. Vysotsky, I. V. Yashchenko. - M: MTsNMO: JSC "Moscow Textbooks", 2004.- 256 p.: ill. ISBN 5-94057-161-1 djvu version found by Guest Download (djvu, 4.58 MB) ifolder.ru || mediafire.com || rghost.ru pdf version (copied page by page from reshlib) provided jagger777 Download (pdf, 8.6 MB) ifolder.ru || rghost.ru

Tyurin Yu. N. et al. Probability theory and statistics / Yu. N. Tyurin, A. A. Makarov, I. R. Vysotsky, I. V. Yashchenko. - 2nd ed., revised. - M.: MTsNMO: OJSC "Moscow Textbooks", 2008. -256 p.: ill. ISBN 987-5-94057-319-7 The textbook on the basics of probability theory and statistics is designed for students in grades 7-9 of general education institutions. It can also be used in senior secondary schools. This book pays equal attention to statistics and probability theory and their role in the study of phenomena in the surrounding world. The book is intended for initial acquaintance of students with the forms of presentation and description of data in statistics, talks about random events, probabilities and their properties. The appendices provide approximate independent and test papers for grades 7, 8 and 9, explanations are written for the terms encountered. The authors sought to make the presentation simple and did not abuse mathematical formalism. Found on the web Download (djvu, 1.8 MB)mediafire.com ||ifolder.ru

Sheveleva N.V. T.A. Koreshkova, V.V. Miroshin Mathematics (algebra, elements of statistics and probability theory). 9th grade M.: National education, 2011. - 144 p. : ill. - (Short course). ISBN 978-5-905084-55-3 The manual is intended for 9th grade students. It provides information on the main topics of the 9th grade algebra course, as well as on the section “Elements of Statistics and Probability Theory” in a concise and accessible form. Particular attention is paid to the analysis of solutions to typical problems. The book will be useful to students in the learning process, as well as when repeating material in order to systematize it and in preparation for the exam. Book provided Robot Download (djvu/rar, 1.96 MB) ifolder.ru || rghost

Shor E. In the world of accidents. Chisinau, Publishing house “Kartya Moldovenyaska”, 1977. - 90 p. The reader will take a journey into demography, mathematical statistics, psycholinguistics, and, together with the heroes of Poe, will take part in solving the mysterious text. For the success of such an excursion, he will first gain an understanding of probability and methods of calculating it, and no special mathematical training is required from him. The reader will return from the trip enriched with the concepts and methods of probability theory and knowledge of the areas of its application. The brochure will be useful to anyone interested in the world of chance. Found on the web Download (djvu, 1.05 MB) ifolder.ru || mediafire.com

Books on combinatorics
A selection of books by N. Ya Vilenkin
Vilenkin N.Ya Combinatorics. - M., Nauka, 1969. -328 p.
Vilenkin N.Ya. Popular combinatorics. -M., Nauka, 1975. - 208 p.
Vilenkin N.Ya., Vilenkin A.N., Vilenkin P.A. Combinatorics. - M.: FIMA, MTsNMO, 2006. - 400 p.
is in the section Literature on probability theory and mathematical statistics (part 1) .
Next books
Vilenkin N, Ya. Induction. Combinatorics. Manual for teachers. M., “Enlightenment”, 1976
Ezhov I. I., Skorokhod A. V., Yadrenko M. K. Elements of combinatorics. - M., Main editorial office of physical and mathematical literature of the Nauka publishing house, 1977
Savelyev L.Ya. (ed) Olympics. Algebra. Combinatorics. - Novosibirsk, NSU, 1979.
are in the section Literature on preparation for mathematical Olympiads (part II).

Tests in mathematics (probability theory and statistics) for grade 7



Test on probability theory and statistics dated 04/24/2008
(contains options 1-2 of the test on probability theory and statistics, answers and evaluation criteria for the test on probability theory and statistics, and demo versions of the test on probability theory and statistics for grade 7)
Download (pdf/rar, 1.38 MB) ifolder.ru || onlinedisk

Tests in mathematics (probability theory and statistics) for grade 8, compiled by the Moscow Institute of Open Education (MIOO).
Test on probability theory and statistics dated 05/12/2011
Test on probability theory and statistics dated 05/19/2010
Test on probability theory and statistics dated 05/19/2009
(contains options 1-2 of the test on probability theory and statistics, answers and evaluation criteria for the test on probability theory and statistics, and demo versions of the test on probability theory and statistics for grade 8)
Download (pdf/rar, 1.01 MB) ifolder.ru || onlinedisk

Wanted
Afanasyev V. For schoolchildren about probability in games. Introduction to Probability Theory
Ivashev-Musatov O.S. Beginnings of probability theory for schoolchildren
Prosvetov G.I. Probability theory and statistics for schoolchildren: problems and solutions

The course develops in children the need for systematic reading, analysis of what they read and application of acquired knowledge in practice. During classes, schoolchildren perform search, browsing, and selective reading tasks, and learn to present information in a condensed form or in the form of a table. Textbooks for grades 1 and 2 are structured according to a thematic principle, and grades 3 and 4 - according to genre and author, which gives students the opportunity to get acquainted with the diversity of literature and learn to compare texts according to various signs. Each section contains tasks for developing communication skills and special reminders to help complete learning tasks.

Program " Literary reading. grades 1–4" by Efrosinina L.A., reflects the content of teaching literary reading in modern primary school, contains the planned learning outcomes at the primary level of education.

Textbooks for grades 1–4 are intended for mastering the course of literary reading at the level of primary general education and belong to a completed subject line of textbooks, developed in accordance with the requirements regulated by the Federal State Educational Standard of the NEO.

The textbooks contain tasks that develop motivation for reading and learning, Creative skills students, as well as tasks that form junior schoolchildren the need for systematic reading and application of acquired knowledge in practical activities. Throughout the course of literary reading, students complete tasks for selective, search, viewing, studying reading, presenting information from the text read and listened to in a condensed form, in the form of a plan, a simple table, which is necessary at this stage of the formation of semantic reading skills.

The material in textbooks for grades 1 and 2 is organized on a thematic basis, since during this period the main task is to create motivation for reading fiction and accumulation of reading experience. This is reflected in the structure of the sections of the textbook, the methods of selecting material and the sequence of its presentation.

Textbooks for students in grades 3 and 4 are built according to the genre-author principle. In genre blocks, students have the opportunity to compare works of the same genre (folk and author's), generalize genre characteristics; in author's blocks - get an idea of ​​the diversity of one author's work, consolidating ideas about genres and forms of fiction, as well as some features of the author's style. Such a transition from one principle of organizing material to another is traditional for textbooks on literary reading in primary school, corresponds to the psychophysiological capabilities of students and allows for continuity between the primary and main levels of general education, in which the monographic principle becomes the leading one.

For each work, a system of tasks has been developed that organizes the work of students. The clarity of the selection and repeatability of the components of the textbook structure is supported by the system symbols, uniform for the entire line of textbooks. This allows students to easily navigate the textbook when working independently with texts and exercise self-control.

The line's textbooks implement a systemic activity approach: each section of the textbook includes tasks that allow for the comprehensive development of students' communication skills. Tasks that help personalize the learning process, establish interdisciplinary connections, and enrich the vocabulary are also aimed at this. The textbooks contain algorithms (memos) that serve to form regulatory educational actions. Memos help students master and consciously apply methods for solving certain (typical) educational tasks(independent work with the work, preparation expressive reading, reading by heart and by role, detailed and brief retelling, story about the hero of the work, writing a review of the book).

Workbooks on literary reading contain a system of exercises for students to independently work with the text of works included in the textbook and educational anthology. The notebooks contain a variety of developmental and creative tasks that develop a sense of words, enrich speech, and allow for differentiated learning in literary reading lessons.

The educational teaching materials line includes teaching aids, the structure and content of which correspond to the structure and content of textbooks for grades 1–4. Methodical manuals include the course program (according to the class), approximate lesson planning, necessary methodological comments on the lessons, recommendations for monitoring the level of achievement of planned results and for organizing training.

Notebooks for tests and tests on literary reading include current and final comprehensive tests, as well as test tasks based on the works studied and material for self-testing reading ability, allowing you to evaluate learning results.

The Bookman dictionary includes Dictionary concepts and reference material for the course “Literary reading. grades 1–4,” which will help improve students’ learning and reading activities, deepen and generalize the knowledge gained in the lessons. Released as a printed publication and on CD (electronic educational resource), for use in the classroom using an interactive or projection board or at home on a personal computer.