Ministry of Education and Science of the Russian Federation

federal state budgetary educational institution

higher vocational education

"Tula State Pedagogical University. L. N. Tolstoy»

(FGBOU VPO "TSPU named after L. N. Tolstoy")

Department of Algebra, Mathematical Analysis and Geometry

COURSE WORK

in the discipline "Methods of teaching subjects: methods of teaching mathematics"

on the topic:

"METHOD OF STUDYING THE THEORY OF PROBABILITY IN THE SCHOOL COURSE OF MATHEMATICS"

Completed:

3rd year student of group 120922

Faculty of Mathematics, Physics and Informatics

direction "Pedagogical education"

profiles "Physics" and "Mathematics"

Nichepurenko Natalya Alexandrovna

Supervisor:

assistant

Rarova E.M.

Tula 2015

Introduction…………………………………………………………………………...3

Chapter 1: Basic Concepts…………………………………………………………6

1.1 Elements of combinatorics…………………………………………………………6

1.2 Probability theory…………………………………………………………….8

Chapter 2: Methodical aspects of studying the "Probability Theory" in the school course of algebra……………………………………………………….….24

Chapter 3: A fragment of an algebra lesson on the topic “Probability Theory”……….32

Conclusion

Literature

INTRODUCTION

The question of improvement mathematics education in the domestic school was staged in the early 60s of the 20th century by the 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-statistical knowledge into the school curriculum of mathematics is long overdue and does not tolerate further delay. The laws of rigid determination, on the study of which our school education, only one-sidedly reveal the essence of the surrounding world. The random nature of many phenomena of reality is beyond the attention of our schoolchildren. As a result, their ideas about the nature of many natural and social processes are one-sided and inadequate. modern science. They need to be introduced to statistical laws revealing the multifaceted connections of the existence of objects and phenomena.

IN AND. Levin wrote: “... The statistical culture necessary for ... activity must be brought up with early years. It is no coincidence that in developed countries much attention is paid to this: students get acquainted with elements of probability theory and statistics from the very first school years and throughout the training they learn probabilistic-statistical approaches to the analysis of common situations encountered in everyday life.

By the reform of the 1980s, elements of probability theory and statistics were included in the programs of specialized classes, in particular, physics, mathematics and natural sciences, as well as in an optional course in the study of mathematics.

Considering the urgent need to develop individual qualities of students' thinking, author's developments appear optional courses on the theory of probability. An example of this can be the course of N.N. Avdeeva on statistics for grades 7 and 9 and a course of elements of mathematical statistics for grade 10 of high school. In the 10th grade, tests were carried out, the results of which, as well as the observations of teachers and a survey of students, showed that the proposed material was quite accessible to students, caused them great interest showing specific application mathematics to solving practical problems of science and technology.

The process of introducing elements of probability theory into compulsory course school mathematics turned out to be very hard work. There is an opinion that in order to assimilate the principles of probability theory, a preliminary stock of ideas, ideas, habits is needed, which are fundamentally different from those that schoolchildren develop during traditional education as part of familiarization with the laws of strictly conditioning phenomena. Therefore, according to a number of teachers - mathematicians, the theory of probability should enter school mathematics as 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 was recently introduced into the school curriculum, there are currently problems with the implementation of this material in school textbooks. Also, due to the specificity of this course, the number methodological literature also still small. According to the approaches outlined in the vast 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 training 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 form useful skills, and the ability to prove theorems does not apply to such skills.

The origin of probability theory 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?

Assessing the possibility of an event, we often say: "It is very possible", "It will certainly happen", "It is unlikely", "It will never happen". By buying a lottery ticket, you can win, but you can not win; tomorrow at the math lesson you may or may not be called to the blackboard; in the next election, the ruling party may or may not win.

Let's consider a simple example.How many people do you think should be in certain group so that at least two of them have the same birthday with a probability of 100% (meaning the day and month without taking into account the year of birth)? This does not mean leap year, i.e. a year with 365 days. The answer is obvious - there should be 366 people in the group. Now another question: how many people should there be 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!

Here, in order to carry out such interesting calculations andmake unusual discoveries for ourselves, we will study such a section of mathematics "Probability Theory".

The purpose of the course work is to study the foundations of the theory of probability in the school course of mathematics. To achieve this goal, the following tasks were formulated:

1. Consider the methodological aspects of the study"Theory of Probability" in the school course of algebra.
   1. Get acquainted with the basic definitions and theorems on the "Probability Theory" in the school course.
      1. Consider detailed solution tasks on the topic of the course work.
      2. Develop a fragment of the lesson on the topic of the course work.

Chapter 1: Basic Concepts

1.1 Elements of combinatorics

The study of the course should begin with the study of the basics of combinatorics, and the theory of probability should be studied in parallel, since combinatorics is used in calculating probabilities.Combinatorics methods are widely used in physics, chemistry, biology, economics and other fields of knowledge.

In science and practice, there are often problems, solving which you have 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 is the study of ways to count the number of elements in finite sets. Combinatorics formulas are used to calculate probabilities.

Consider some set X, consisting of n elements. We will choose from this set various ordered subsets Y of k elements.

An arrangement of n elements of the set X by k elements is any ordered set () of elements of the set X.

If the choice of elements of the 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 (placement with repetitions).

If the choice is made without a return, i.e. each element of the set X can be chosen only once, then the number of placements from n to k is denoted and determined by the equality

(placement without repetition).

A special case of placement for n=k is called permutation of n elements. The number of all permutations of n elements is

Now let an unordered subset be chosen from the set X Y (the order of the elements in the subset doesn't matter). Combinations of n elements by k are 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 equal to

Valid equalities: ,

When solving problems, combinatorics use the following rules:

Sum rule. If some object A can be chosen from a collection of objects in m ways, and another object B can be chosen in n ways, then either A or B can be chosen in m + n ways.

Product rule. If object A can be chosen from a set of objects in m ways, and after each such choice object B can be chosen in n ways, then the pair of objects (A, B) in the specified order can be chosen in m * n ways.

1.2 Probability theory

In everyday life, in practical and scientific activity we often observe certain phenomena, conduct certain experiments.

An event that may or may not occur during an observation or experiment is calledrandom event. For example, a light bulb hangs under the ceiling - no one knows when it will burn out.Every random event- there 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 of all these causes 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 calledprobability theory.

Probability theory does not set itself the task of predicting whether a single event will occur or not - it simply cannot do it. If we are talking about massive homogeneous random events, then they obey certain laws, namely, probabilistic laws.

First, let's look at the classification of events.

Distinguish events joint and non-joint . 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, toss two dice. 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 of a different color. Event A - a box taken at random will be with black shoes, event B - the box will be with shoes Brown, A and B are incompatible events.

The event is called authentic if it necessarily occurs under the conditions of the given experiment.

The event is called impossible if it cannot occur under the conditions of the given experiment. For example, the event that a standard part is taken from a batch of standard parts is certain, but a non-standard part is impossible.

The event is called possible or random , if as a result of experience it may or may not appear. An example of a random event is the detection of product defects during the control of a batch of finished products, the discrepancy between the size of the processed product and the given one, the failure of one of the links of the automated control system.

The events are calledequally possibleif, under the conditions of the test, none of these events is objectively more likely than the others. For example, suppose a store is supplied with light bulbs (and in equal quantities) by several manufacturers. Events consisting in buying a light bulb from any of these factories are equally probable.

An important concept isfull group of events. Several events in a given experiment form a complete group if at least one of them necessarily appears as a result of the experiment. For example, there are ten balls in an urn, of which six are red and four are white, five of which are numbered. A - the appearance of a red ball in one drawing, B - the appearance of a white ball, C - the appearance of a ball with a number. Events A,B,C form a complete group of joint events.

The event may beopposite, or additional . 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 ones. They form a complete group of events. For example, if a batch of manufactured items consists of good and defective items, then when one item is removed, it can turn out to be either good - event A, or defective - event.

Consider an example. They throw a dice (i.e. a small cube, on the sides of which points 1, 2, 3, 4, 5, 6 are knocked out). When a dice is thrown at his upper face one point, two points, three points, etc. can fall out. Each of these outcomes is random.

Such a test has been carried out. The dice was thrown 100 times and observed how many times the event "6 points fell on the die" occurred. It turned out that in this series of experiments, the “six” fell out 9 times. The number 9, which shows how many times in this trial the event in question occurred, is called the frequency of this event, and the ratio of frequency to total number tests, equal, is called the relative frequency of this event.

In general, let a certain test be carried out repeatedly under the same conditions, and at the same time, each time it is fixed whether the event of interest to us has occurred or not. A. The probability of an event is denoted by a capital letter P. Then the probability of an event A will be denoted: P(A).

The classical definition of probability:

Event Probability A is equal to the ratio of the number of cases m favorable to him, out of the total n the only possible, equally possible and incompatible cases to the number n, i.e.

Therefore, to find the probability events are required:

1. consider different test outcomes;
2. find a set of unique, equally possible and incompatible cases, calculate their total number n , number of cases m favorable to this event;
3. perform a formula calculation.

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 the proportion of the favorable number of cases from the total number of cases:

Let's consider one more example.There are 10 balls in a box. 3 of them are red, 2 are green, the rest are white. Find the probability that a ball drawn at random is red, green, or white. The appearance of red, green and white balloons constitute a complete group of events. Let us denote the appearance of a red ball - event A, the appearance of a green one - event B, the appearance of a white one - event C. Then, in accordance with the formulas written above, we obtain:

Note that the probability of occurrence of one of two pairwise incompatible events is equal to the sum of the probabilities of these events.

Relative frequencyevent 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 lies in the fact that the probability is calculated without the direct product of the experiments, and the relative frequency - after the experience.

So in the above example, if 5 balls are randomly drawn from the box and 2 of them turn out to be red, then the relative frequency of the appearance of a red ball is:

As can be seen, this value does not coincide with the found probability. When enough large numbers In the experiments carried out, the relative frequency changes little, fluctuating around one number. This number can be taken as the probability of the event.

geometric probability.The classical definition of probability assumes that the number of elementary outcomes certainly which also limits its application in practice.

In the event that a test with endless the number of outcomes, use the definition of geometric probability - hitting a point in an area.

When determining geometric probabilities assume that there is an area N and it has a smaller area M. To area N throw a point at random (this means that all points in the area N are “equal” with respect to hitting a randomly thrown point there).

Event A – “hit of the thrown point on the area M". Region M called an auspicious event A.

Probability of hitting any part of the area N 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:

1. segment (the measure is length)
2. geometric figure on a plane (area is the measure)
3. geometric body in space (the measure is volume)

Let us define the geometric probability for the case flat figure.

Let the area M is part of the area N. Event A consists in hitting a randomly thrown on the area N points into area M . geometric probability events A is called the area ratio M to area area N :

In this case, the probability of a randomly thrown point hitting the boundary of the region is considered to be equal to zero.

Consider an example: A mechanical watch with a twelve o'clock dial broke and stopped working. Find the probability that the hour hand is frozen at 5 o'clock but not at 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 occurrence of event A entails the occurrence of event B and vice versa.

Union or sum event is called event A, which means the occurrence of at least one of the events.

Intersection or product events is called event A, which consists in the implementation of all events.

A =∩

difference 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 - “rolled 2; 4; 6 or 3 points"

A ∙ B – “6 points fell out”

A-B – “dropped 2 and 4 points”

Additional event A is called an event, meaning that event A does not occur.

elementary outcomesexperience are called such results of experience that mutually exclude each other and as a result of the experience one of these events occurs, also whatever the event A is, according to the elementary outcome that has come, one can judge whether this event occurs or does not occur.

The totality of all elementary outcomes of experience is calledspace of elementary events.

Probability properties:

Property 1. If all cases are favorable to the given event A , then this event must occur. Therefore, the event in question is authentic

Property 2. If there is no case favorable for this event A , then this event cannot occur as a result of the experiment. Therefore, the event in question 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 the opposite event occurring is defined in the same way as the probability of the occurrence of the event A :

where (n - m ) is the number of cases favoring the occurrence of the opposite event. Hence, the probability of the opposite event occurring is equal to the difference between unity and the probability of the event occurring A :

Addition and multiplication of probabilities.

Event A is called special case event B, if when A occurs, B also occurs. That A is special case of B, we write A ⊂ B .

Events A and B are called equal if each is a special case of the other. The equality of events A and B is written A = B.

sum events A and B is called the event A + B, which occurs if and only if at least one of the events occurs: A or B.

Addition Theorem 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 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

work events A and B is called the event AB, which occurs if and only if both events occur: A and B at the same time. Random events A and B are called joint if both of these events can occur during a given test.

Addition Theorem 2. The probability of the sum of joint events is calculated by the formula

P=P+P-P

Examples of problems on the addition theorem.

1. In the geometry exam, the student gets one question from the list 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 related to these two topics at the same time. Find the probability that the 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.

1. AT 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 by the end of the day there will be coffee left in both vending machines.
   Decision. Consider eventsA - "coffee will end in the first machine", B - "coffee will end in the second machine". Then A B - "coffee will end in both vending machines", A + B - "coffee will end 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) \u003d P (A) + P (B) - P (A B) \u003d 0.3 + 0.3 - 0.12 \u003d 0.48.

Therefore, the probability of the opposite event, that coffee will remain in both machines, is equal to 1 − 0.48 = 0.52.

Answer: 0.52.

The events of events A and B are called independent if the occurrence of one of them does not change the probability of occurrence of the other. Event A is called dependent from event B if the probability of event A changes depending on whether event B occurred or not.

Conditional Probability P(A|B ) event A is called the probability calculated under the condition that event B occurred. Likewise, through P(B|A ) is denoted conditional probability 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

Probability of product of dependent eventsis equal to the product of the probability of one of them by the conditional probability of the other, provided that the first happened:

P(A ∙ B) = P(A) ∙ P(B|A) P(A ∙ B) = P(B) ∙ P(A|B)

(depending on which event happened first).

Consequences from the theorem:

Multiplication theorem for independent events. The probability of producing 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 (;), (; B), (A;) are also independent.

Examples of tasks on the multiplication theorem:

1. If grandmaster A. plays white, then he wins grandmaster B. with a probability of 0.52. If A. plays black, then A. beats 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.

1. The store has two payment machines. Each of them can be faulty with a probability of 0.05, regardless of the other automaton. Find the probability that at least one automaton is serviceable.

Decision. 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.
An event consisting in the fact that at least one automaton is serviceable is the opposite. Therefore, its probability is 1 − 0.0025 = 0.9975.

Answer: 0.9975.

Formula full probability

A consequence of the theorems of addition and multiplication of probabilities is the formula for the total probability:

Probability P (A) event A, which can occur only if one of the events (hypotheses) B occurs 1 , V 2 , V 3 … V n , 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) B 1 , V 2 , V 3 , …, V n on the corresponding conditional probabilities of event A:

P (A) \u003d P (B 1)  P (A | B 1) + P (B 2)  P (A | B 2) + P (B 3)  P (A | B 3) + ... + P (В n )  P (A | B n )

Consider an example:The automatic line makes batteries. The probability that a finished battery is defective is 0.02. Before packaging, each battery goes through a control system. The probability that the system will reject a bad battery is 0.99. The probability that the system will mistakenly reject a good battery is 0.01. Find the probability that a randomly selected battery will be rejected.

Decision. The situation in which the battery will be rejected may arise as a result of the events: A - "the battery is really defective and rejected fairly" or B - "the battery is good, but 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) \u003d P (A) + P (B) \u003d 0.02  0.99 + 0.98  0,01 = 0,0198 + 0,0098 = 0,0296.

Answer: 0.0296.

Chapter 2: Methodological Aspects of Studying "Probability Theory" in the School Algebra Course

In 2003, a decision was made to include elements of probability theory in the school mathematics course of a general education school (instructive letter No. 03-93in / 13-03 of September 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 mathematical education elementary school”, “Mathematics at School”, No. 9, 2003). By this time, elements of probability theory had been present in various forms in well-known school textbooks of algebra for more than ten years. 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 grades 10–11 of educational institutions” G.V. Dorofeev, L. V. Kuznetsova, E.A. Sedova”), and in the form of separate teaching aids. However, the presentation of the material on the theory of probability in them, as a rule, was not of a systematic nature, 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, enabling the teaching community to prepare for the corresponding changes.

In 2004–2008 A number of textbooks are being published to complement 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 Teacher's Guide", Makarychev Yu.N., Mindyuk N.G. Algebra: elements of statistics and probability theory: textbook. A guide for students in grades 7–9. general education institutions”, Tkacheva M.V., Fedorova N.E. "Elements of statistics and probability: Proc. Allowance for 7–9 cells. general education institutions." They also came out to help teachers. teaching aids. 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 sections of statistics and probability theory have taken their place in curricula 7-9 grades, analysis and understanding of the consistency of the main definitions and designations 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 for significant terminological inconsistency and differences in the designations of basic concepts and formulas. An analysis of the content of the above school textbooks shows that today they have inherited these features from higher school textbooks. With more accuracy, it can be argued that the choice of a particular educational material according to sections of mathematics new to the school, concerning the concept of "random", occurs in this moment in the most random way, down to names and designations. Therefore, the teams of authors of the leading school textbooks on probability theory and statistics decided to join their efforts under the auspices of the Moscow Institute of Open Education to develop agreed positions on the unification of the main definitions and notation used in school textbooks 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 Probability Theory", highlighted in the "Program for educational institutions. Mathematics", provides further development in students of their mathematical abilities, orientation to professions, significantly related to mathematics, preparation for studying at a university. The specificity of the mathematical content of the topic under consideration makes it possible to concretize the selected main task in-depth study mathematics as follows.

1. Continue the disclosure of the content of mathematics as a deductive system of knowledge.

Build a system of definitions of basic concepts;

Reveal additional properties of the introduced concepts;

Establish connections between the introduced and previously studied concepts.

2. Systematize some probabilistic ways of solving problems; reveal the operational composition of the search for solutions to problems of certain types.

3. Create conditions for students to understand and understand the main idea practical significance 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. Form an idea of ​​​​the 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; determine the limits of the use of these theorems. Show their applications for the derivation of full 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 total probability formula.

5. Form a prescription that allows you to rationally choose one of the algorithms when solving a specific problem.

Dedicated educational goals to study the elements of probability theory, we will supplement the setting of developmental and educational goals.

Development goals:

• to form in students a steady interest in the subject, to identify and develop mathematical abilities;
• in the process of learning to develop speech, thinking, emotional-volitional and concrete-motivational areas;
• independent finding by students of new ways of solving problems and tasks; application of knowledge in new situations and circumstances;
• develop the ability to explain facts, connections between phenomena, convert material from one form of representation 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;
• 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's analyze the textbook on algebra 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 related exercises. At the end of the paragraph, exercises for repetition are given. For each paragraph, additional exercises of a higher level of complexity are given in comparison with the main exercises.

According to the “Program for General Educational Institutions”, 15 hours are allotted 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.On the simple examples the solution of combinatorial problems by the method of enumeration of possible variants is demonstrated. This method is illustrated by building a tree of possible options. The rule of multiplication is considered.

Permutations. The concept itself and the formula for counting permutations are introduced.

Accommodations. The concept is introduced on a concrete example. The formula for the number of placements is derived.

Combinations. The concept and formula of the number of combinations.

The purpose of this section is to give students different ways of describing all the possible elementary events in various types random experience.

§4 "Initial information from the theory of probability".

The presentation of the material begins with a consideration of the experiment, after which the concepts of "random event" and "relative frequency of a random event" are introduced. A statistical and classical definition of probability is introduced. The paragraph ends with the point "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 intended for students with an interest and aptitude for mathematics and can be used to individual work or at extracurricular activities with students.

Guidelines to 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 to avoid mistakes when working with this textbook.
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 existing textbooks in the school is the textbookMordkovich A.G., Semenov P.V. "Events, probabilities, statistical processing data", it also has additional chapters for grades 7-9. Let's analyze it.

According to the Algebra Work Program, 20 hours are allotted for the study of the topic “Elements of Combinatorics, Statistics and Probability Theory”.

The material on the topic "Probability Theory" is disclosed in the following paragraphs:

§ 1. The simplest combinatorial problems. Multiplication rule and tree of variants. Permutations.It starts with a simple combinatorial problem, then considers a table of possible options, which shows the principle of the multiplication rule. Then trees of possible variants and permutations are considered. After theoretical material there are exercises for each of the sub-items.

§ 2. Selection of several elements. Combinations.First, a formula is derived for 2 elements, then for three, and then a 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 deal with tables and trees of options. These points are necessary because it is tables and option trees that teach students about 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. In terms of combinatorics, the material is presented just as successfully. Each paragraph contains exercises, which allows you to consolidate the material. Comments on this tutorial are contained in the article by Studenetskaya and Fadeeva.

In grade 10, three paragraphs are given on this topic. In the first of them “The rule of multiplication. Permutations and factorials”, in addition to the multiplication rule itself, the main emphasis was placed on the derivation of two basic combinatorial identities from this rule: for the number of permutations and for the number of possible subsets of the set consisting of n elements. At the same time, 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 class 10 “Selecting multiple 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 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 test with two outcomes. This is the most important probabilistic model from a practical point of view (Bernoulli trials), which has a significant number of applications. The latter material formed a transition between the content of the educational material in grades 10 and 11.

In the 11th grade, the topic "Elements of Probability Theory" is devoted to two paragraphs of the textbook and the problem book. ATSection 22 deals with geometric probabilities, § 23 repeats and expands knowledge about independent repetitions of trials with two outcomes.

Chapter 3: A fragment of an algebra lesson on the topic "Probability Theory"

Grade: 11

Lesson topic: "Analysis of task C6".

Lesson type: problem solving.

Formed UUD

Cognitive: analyze,

draw conclusions, compare objects according to the methods of action;

Regulatory: determine the goal, problem, put forward versions, plan activities;

Communicative: express your opinion, use speech means;

Personal: be aware of your emotions, develop a respectful attitude towards classmates

Planned results

Subject: the ability to use a formula to solve problems for calculating probability.

Meta-subject: the ability to put forward hypotheses, assumptions, see

different ways of solving the problem.

Personal: the ability to correctly express one's thoughts, understand the meaning

assigned task.

Task: Each of the group of students went to the cinema or to the theater, while it is possible that one of them could go to both the cinema and the theater. It is known that there were no more than 2/11 of the total number of students in the group who visited the theater in the boys' theater, and no more than 2/5 of the total number of students in the group who visited the cinema were in the cinema.
a) Could there be 9 boys in the group if it is additionally known that there were 20 students in total in the group?
b) What is the maximum number of boys in the group, if it is additionally known that there were 20 students in the group?
c) What was the smallest proportion of girls in the total number of students in the group without the additional condition of points a) and b)?

Parsing the task:

First, let's deal with the condition:

(In parallel with the explanation, the teacher depicts everything on the blackboard).

Suppose we have a lot of guys who went to the movies and a lot of guys who went to the theater. Because it is said that they all went, then the whole group is either in the set of guys who went to the theater, or in the set of guys who went to the movies. What is the place where these sets intersect?

It means that these guys went to the cinema and the theater at the same time.

It is known that the boys who went to the theater were no more than 2/11 of the total number of those who went to the theater. The teacher asks one of the students to draw this on the board.

And there could have been more boys who went to the cinema - no more than 2/5 of the total number of students in the group.

Now let's move on to the solution.

a) We have 9 boys, total students, let's denote N =20, all conditions must be met. If we have 9 boys, girls, respectively, 11. Item a) can be solved in most cases by enumeration.

Suppose that our boys went either only to the cinema or to the theater.

And the girls went back and forth. (Blue shows many boys and black shading shows girls)

Since we have only 9 boys and, by condition, went to the theater fewer boys, we assume that 2 boys went to the theater, and 7 to the cinema. And let's see if our condition is met.

Let's check it first on the example of the theater. We take the number of boys who went to the theater to all who went to the theater and plus the number of girls and compare this with: . Multiply this by 18 and by 5: .

Therefore the fraction is 7/18 2/5. Hence, the condition is satisfied for the cinema.

Now let's see if this condition is met for the theater. Independently, then one of the students writes the solution on the board.

Answer: If the group consists of 2 boys who visited only the theater, 7 boys who visited only the cinema, and 11 girls who went to both the theater and the cinema, then the condition of the problem is fulfilled. This means that in a group of 20 students there could be 9 boys.

b) Suppose there were 10 or more boys. Then there were 10 girls or less. The theater was attended by no more than 2 boys, because if there were 3 or more, then the proportion of boys in the theater would be no less = which is more.

Similarly, no more than 7 boys visited the cinema, because then at least one boy did not visit either the theater or the cinema, which contradicts the condition.

In the previous paragraph, it was shown that there could be 9 boys in a group of 20 students. So, the largest number of boys in the group is 9.

c) Suppose a certain boy went to both the theater and the cinema. If instead of him there were two boys in the group, one of whom visited only the theater, and the other only the cinema, then the share of boys in both the theater and the cinema would remain the same, and the total share of girls would become smaller. Hence, to estimate the smallest proportion of girls in the group, we can assume that each boy went either only to the theater or only to the cinema.

Let in the group of boys who visited the theater, boys who visited the cinema, and d girls.

Let us estimate the share of girls in this group. It is zero to assume that all the girls went to both the theater and the cinema, since their share in the group will not change from this, and the share in the theater and cinema will not decrease.

If the group consists of 2 boys who visited only the theater, 6 boys who visited only the cinema, and 9 girls who went to both the theater and the cinema, then the condition of the problem is satisfied, and the share of girls in the group is equal.

In everyday life, in practical and scientific activities, we often observe certain phenomena, conduct certain experiments. An event that may or may not occur during an observation or experiment is called a random event. For example, a light bulb hangs under 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 of all these causes 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 it. If we are talking about massive homogeneous random events, then they obey certain patterns, namely, probabilistic patterns. First, let's look at the classification of events. Distinguish 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 - 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 of a different color. Event A - a box taken at random will be with black shoes, event B - the box will be with brown shoes, A and B are incompatible events. An event is called certain if it will necessarily occur under the conditions of a given experiment. An event is said to be impossible if it cannot occur under the conditions of the given experience. For example, the event that a standard part is taken from a batch of standard parts is certain, but a non-standard part is impossible. An event is called possible or random if, as a result of experience, it may or may not occur. An example of a random event is the detection of product defects during the control of a batch of finished products, the discrepancy between the size of the processed product and the given one, the failure of one of the links of the automated control system. Events are said to be equally likely if, under the conditions of the test, none of these events is objectively more likely than the others. For example, suppose a store is supplied with light bulbs (and in equal quantities) by several manufacturers. Events consisting in buying a light bulb from any of these factories are equally probable. An important concept is the complete group of events. Several events in a given experiment form a complete group if at least one of them necessarily appears as a result of the experiment. For example, there are ten balls in an urn, of which six are red and four are white, five of which are numbered. A - the appearance of a red ball in one drawing, B - the appearance of a white ball, C - the appearance of a ball with a number. Events A,B,C form a complete group of joint events. The event may be opposite, or additional. 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 ones. They form a complete group of events. For example, if a batch of manufactured items consists of good and defective items, then when one item is removed, it can turn out to be either good - event A, or defective - event. Consider an example. They throw a dice (i.e. a small cube, on the sides of which points 1, 2, 3, 4, 5, 6 are knocked out). When throwing a dice, one point, two points, three points, etc. can fall on its top face. Each of these outcomes is random. Such a test has been carried out. The dice was thrown 100 times and observed how many times the event "6 points fell on the die" occurred. It turned out that in this series of experiments, the “six” fell out 9 times. The number 9, which shows how many times in this trial the event in question occurred, is called the frequency of this event, and the ratio of the frequency to the total number of trials, which is 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 fixed whether the event A of interest to us has occurred or not. The probability of an event is denoted by a capital Latin letter P. Then the probability of an event A will be denoted: P (A). The classical definition of probability: The probability of an event A is equal to the ratio of the number of cases m that favor it, out of the total number n of the only 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 the totality of the only possible, equally possible and incompatible cases, calculate their total number n, the number of cases m that favor the given event; perform a formula calculation. 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 the proportion of the favorable number of cases from the total number of cases: Consider another example. There are 10 balls in a box. 3 of them are red, 2 are green, the rest are white. Find the probability that a ball drawn at random is red, green, or white. The appearance of the red, green and white balls constitute a complete group of events. Let us denote the appearance of a red ball - event A, the appearance of a green one - event B, the appearance of a white one - event C. Then, in accordance with the formulas written above, we obtain: ; ; Note that the probability of occurrence of one of 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 experiences that resulted in event A to the total number of experiences. The difference between the relative frequency and the probability lies in the fact that the probability is calculated without the direct product of the experiments, and the relative frequency - after the experience. So in the above example, if 5 balls are randomly drawn 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 probability of the 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 there is a test with an infinite number of outcomes, the definition of geometric probability is used - hitting a point in an area. When determining the geometric probability, it is assumed that there is a region N and a smaller region M in it. A point is randomly thrown onto the region N (this means that all points in the region N are “equal” in terms of getting a randomly thrown point there). Event A - "the thrown point hits the area M". The area M is called favorable to the event A. The probability of getting into any part of the 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 the plane (the measure is the area) a geometric body in space (the measure is the volume) Let's 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 randomly thrown point on the area N in 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 hitting the border of the area is considered to be equal to zero . Consider an example: A mechanical watch with a twelve o'clock dial broke and stopped working. Find the probability that the hour hand is frozen at 5 o'clock but not at 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 occurrence of event A entails the occurrence of event B and vice versa. A union or sum of events is an event A, which means the occurrence of at least one of the events. A = Intersection or product of events is called 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 fell out; 4; 6 or 3 points” A B - “6 points rolled” A - B - “2 and 4 points rolled out” 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, and no matter what the event A is, it can be judged by the elementary outcome that 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 definitely occur. Therefore, the event under consideration is certain, and the probability of its occurrence, since in this case Property 2. If there is not a single case favorable for this event A, then this event cannot occur as a result of the experiment. Therefore, 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 that form a complete group is equal to one. Property 4. The probability of the occurrence of the opposite event is determined in the same way as the probability of the occurrence of the event A: where (n-m) is the number of cases that favor the occurrence of the opposite event. Hence, the probability of the opposite event occurring is equal to the difference between one and the probability of the event occurring 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. We write the equality of events A and B as A \u003d 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. Probability addition theorem 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 and only when both events occur: A and B at the same time. Random events A and B are called joint if both of these events can occur during a given test. The addition theorem 2. The probability of the sum of joint events is calculated by the formula P=P+P-P Examples of tasks on 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 related to these two topics at the same time. Find the probability that the 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. Two identical vending machines sell coffee in the mall. 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 by the end of the day there will be coffee left in both vending machines. Decision. Consider the events A - "coffee will end in the first machine", B - "coffee will end in the second machine". Then A B - "coffee will end in both vending machines", A + B - "coffee will end 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(A B) = 0.3 + 0.3? 0.12 = 0.48. Therefore, the probability of the opposite event, consisting in the fact that the 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 occurrence of one of them does not change the probability of 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 occurred or not. The conditional probability P(A|B) of an event A is the probability calculated assuming that event B has occurred. Similarly, P(B|A) denotes the conditional probability of an 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 product of dependent events is equal to the product be0.01 = 0.0198 + 0.0098 = 0.0296. Answer: 0.0296.

In 2003, a decision was made to include elements of probability theory in the school course of mathematics of a general education school (instructive letter No. 03-93in / 13-03 of September 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 mathematical education elementary school”, “Mathematics at School”, No. 9, 2003). By this time, elements of probability theory had been present in various forms in well-known school textbooks of algebra for different classes for more than ten years (for example, I.F. “Algebra: Textbooks for grades 7-9 of educational institutions” edited by G.V. Dorofeev; “ Algebra and the Beginnings of Analysis: Textbooks for Grades 10-11 of General Educational Institutions "G.V. Dorofeev, L.V. Kuznetsova, E.A. Sedova"), and in the form of separate teaching aids. However, the presentation of the material on the theory of probability in them, as a rule, was not of a systematic nature, 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, enabling the teaching community to prepare for the corresponding changes. In 2004-2008 A number of textbooks are being published to complement 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 Teacher's Guide", Makarychev Yu.N., Mindyuk N.G. Algebra: elements of statistics and probability theory: textbook. Allowance for students 7-9 cells. general education institutions”, Tkacheva M.V., Fedorova N.E. "Elements of statistics and probability: Proc. Allowance for 7-9 cells. general education institutions." Teaching aids are also available 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 sections of statistics and probability theory have taken their place in the curricula of grades 7-9, analysis and understanding of the consistency of the main definitions and designations 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 of higher education, often allowed significant terminological inconsistencies and differences in the designations of basic concepts and 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, it can be argued that the choice of specific educational material for new sections of mathematics for the school, concerning the concept of "random", is currently happening in the most random way, up to names and notation. Therefore, the teams of authors of the leading school textbooks on probability theory and statistics decided to join their efforts under the auspices of the Moscow Institute of Open Education to develop agreed positions on the unification of the main definitions and notation used in school textbooks on probability theory and statistics. Let's analyze the introduction of the topic "Probability Theory" in school textbooks. General characteristics: The content of teaching the topic "Elements of the Theory of Probability", highlighted in the "Program for General Educational Institutions. Mathematics", ensures the further development of students' mathematical abilities, orientation to professions that are significantly related to mathematics, and preparation for studying at a university. The specificity of the mathematical content of the topic under consideration makes it possible to concretize the identified main task of in-depth study of mathematics as follows. 1. Continue the disclosure of 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; - to establish connections between the introduced and previously studied concepts. 2. Systematize some probabilistic ways of solving problems; reveal the operational composition of the search for solutions to problems of certain types. 3. To create conditions for students to understand and comprehend the main idea of ​​the practical significance of probability theory by analyzing the main 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 ​​​​the various ways of determining 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; determine the limits of the use of these theorems. Show their applications for the derivation of full 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 packaging, each battery goes through a control system. The probability that the system will reject a bad battery is 0.99. The probability that the system will mistakenly reject a good battery is 0.01. Find the probability that a randomly selected battery will be rejected. Decision. The situation in which the battery will be rejected may arise as a result of the following events: A - “the battery is really defective and fairly rejected” or B - “the battery is good, but 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 ) + P(B2) of the probability of one of them by the conditional probability of the other, provided that the first one happened: P(A B) = P(A) P(B|A) P(A B) = P(B) P(A| B) (depending on which event happened first). Consequences from 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 also independent: (;), (; B), (A;). Examples of tasks on the multiplication theorem: If grandmaster A. plays white, then he wins grandmaster B. with a probability of 0.52. If A. plays black, then A. beats 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. The store has two payment machines. Each of them can be faulty with a probability of 0.05, regardless of the other automaton. Find the probability that at least one automaton is serviceable. Decision. 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. An event consisting in the fact that at least one automaton is serviceable is the opposite. Therefore, its probability is 1? 0.0025 = 0.9975. Answer: 0.9975. Total probability formula The consequence of the theorems of addition and multiplication of probabilities is the formula of total probability: Probability P(A) of event A, which can occur only if one of the events (hypotheses) B1, B2, B3 ... Bn occurs, 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 and the corresponding conditional probabilities of the event A: P(A) = P(B1) of the total probability. 5. Form a prescription that allows you to rationally choose one of the algorithms when solving a specific problem. The selected educational goals for studying the elements of probability theory will be supplemented by setting developmental and educational goals. Developing goals: to form in students a steady interest in the subject, to identify and develop mathematical abilities; in the process of learning to develop speech, thinking, emotional-volitional and concrete-motivational areas; independent finding by students of new ways of solving problems and tasks; application of knowledge in new situations and circumstances; develop the ability to explain facts, connections between phenomena, convert material from one form of representation 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; to educate a person capable of self-education and self-education. Let's analyze the textbook on algebra 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 related exercises. At the end of the paragraph, exercises for repetition are given. For each paragraph, additional exercises of a higher level of complexity are given in comparison with the main exercises. According to the “Program for General Educational Institutions”, 15 hours are allotted for studying the topic “Probability Theory and Statistics” in the school algebra course. The material on this topic falls on the 9th grade 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 enumeration of possible options. This method is illustrated by building a tree of possible options. The rule of multiplication is considered. Permutations. The concept itself and the formula for counting permutations are introduced. Accommodations. The concept is introduced on a concrete example. The formula for the number of placements is derived. Combinations. The concept and formula of the number of combinations. The purpose of this section is to give students different ways of describing all possible elementary events in different types of random experience. §4 "Initial information from the theory of probability". The presentation of the material begins with a consideration of the experiment, after which the concepts of "random event" and "relative frequency of a random event" are introduced. A statistical and classical definition of probability is introduced. The paragraph ends with the point "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”, “Introductory information from probability theory 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 to avoid mistakes when working with this textbook. 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 existing textbooks in the school is the textbook of 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 Algebra Work Program, 20 hours are allotted 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. Multiplication rule and tree of variants. Permutations. It starts with a simple combinatorial problem, then considers a table of possible options, which shows the principle of the multiplication rule. Then trees of possible variants 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 for 2 elements is displayed, 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 deal with tables and trees of options. These points are necessary because it is tables and option trees that teach students about 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. In terms of combinatorics, the material is presented just as successfully. Each paragraph contains exercises, which allows you to consolidate the material. Comments on this tutorial are contained in the article by Studenetskaya and Fadeeva. In grade 10, three paragraphs are given on this topic. 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 of two basic combinatorial identities from this rule: for the number of permutations and for the number of possible subsets of a set consisting of n elements. At the same time, 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 class 10 “Selecting multiple 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 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 test with two outcomes. This is the most important probabilistic model from a practical point of view (Bernoulli trials), which has a significant number of applications. The latter material formed a transition between the content of the educational material in grades 10 and 11. In the 11th grade, the topic "Elements of Probability Theory" is devoted to two paragraphs of the textbook and the problem book. § 22 deals with geometric probabilities, § 23 repeats and expands knowledge about independent repetitions of trials with two outcomes.

Events that occur in reality or in our imagination can be divided into 3 groups. These are certain events that are bound to happen, impossible events, and random events. Probability theory studies random events, i.e. events that may or may not occur. This article will be presented in summary probability theory formulas and examples of solving problems in probability theory, which will be in the 4th task of the USE in mathematics (profile level).

Why do we need the theory of probability

Historically, the need to study these problems arose in the 17th century in connection with the development and professionalization of gambling and the advent of the casino. It was a real phenomenon that required its study and research.

Playing cards, dice, roulette created situations where any of a finite number of equally probable events could occur. There was a need to give numerical estimates of the possibility of the occurrence of an event.

In the 20th century, it turned out that this seemingly frivolous science plays important role in the knowledge of the fundamental processes occurring in the microworld. The modern theory of probability was created.

Basic concepts of probability theory

The object of study of probability theory is events and their probabilities. If the event is complex, then it can be broken down into simple components, the probabilities of which are easy to find.

The sum of events A and B is called event C, which consists in the fact that either event A, or event B, or events A and B happened at the same time.

The product of events A and B is the event C, which consists in the fact that both the event A and the event B happened.

Events A and B are said to be incompatible if they cannot happen at the same time.

An event A is said to be impossible if it cannot happen. Such an event is denoted by the symbol .

An event A is called certain if it will definitely occur. Such an event is denoted by the symbol .

Let each event A be assigned a number P(A). This number P(A) is called the probability of the event A if the following conditions are satisfied with such a correspondence.

An important special case is the situation when there are equally probable elementary outcomes, and arbitrary of these outcomes form events A. In this case, the probability can be introduced by the formula . The probability introduced in this way is called the classical probability. It can be proved that properties 1-4 hold in this case.

Problems in the theory of probability, which are found on the exam in mathematics, are mainly related to classical probability. Such tasks can be very simple. Particularly simple are problems in probability theory in demo versions. It is easy to calculate the number of favorable outcomes, the number of all outcomes is written directly in the condition.

We get the answer according to the formula.

An example of a task from the exam in mathematics to determine the probability

There are 20 pies on the table - 5 with cabbage, 7 with apples and 8 with rice. Marina wants to take a pie. What is the probability that she will take the rice cake?

Decision.

There are 20 equiprobable elementary outcomes in total, that is, Marina can take any of the 20 pies. But we need to estimate the probability that Marina will take the rice patty, that is, where A is the choice of the rice patty. This means that we have a total of 8 favorable outcomes (choosing rice pies). Then the probability will be determined by the formula:

Independent, Opposite, and Arbitrary Events

However, more complex tasks began to appear in the open bank of tasks. Therefore, let us draw the reader's attention to other questions studied in probability theory.

Events A and B are called independent if the probability of each of them does not depend on whether the other event occurred.

Event B consists in the fact that event A did not occur, i.e. event B is opposite to event A. The probability of the opposite event is equal to one minus the probability of the direct event, i.e. .

Addition and multiplication theorems, formulas

For arbitrary events A and B, the probability of the sum of these events is equal to the sum of their probabilities without the probability of their joint event, i.e. .

For independent events A and B, the probability of the product of these events is equal to the product of their probabilities, i.e. in this case .

The last 2 statements are called the theorems of addition and multiplication of probabilities.

Not always counting the number of outcomes is so simple. In some cases, it is necessary to use combinatorics formulas. The most important thing is to count the number of events that meet certain conditions. Sometimes such calculations can become independent tasks.

In how many ways can 6 students be seated in 6 empty seats? The first student will take any of the 6 places. Each of these options corresponds to 5 ways to place the second student. For the third student there are 4 free places, for the fourth - 3, for the fifth - 2, the sixth will take the only remaining place. To find the number of all options, you need to find the product, which is denoted by the symbol 6! and read "six factorial".

AT general case the answer to this question is given by the formula for the number of permutations of n elements. In our case, .

Consider now another case with our students. In how many ways can 2 students be seated in 6 empty seats? The first student will take any of the 6 places. Each of these options corresponds to 5 ways to place the second student. To find the number of all options, you need to find the product.

In the general case, the answer to this question is given by the formula for the number of placements of n elements by k elements

In our case .

And last case from this series. How many ways are there to choose 3 students out of 6? The first student can be chosen in 6 ways, the second in 5 ways, and the third in 4 ways. But among these options, the same three students occur 6 times. To find the number of all options, you need to calculate the value: . In the general case, the answer to this question is given by the formula for the number of combinations of elements by elements:

In our case .

Examples of solving problems from the exam in mathematics to determine the probability

Task 1. From the collection, ed. Yashchenko.

There are 30 pies on a plate: 3 with meat, 18 with cabbage and 9 with cherries. Sasha randomly chooses one pie. Find the probability that he ends up with a cherry.

.

Answer: 0.3.

Problem 2. From the collection, ed. Yashchenko.

In each batch of 1000 light bulbs, an average of 20 defective ones. Find the probability that a light bulb chosen at random from a batch is good.

Solution: The number of serviceable light bulbs is 1000-20=980. Then the probability that a light bulb taken at random from the batch will be serviceable is:

Answer: 0.98.

The probability that student U. correctly solves more than 9 problems on a math test is 0.67. The probability that U. correctly solves more than 8 problems is 0.73. Find the probability that U. correctly solves exactly 9 problems.

If we imagine a number line and mark points 8 and 9 on it, then we will see that the condition "U. correctly solve exactly 9 problems” is included in the condition “U. correctly solve more than 8 problems", but does not apply to the condition "W. correctly solve more than 9 problems.

However, the condition "U. correctly solve more than 9 problems" is contained in the condition "U. correctly solve more than 8 problems. Thus, if we designate events: “W. correctly solve exactly 9 problems" - through A, "U. correctly solve more than 8 problems" - through B, "U. correctly solve more than 9 problems ”through C. Then the solution will look like this:

Answer: 0.06.

In the geometry exam, the student answers one question from the list of exam questions. The probability that this is a trigonometry question is 0.2. The probability that this is an Outer Corners question is 0.15. There are no questions related to these two topics at the same time. Find the probability that the student will get a question on one of these two topics on the exam.

Let's think about what events we have. We are given two incompatible events. That is, either the question will relate to the topic "Trigonometry", or to the topic "External angles". According to the probability theorem, the probability of incompatible events is equal to the sum of the probabilities of each event, we must find the sum of the probabilities of these events, that is:

Answer: 0.35.

The room is illuminated by a lantern with three lamps. The probability of one lamp burning out in a year is 0.29. Find the probability that at least one lamp does not burn out within a year.

Let's consider possible events. We have three light bulbs, each of which may or may not burn out independently of any other light bulb. These are independent events.

Then we will indicate the variants of such events. We accept the notation: - the light bulb is on, - the light bulb is burned out. And immediately next we calculate the probability of an event. For example, the probability of an event in which three independent events“light bulb burned out”, “light bulb on”, “light bulb on”: where the probability of the event “light bulb on” is calculated as the probability of an event opposite to the event “light bulb off”, namely: .

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NEW. Korolyuk V.S., Portenko N.I., Skorokhod A.V. Turbin A.F. Handbook of probability theory and mathematical statistics. 2nd ed. revised add. 1985 640 pp. djvu. 13.2 MB.
The handbook is an expanded and revised edition of the book "Handbook of Probability Theory and Mathematical Statistics" edited by V. S. Korolyuk, published in 1978 by the Naukova Dumka publishing house. In terms of the breadth of coverage of the main ideas, methods, and specific results of modern probability theory, the theory of random processes, and partly mathematical statistics, the Handbook is the only publication of its kind.
For scientists and engineers.

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NEW. F. Mosteller, R. Rourke, J. Thomas. Probability. 1969 432 pp. pdf. 12.6 MB.
This book, written by a group of famous American mathematicians and educators, is an elementary introduction to probability theory and statistics - branches of mathematics that are now finding more and more use in science and in practice. Written in a lively and vivid language, it contains many examples taken for the most part from the realm of everyday life. Despite the fact that to read the book it is enough to have a knowledge of mathematics in the volume of the school, it is a completely correct introduction to the theory of probability. I read in this book what I have never seen in others.

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Andronov A.M., Kopytov E.A., Greenglaz L.Ya. Probability theory and math statistics. 2004 460 pages djvu. 6.7 MB.
From the publisher:
Before you - an extended textbook on probability theory and mathematical statistics. The traditional material is supplemented with such questions as the probabilities of combinations of random events, random walks, linear transformations random vectors, numerical determination of non-stationary probabilities of states of discrete Markov processes, application of optimization methods for solving problems of mathematical statistics, regression models. The main difference between the proposed book and well-known textbooks and monographs on probability theory and mathematical statistics lies in its focus on the constant use of a personal computer when studying the material. The presentation is accompanied numerous examples solution of the considered problems in the environment of packages Mathcad and STATISTICA. The book is written on the basis of more than thirty years of experience of the authors in teaching the disciplines of probability theory, mathematical statistics and the theory of random processes for students of various specialties of higher educational institutions. It is of practical interest both for students and university professors, and for everyone who is interested in the application of modern probabilistic-statistical methods.

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Agekyan. Probability theory for astrons and physicists. 260 pages. Size 1.7 Mb. The book contains material in such a way as to be used in the processing of measurement results by physicists and astronomers. A useful book for calculating errors.

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I.I. Bavrin. Probability theory mathematical statistics. 2005 year. 161 pp. djv. 1.7 MB.
The fundamentals of probability theory and mathematical statistics are outlined in applications to physics, chemistry, biology, geography, ecology, exercises for independent work All basic concepts and provisions are illustrated by analyzed examples and tasks
For students of natural sciences pedagogical universities Can be used by students from other universities

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Borodin A. N. Elementary Course in Probability Theory and Mathematical Statistics. 1999 224 pp. djvu. 3.6 MB.
The textbook contains a systematic presentation of the main sections of the elementary course in probability theory and mathematical statistics. One new section has been added to the traditional sections - "Procedure of recursive estimation", in view of the special importance of this procedure for applications. The theoretical material is accompanied large quantity examples and tasks from different fields of knowledge.

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Bocharov P. P., Pechinkin A. V. Probability Theory. Math statistics. 2005 year. 296 pp. djvu. 2.8 MB.
The first part deals with the basic concepts of probability theory, using relatively simple mathematical constructions, but, nevertheless, the presentation is based on the axiomatic construction proposed by Academician A. N. Kolmogorov. The second part outlines the basic concepts of mathematical statistics. The most common problems of estimating unknown parameters and testing statistical hypotheses are considered, and the main methods for their solution are described. Each given position is illustrated by examples. The material presented as a whole corresponds to the state educational standard.
Students, graduate students and university professors, researchers of various specialties and those who want to get a first idea of ​​probability theory and mathematical statistics.

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V.N. Vapnik. Restoration of dependences on empirical data. 1979 449 pp. djvu. 6.3 MB.
The monograph is devoted to the problem of recovering dependencies from empirical data. It explores a risk minimization method on samples of a limited size, according to which, when restoring a functional dependence, one should choose a function that satisfies a certain compromise between the value that characterizes its “complexity” and the value that characterizes the degree of its approximation to the totality of empirical data. The application of this method to three main problems of dependency recovery is considered: the problem of learning pattern recognition, regression recovery, and interpretation of the results of indirect experiments. It is shown that taking into account the limited volume of empirical data allows solving problems of pattern recognition with a large dimension of the feature space, restoring regression dependencies in the absence of a model of the restored function, and obtaining stable solutions to incorrect problems of interpreting the results of indirect experiments. The corresponding algorithms for recovering dependencies are given.

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A.I. Volkovets, A.B. Gurinovich. Theory of Probability and Mathematical Statistics. Lecture notes. 2003 84 pp. PDF. 737 Kb.
The abstract of lectures on the course "Probability Theory and Mathematical Statistics" includes 17 lectures on topics defined by the typical work program for studying this discipline. The purpose of the study is to master the basic methods of formalized description and analysis of random phenomena, processing and analysis of the results of physical and numerical experiments. To study this discipline, the student needs the knowledge gained in the study of the sections "Series", "Sets and operations on them", "Differential and integral calculus" of the course of higher mathematics.

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Volodin. Lectures on the theory of probability and mathematical statistics. 2004 257 pages. Size 1.4 Mb. PDF. Theorver emphasizes methods for constructing probabilistic models and the implementation of these methods on real tasks natural sciences. In statistics, the focus is on methods for calculating the risk of specific statistical rules.

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Wentzel, Ovtcharov. Probability theory and its engineering applications. year 2000. 480 pages djvu. 10.3 MB.
The book gives a systematic presentation of the foundations of the theory of probability from the point of view of their practical applications in the specialties: cybernetics, Applied Mathematics, computers, automated control systems, theory of mechanisms, radio engineering, reliability theory, transport, communications, etc. Despite the variety of areas to which applications belong, they are all permeated with a single methodological basis.
For students of higher technical educational institutions. It can be useful for teachers, engineers and scientists of various profiles, who in their practical activities are faced with the need to set and solve problems related to the analysis of random processes.

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Wentzel, Ovtcharov. Probability Theory. 1969 365 pp. djvu. 8.3 MB.
The book is a collection of tasks and exercises. All problems have an answer, and most have solutions.

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N. Ya. VILENKIN, V. G. POTAPOV. PROBLEM-WORKSHOP ON PROBABILITY THEORY WITH ELEMENTS OF COMBINATORICS AND MATHEMATICAL STATISTICS. Tutorial. 1979 113 pp. djvu. 1.3 MB.
The book brought to the attention of the reader is a practical workbook for the course "Probability Theory". The task book consists of three chapters, which in turn are divided into paragraphs. At the beginning of each paragraph, the main theoretical information is given as briefly as possible, then typical examples are analyzed in detail, and, finally, problems for independent solution are proposed, provided with answers and instructions. The taskbook also contains texts laboratory work, the implementation of which will help a part-time student to better understand the basic concepts of mathematical statistics.

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Gmurman. Theory of Probability and Mathematical Statistics. 2003 480 pp. DJVU. 5.8 MB.
The book contains basically all the material of the program on the theory of probability and mathematical statistics. great attention devoted to statistical methods of processing experimental data. At the end of each chapter there are problems with answers. It is intended for university students and persons using probabilistic and statistical methods in solving practical problems.

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Kolmogorov. Probability Theory. Size 2.0 Mb.

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Kibzun et al. Theory of Probability and Mathematical Statistics. Uch. allowance. Basic course with examples and tasks. Size 1.7 Mb. djvu. 225 pp.

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M. Katz. Statistical independence in probability theory, analysis and number theory. 152 pages djv. 1.3 MB.
The book is presented in a very accessible and fascinating form application of some ideas of probability theory in other areas of mathematics. The main part of the book is devoted to the concept of statistical independence.
The book will be useful and interesting for students, mathematicians, physicists, engineers.

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M. Katz. Probability and related questions in physics. 408 pp. djv. 3.8 MB.
The author is familiar to the Soviet reader from the translation of his work "Statistical Independence in Probability Theory, Analysis and Number Theory" (IL, 1963). His A new book mainly dedicated to one of the most interesting tasks physics: to describe how a system of a very large number of particles (a gas in a vessel) comes to a state of equilibrium, and to explain how the irreversibility of this process in time is consistent with the reversibility in time of the original equations. The greatest attention is paid to the probabilistic aspect of the problem; Statistical models are considered that imitate the main features of the problem. The first two chapters are also of independent interest - using well-chosen examples, the author shows how the concept of probability arises in mathematical and physical tasks and what analytical apparatus is used by the theory of probability. This edition includes articles by Katz and other authors relating to the issues raised in the book.

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Kendall. Stuart. Multivariate statistical analysis and time series. 375 pp. DJVU. 8.2 MB.
The book is the last volume of a three-volume course in statistics by M. Kendall and A. Stewart, the first volume of which was published in 1966 under the title "Theory of distributions:", and the second - in 1973 under the title "Statistical inferences and connections".
The book contains information on analysis of variance, design of experiments, theory sample surveys, multidimensional analysis and time series.
Like the first two volumes, the book contains many practical recommendations and examples of their application, and the presentation combines a more or less detailed derivation of the main results with a relatively brief enumeration. a large number more private information.
The book will be of interest to undergraduate and graduate students specializing in mathematical statistics, as well as to a wide range of scientists dealing with its applications.

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Kendall. Stuart. THE THEORY OF DISTRIBUTIONS. Volume 1. 590 pages 10.3 Mb. 6.1 MB.
Contents: Frequency distributions. Measures of location and dispersion. Moments and semi-invariants. Characteristic functions. standard distributions. Calculus of Probabilities. Probability and statistical inference. Random selection. standard errors. Accurate sampling distributions. Approximation of sample distributions. Approximation of sample distributions. Ordinal statistics. Multivariate normal distribution and quadratic forms. Distributions related to the normal.

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Kendall. Stuart. STATISTICAL CONCLUSIONS AND RELATIONS. Volume 2. 900 pages djvu. 10.3 MB.
The book contains information on the theory of estimation, hypothesis testing, correlation analysis, regression, non-parametric methods, sequential analysis.

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N.Sh. Kremer. Theory of Probability and Mathematical Statistics. Textbook. 2nd ed., revised. add. 2004 575 pp. djvu. 12.2 MB.
This is not only a textbook, but also a short guide to solving problems. The stated foundations of the theory of probability and mathematical statistics are accompanied by a large number of problems (including economic ones), given with solutions and for independent work. At the same time, the emphasis is on the basic concepts of the course, their theoretical and probabilistic meaning and application. Examples are given of the use of probabilistic and mathematical-statistical methods in problems queuing and financial market models.
For students and graduate students of economic specialties and areas, as well as university professors, researchers and economists.

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Kobzar A.I. Applied mathematical statistics. For engineers and scientists. 2006 814 pp. djvu. 7.7 MB.
The book discusses ways to analyze observations by methods of mathematical statistics. Consistently in a language accessible to a specialist - not a mathematician, modern methods analysis of probability distributions, evaluation of distribution parameters, testing of statistical hypotheses, evaluation of relationships between random variables, planning of a statistical experiment. The main attention is paid to the explanation of examples of applying the methods of modern mathematical statistics.
The book is intended for engineers, researchers, economists, physicians, graduate students and students who want to quickly, economically and at a high professional level use the entire arsenal of modern mathematical statistics to solve their applied problems.

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M.L. Krasnov. Probability Theory. Textbook. year 2001. 296 pp. djvu. 3.9 MB.
When studying various phenomena in nature and society, the researcher is faced with two types of experiments - those whose results are unambiguously predictable under given conditions, and those whose results cannot be unambiguously predicted under conditions controlled by the researcher, but one can only make an assumption about the spectrum of possible results. In the first case, one speaks of deterministic phenomena, in the second - of phenomena that bear random character. At the same time, they mean that a priori (in advance, before the experiment is carried out or the observation of the phenomenon is completed) in the first case we are able to predict the result, but in the second we are not. For what follows, it is not important what caused such unpredictability - the laws of nature underlying the phenomenon under study or the incompleteness of information about the processes that cause this phenomenon. An important circumstance is the presence of the very fact of unpredictability. The theory of probability, the foundations of which this section is devoted to, is designed to enable the researcher to describe such experiments and phenomena and provides him with a reliable tool for studying reality in situations where a deterministic description is impossible.

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E.L. Kuleshov. Probability Theory. Lectures for physicists. 2002 116 pages djvu. 919 Kb.
For senior students.

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Lazakovich, Stashulenok, Yablonsky. Probability theory course. Tutorial. 2003 322 pp. PDF. 2.9 MB.
The training manual is based on annual rate lectures given by the authors for a number of years to students of the Faculty of Mechanics and Mathematics of the Belarusian State University. The book contains the following sections: probability spaces, independence, random variables, numerical characteristics random variables, characteristic functions, limit theorems, fundamentals of the theory of random processes, elements of mathematical statistics and applications, which contain tables of the main probability distributions and the values ​​of some of them. Most of the chapters include appendices, which contain supporting material and topics for self-study.
The presentation is accompanied by a large number of examples, exercises and problems illustrating the basic concepts and explaining the possible applications of the proven statements.
For students of mathematical specialties of universities.

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Loev M. Probability Theory. 1962 449 pp. djvu. 6.2 MB.
The book is an extensive systematic course in modern probability theory, written at a high theoretical level. On the basis of measure theory, the author studies random events, random variables and their sequences, distribution functions and characteristic functions, limit theorems of probability theory and random processes. The presentation is accompanied by a large number of tasks varying degrees difficulties.
A book for undergraduate and graduate students - mathematicians studying theory.

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Lvovsky B.N. Statistical methods for constructing empirical formulas: Proc. allowance. 2nd ed., revised. add. 1988 239 pp. djvu. 2.3 MB.
The 2nd edition of the manual outlines the main methods for processing experimental data. The methods of preliminary processing of the results of observations are described in detail. Statistical methods for constructing empirical formulas, the maximum Likelihood method, the method of means and co-fluent analysis are considered. The methodology for planning and processing active experiments is covered. The basics of dispersion analysis are given.

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Yu.D. Maksimov editor. Probabilistic branches of mathematics. Textbook. year 2001. 581 pp. djvu. 7.4 MB.
Sections: !. Probability Theory. 2. Mathematical statistics. 3. Theory of random processes. 4. Theory of queuing.
Textbook for bachelors of technical wrong.

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Maksimov Yu.D. Mathematics. Vshusk 9. Probability Theory. Detailed summary. Handbook of One-Dimensional Continuous Distributions. 2002 98 pages djv. 4.3 MB.
The manual complies with the state educational standard and the current programs of the discipline "Mathematics" for bachelor's studies in all general technical and economic areas. It is a detailed abstract of lectures on probability theory, basically corresponding to the basic abstract (issue 7 of the series of basic abstracts in mathematics published by the SPBPU publishing house). In contrast to the reference abstract, here are proofs of theorems and derivations of formulas omitted in the reference abstract, and a reference book on one-dimensional continuous distributions.

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J. Neveu. Mathematical Foundations probability theory. 1969 310 pp. djv. 3.0 MB.
The author of the book is known for his work on the application of the methods of functional analysis and measure theory to questions of probability theory. Masterfully written book contains a compact and at the same time a complete presentation of the foundations of the theory of probability. Included a lot useful additions and exercise.
The book can serve a good textbook for students and graduate students who want to seriously study the theory of random processes, and an excellent reference book for specialists.

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D.T. Writing. Lecture notes on probability theory and mathematical statistics. 2004 256 pp. djvu. 1.4 MB.
This book is a course of lectures on probability theory and mathematical statistics. The first part of the book contains the basic concepts and theorems of probability theory, such as random events, probability, random functions, correlation, conditional probability, the law of large numbers, and limit theorems. The second part of the book is devoted to mathematical statistics, it sets out the basics) of the sampling method, the theory of estimates and hypothesis testing. The presentation of the theoretical material is accompanied by a consideration of a large number of examples and problems, and is conducted in an accessible, if possible, strict language.
Designed for students of economic and technical universities.

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Poddubnaya O.N. Lectures on the theory of probability. 2006 125 pp. pdf. 2.0 Mb.
Clearly written. The advantages of the course, for example, include the fact that theoretical statements are explained by examples.

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Yu.V. Prokhorov, Yu.A. Rozanov. Probability Theory. Basic concepts. Limit theorems. random processes. 1967 498 pp. djvu. 7.6 MB.
The book was written by well-known American mathematicians and is devoted to one of the important modern directions in the theory of probability, which is not sufficiently reflected in the literature in Russian. The authors gravitate toward meaningful results, rather than maximum generality, and consider a number of examples and applications. The book successfully combines a high scientific level of presentation and, at the same time, accessibility for a student audience.
For specialists in probability theory, physicists, engineers, graduate students and university students.

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Poincare A. Probability Theory. 1999 284 pp. djv. 700 Kb.
The book is one of the parts of the course of lectures by A. Poincaré. It discusses both the general foundations of the theory of probability and non-traditional issues that are practically not contained in any course. Various applications to physics, mathematics and mechanics are considered.
The book is useful to a wide range of readers - physicists, mathematicians, historians of science.

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Pyt'ev Yu. P. Shishmarev IA A course in the theory of probability and mathematical statistics for physicists. Proc. allowance. Moscow State University 1983. 256 pp. djvu. 4.6 MB.
The book is based on a six-month course of lectures, read by authors at the Faculty of Physics. Much space is given to the theory of random processes: Markov and stationary. The presentation is mathematically rigorous, although not based on the use of the Lebesgue integral. The part of the course devoted to mathematical statistics contains sections focused on applications to the tasks of automating the planning, analysis and interpretation of physical experiments. The statistical theory of the measuring and computing complex "instrument + computer" is presented, which makes it possible to significantly improve the parameters of real experimental equipment by processing data on a computer. Theory elements included statistical check hypotheses used in the problem of interpretation of experimental data.

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Saveliev. elementary theory probabilities. Textbook, Novosibirsk State University, 2005.
Part 1 is devoted to theory. Size 660 Kb. Part 2 is devoted to the analysis of examples. Size 810 Kb. Part 3. Riemann and Stieltjes integrals. 240 pages djvu. 5.0 Mb. Part 3 of the manual details the elements of differential and integral calculus, which were used in Part I. Combined material from the author's manuals "Lectures on mathematical analysis, 2.1" (Novosibirsk, Novosibirsk State University, 1973) and "Integration of uniformly measurable functions" (Novosibirsk, Novosibirsk State University, 1984). The main object is the Stieltjes integral. It is defined as a bounded linear functional on the space of functions without complex discontinuities, which was considered in Part 1. The Stieltjes integral is widely used not only in probability theory, but also in geometry, mechanics, and other areas of mathematics. The appendix in part 3 of the manual supplements the appendix in part 2. For completeness, some places from part 1 are repeated in part 3. The appendix retains the numbering of pages and paragraphs of the author's manual "Lectures on Mathematical Analysis".

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Download part 1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Download part 2

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Savrasov Yu.S. Optimal solutions. Lectures on measurement processing methods. year 2000. 153 pp. djvu. 1.1 Mb.
Methods for processing measurements that provide the most complete extraction are considered. useful information about measured parameters or observed phenomena. The methods presented relate to the field of probability theory, mathematical statistics, decision theory, utility theory, filtering theory for dynamic systems with discrete time. The material of the book is based on lectures given by the author in 1994-1997. third-year students of the basic department of "Radiophysics" of the Moscow Institute of Physics and Technology. In the proposed form, the book will be useful to students of physical and technical specialties, engineers in the field of radar, information processing and automated control systems.
Many examples have been analyzed.

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Samoilenko N.I., Kuznetsov A.I., Kostenko A.B. Probability Theory. Textbook. year 2009. 201 pp. PDF. 2.1 MB.
The textbook introduces the basic concepts and methods of probability theory. The given methods are illustrated by typical examples. Each topic ends with a practical section for self-acquisition of skills on the use of methods of probability theory in solving stochastic problems.
For university students.
Examples from the textbook: tossing a coin is an experience, falling heads or tails is an event; pulling out a card from a preference deck - experience, the appearance of a red or black suit - events; giving a lecture is an experience, the presence of a student at a lecture is an event.

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Sekey. Paradoxes of probability theory and mathematical statistics. Size 3.8 Mb. djv. 250 pages

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Sevastyanov B.A. Course of Probability Theory and Mathematical Statistics. Textbook. 1982 255 pp. djvu. 2.8 MB.
The book is based on a one-year course of lectures given by the author for a number of years at the Department of Mathematics of the Faculty of Mechanics and Mathematics of Moscow State University. The basic concepts and facts of probability theory are introduced initially for a finite scheme. The mathematical expectation is generally defined in the same way as the Lebesgue integral, but the reader is not expected to have any prior knowledge of Lebesgue integration.
The book contains the following sections: independent tests and Markov chains, limit theorems of Moivre - Laplace and Poisson, random variables, characteristic and generating functions, law of large numbers, central limit theorem, basic concepts of mathematical statistics, testing of statistical hypotheses, statistical estimates, confidence intervals .
For undergraduate students of universities and technical colleges studying probability theory.

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A.N. Sobolevsky. Probability theory and mathematical statistics for physicists. 2007 47 pages djv. 515 Kb.
The textbook contains a presentation of the fundamentals of probability theory and mathematical statistics for physics students of theoretical specialization. Along with classical material (scheme independent tests Bernoulli, finite homogeneous chains Markov, diffusion processes), considerable attention is paid to such topics as the theory of large deviations, the concept of entropy in its various options, stable laws and decreasing probability distributions, stochastic differential calculus. The textbook is intended for students specializing in various sections of theoretical and mathematical physics.

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Tarasov L. V. Patterns of the surrounding world. In 3 books. 2004 djvu.
1. Chance, necessity, probability. 384 pages 6.8 Mb.
This book is a fairly popular and at the same time strictly scientific detailed introduction to probability theory, which includes detailed analysis of the problems under consideration, broad generalizations of the philosophical plan, digressions of a historical nature. The book has a clearly defined educational character; its material is strictly structured, built on an evidence-based basis, provided with a large number of graphs and diagrams; a significant number of original problems are given, some of which are dealt with in the book, and some are offered to the reader for independent solution. The book is a finished work and at the same time is the first book of the author's three-volume set.
2. Probability in modern society. 360 pages 4.5 Mb.
This book demonstrates the fundamental role of probability theory in modern society, which is based on highly developed information technology. The book is quite a popular and at the same time strictly scientifically detailed introduction to operations research and information theory. It has a clearly defined educational character; its material is strictly structured, built on an evidence-based basis, provided with a large number of graphs and diagrams; a significant number of tasks are given, some of which are dealt with in the book, and some are offered to the reader for independent solution.
3. 440 pages 7.5 Mb. The evolution of natural science knowledge.
Here, in a popular and systematized form, the evolution of natural-science pictures of the world is analyzed: from scientific programs antiquity to the mechanical picture, then to the electromagnetic picture, and finally to contemporary painting. The transition from dynamic (rigidly determined) regularities to statistical (probabilistic) regularities is demonstrated as the scientific comprehension of the surrounding world gradually deepens. The evolution of the concepts of quantum physics, elementary particle physics, and cosmology is considered in sufficient detail. In conclusion, the ideas of self-organization of open nonequilibrium systems (the emergence of dissipative structures) are discussed.
For a wide range of readers, and primarily for high school students (starting from the 9th grade), as well as for students of technical schools and higher educational institutions.

The study of elements of statistics and probability theory begins in the 7th grade. The inclusion in the course of algebra of initial information from statistics and probability theory is aimed at developing in students such important skills in modern society as understanding and interpreting the results of statistical studies, which are widely presented in the media. mass media. In modern school textbooks, the concept of the probability of a random event is introduced based on life experience and intuition of students.

I would like to note that in grades 5-6, students should already get an idea about random events and their probabilities, so in grades 7-9 it would be possible to quickly get acquainted with the basics of probability theory, expand the range of information reported to them.

Our educational institution is testing the program " elementary School 21st century". And as a mathematics teacher, I decided to continue testing this project in grades 5-6. The course was implemented on the basis of the educational and methodological set of M.B. Volovich “Mathematics. 5-6 classes. In the textbook "Mathematics. Grade 6 ”6 hours are allotted to study the elements of probability theory. Here we give the very first preliminary information about such concepts as testing, the probability of a random event, certain and impossible events. But the most important thing that students must learn is that with a small number of trials, it is impossible to predict the outcome of a random event. However, if there are many tests, then the results become quite predictable. To make students aware that the probability of an event occurring can be calculated, a formula is given to calculate the probability of an event occurring when all the outcomes under consideration are “equal”.

Subject: The concept of "probability". Random Events.

Lesson Objectives:

• to provide an acquaintance with the concept of "test", "outcome", "random event", "certain event", "impossible event", to give an initial idea of ​​​​what the "probability of an event" is, to form the ability to calculate the probability of an event;
• develop the ability to determine the reliability, impossibility of events;
• increase curiosity.

Equipment:

1. M.B. Volovich Mathematics, 6th grade, M.: Ventana-Graf, 2006.
2. Yu.N.Makarychev, N.G.Mindyuk Elements of statistics and probability theory, Moscow: Education, 2008.
3. 1 ruble coin, dice.

DURING THE CLASSES

I. Organizational moment

II. Actualization of students' knowledge

Solve the rebus:

(Probability)

III. Explanation of new material

If a coin, for example, a ruble, is tossed up and allowed to fall to the floor, then only two outcomes are possible: “the coin fell head down” and “the coin fell tails up.” The case when a coin falls on its edge, rolls up to the wall and rests against it, is very rare and usually not considered.
For a long time in Russia they played "toss" - they tossed a coin if it was necessary to solve a controversial problem that had no obviously fair solution, or they played some kind of prize. In these situations, they resorted to chance: some thought of a loss of "heads", others - "tails".
Tossing a coin is sometimes resorted to even when solving very important issues.
For example, the semi-final match for the European Championship in 1968 between the teams of the USSR and Italy ended in a draw. The winner was not revealed either in extra time or in the penalty shootout. Then it was decided that the winner would be determined by His Majesty chance. They threw a coin. The case was favorable to the Italians.
In everyday life, in practical and scientific activities, we often observe certain phenomena, conduct certain experiments.
An event that may or may not occur during an observation or experiment is called random event.
The patterns of random events are studied by a special branch of mathematics called probability theory.

Let's spend experience 1: Petya tossed the coin up 3 times. And all 3 times the “eagle” fell out - the coin fell with the coat of arms up. Guess if it's possible?
Answer: Possibly. "Eagle" and "tails" fall out completely by accident.

Experience 2: (students work in pairs) Toss a 1 ruble coin 50 times and count how many times it comes up heads. Record the results in a notebook.
In the class, calculate how many experiments were conducted by all students and what is the total number of headings.

Experience 3: The same coin was tossed up 1000 times. And all 1000 times the "eagle" fell out. Guess if it's possible?
Let's discuss this experience.
The coin toss is called test. Loss of "heads" or "tails" - outcome(result) of the test. If the test is repeated many times under the same conditions, then information about the outcomes of all tests is called statistics.
Statistics captures as a number m outcomes (results) of interest to us, and the total number N tests.
Definition: The relation is called statistical frequency the result of interest to us.

In the 18th century, a French scientist, an honorary member of the St. Petersburg Academy of Sciences, Buffon, to check the correctness of calculating the probability of falling "eagle", tossed a coin 4040 times. "Eagle" fell out 2048 times.
In the 19th century, the English scientist Pearson tossed a coin 24,000 times. "Eagle" fell out 12,012 times.
Let us substitute into the formula, which allows us to calculate the statistical frequency of occurrence of the result of interest to us, m= 12 012, N= 24,000. We get = 0.5005.

Consider the example of rolling a dice. We will assume that this die has a regular shape and is made of a homogeneous material, and therefore, when it is thrown, the chances of getting any number of points from 1 to 6 on its upper face are the same. They say there are six equally likely outcomes of this challenge: roll points 1, 2, 3, 4, 5 and 6.

The probability of an event is easiest to calculate if all n possible outcomes are "equal" (none of them has advantages over the others).
In this case, the probability P calculated by the formula R= , where n is the number of possible outcomes.
In the coin toss example, there are only two outcomes (“heads” and “tails”), i.e. P= 2. Probability R heading is equal to .
Experience 4: What is the probability that when a dice is thrown, it will come up:
a) 1 point; b) more than 3 points.
Answer: a), b).

Definition: If an event always occurs under the conditions under consideration, then it is called authentic. The probability of a certain event occurring is 1.

There are events that, under the conditions under consideration, never occur. For example, Pinocchio, on the advice of the fox Alice and the cat Basilio, decided to bury his gold coins in the field of Miracles so that a money tree would appear from them. What will be the probability that their planted coins will grow a tree? The probability of a money tree growing from coins planted by Pinocchio is 0.

Definition: If an event never occurs under the conditions under consideration, then it is called impossible. The probability of an impossible event is 0.

IV. Physical education minute

"Magical dream"

Everyone can dance, run, jump and play,
But not everyone knows how to relax, to rest.
They have such a game, very easy, simple.
Movement slows down, tension disappears,
And it becomes clear: relaxation is pleasant.
Eyelashes fall, eyes close
We calmly rest, we fall asleep with a magical dream.
Breathe easily, evenly, deeply.
The tension has flown away and the whole body is relaxed.
It's like we're lying on the grass...
On green soft grass...
The sun is warming now, our hands are warm.
The sun is hotter now, our feet are warm.
Breathe easily, freely, deeply.
The lips are warm and flaccid, but not at all tired.
Lips slightly parted, and pleasantly relaxed.
And our obedient tongue is accustomed to being relaxed.”
Louder, faster, more energetic:
“It was nice to rest, and now it’s time to get up.
Clench your fingers tightly into a fist
And press it to your chest - like that!
Stretch, smile, take a deep breath, wake up!
Open your eyes wide - one, two, three, four!
The children stand up and sing along with teacher pronounce:
“We are cheerful, cheerful again and ready for classes.”

V. Consolidation

Task 1:

Which of the following events are certain and which are impossible:

a) Throw two dice. Dropped 2 points. (authentic)
b) Throw two dice. Dropped 1 point. (impossible)
c) Throw two dice. Dropped 6 points. (authentic)
d) Throw two dice. Number of points rolled less than 13. (valid)

Task 2:

The box contains 5 green, 5 red and 10 black pencils. Got 1 pencil. Compare the probabilities of the following events using the expressions: more likely, less likely, equally likely.

a) The pencil turned out to be colored;
b) the pencil turned out to be green;
c) the pencil is black.

Answer:

a) equally likely;
b) more likely that the pencil turned out to be black;
c) equally likely.

Task 3: Petya rolled a die 23 times. However, 1 point rolled 3 times, 2 points rolled 5 times, 3 points rolled 4 times, 4 points rolled 3 times, 5 points rolled 6 times. In other cases, 6 points fell out. When doing the task, round the decimals to hundredths.

1. Calculate the statistical frequency of occurrence of the highest number of points, the probability that 6 points will fall out, and explain why the statistical frequency differs significantly from the probability of occurrence of 6 points found by the formula.
2. Calculate the statistical frequency of occurrence of an even number of points, the probability that even number points, and explain why the statistical frequency is significantly different from the probability of an even number of points found by the formula.

Task 4: To decorate the Christmas tree, they brought a box containing 10 red, 7 green, 5 blue and 8 gold balls. One ball is drawn at random from the box. What is the probability that it will be: a) red; b) gold; c) red or gold?

VI. Homework

1. 1 ball is taken from the box containing red and green balls and then put back into the box. Is it possible to consider that taking the ball out of the box is a test? What might be the outcome of the test?
2. A box contains 2 red and 8 green balls.

a) Find the probability that a randomly drawn ball is red.
b) Find the probability that a ball drawn at random is green.
c) Two balls are drawn at random from the box. Can it turn out that both balls are red?

VII. Outcome

- You learned the most information from the theory of probability - what is a random event and the statistical frequency of the test result, how to calculate the probability of a random event with equally likely outcomes. But we must remember that it is not always possible to evaluate the results of trials with a random outcome and find the probability of an event even with a large number of trials. For example, it is impossible to find the probability of getting the flu: too many factors each time affect the outcome of this event.