Application of probability theory in engineering. Application of probability theory and mathematical statistics in construction. Collection imprint

Webinar about how to understand probability theory and how to start using statistics in business. Knowing how to work with such information, you can make your own business.

Here is an example of a problem that you will solve without thinking. In May 2015, Russia launched spaceship"Progress" and lost control over it. This pile of metal, under the influence of the Earth's gravity, should have crashed onto our planet.

Attention, the question is: what was the probability that Progress would have fallen on land, and not in the ocean, and whether we should have been worried.

The answer is very simple - the chances of falling on land were 3 to 7.

My name is Alexander Skakunov, I am not a scientist or a professor. I just wondered why we need the theory of probability and statistics, why did we take them at the university? Therefore, in a year I read more than twenty books on this topic - from The Black Swan to The Pleasure of X. I even hired myself 2 tutors.

In this webinar, I will share my findings with you. For example, you'll learn how statistics helped create an economic miracle in Japan and how this is reflected in the script for the movie Back to the Future.

Now I'm going to show you some street magic. I don't know how many of you will sign up for this webinar, but only 45% will turn up.

It will be interesting. Sign up!

3 stages of understanding the theory of probability

There are 3 stages that anyone who gets acquainted with the theory of probability goes through.

Stage 1. “I will win at the casino!”. Man believes that he can predict the outcome of random events.

Stage 2. “I will never win at the casino!..” The person is disappointed and believes that nothing can be predicted.

And stage 3. “Let's try outside the casino!”. A person understands that in the seeming chaos of the world of chances one can find patterns that allow one to navigate well in the world around.

Our task is just to reach stage 3, so that you learn how to apply the basic provisions of the theory of probability and statistics to the benefit of yourself and your business.

So, you will learn the answer to the question "why the theory of probability is needed" in this webinar.

BASIC CONCEPTS AND BRIEF INFORMATION FROM PROBABILITY THEORY

The presented material is intended for students who are familiar with the probabilistic methods of describing and analyzing random phenomena, which form the basis mathematical models general technical course "Reliability of technical systems".

II.1. Application of probability theory in engineering

Probability theory is necessary for solving many technical problems.

The peculiarity of the theory of probability is that it considers phenomena where uncertainty is present in one form or another. Therefore, there is an idea that probabilistic methods solutions to practical problems are considered less preferable than "exact" analysis, since the alleged lack of sufficiently complete information forces one to turn to these methods. In addition, many consider probability theory to be a mysterious area of ​​mathematical science.

The opinions presented are incorrect. First, there is hardly any other area of ​​mathematics that is based so completely on such a limited set of initial ideas (only three axioms, which are almost obvious). Secondly, the dogmatic desire to present physical laws as deterministic and fair under any circumstances. Of course, Ohm's law cannot be denied, but it doesn't hold at the micro level of what's going on - a fact that is obvious to anyone who has ever connected a large resistor to the input of a high gain amplifier and heard the noise that appears as a result of this at the output.

So in best case, immutable laws reflect the "behavior" of nature, so to speak, "on average". In many situations, this "average behavior" is close enough to what is observed in practice, and the existing deviations can be neglected. In other equally important situations, random deviations can be significant, which requires the use of analytical methods based on probabilistic concepts.

Therefore, it becomes clear that the so-called "exact solution" is not always exact and, moreover, is an idealized special case, which almost never occurs in practice. On the other hand, the probabilistic approach is far from the worst substitute for exact solution methods and most fully reflects the physical reality. In addition, it includes the result of the deterministic approach as a special case.

Now it makes sense to describe in general types situations in which the use of probabilistic calculation methods in solving practical problems is the rule rather than the exception.

Random parameters of systems. In some cases, certain parameters of the system may be unknown or change randomly. Typical examples of such systems are electric power networks, the loads of which are unpredictable and vary widely; telephone systems, the number of users of which varies randomly over time; electronic systems, the parameters of which are random, due to the fact that the characteristics of semiconductor devices are set by a range of possible values.

Systems reliability. The composition of any technical system includes a large number of different elements, the failure of one or more of them can cause the failure of the entire system. As the complexity and cost of systems increase at the design stage, the task of synthesizing logical block diagrams of reliability and optimizing reliability arises.

Quality control and diagnostics. Improving consumer properties and competitiveness of products can be achieved by final control and diagnostics during operation. This requires rules for checking individual randomly selected elements, probabilistic methods for detecting defects and predicting performance.

Information theory. A quantitative measure of the information content of various messages: numerical and graphic data, technical measurements are of a probabilistic nature. In addition, the throughput of communication channels depends on random noise impacts.

From the brief enumeration, it is clear that when solving a large number technical problems have to meet with uncertainty, and this makes probability theory an indispensable tool for a modern engineer.

II.2. Basic concepts

II.2. 1. Fundamentals of set theory.

Probability theory - mathematical science, studying patterns in random phenomena. One of the basic concepts is the concept of a random event (hereinafter simply an event).

event any fact (outcome) is called, which as a result of experience (trial, experiment) may or may not occur. Each of these events can be associated with a certain number, called its probability and is a measure of the possible occurrence of this event.

The modern construction of probability theory is based on an axiomatic approach and relies on the elementary concepts of set theory.

Lots of is any collection of objects of an arbitrary nature, each of which is called an element of the set. Sets are denoted in different ways: either by a single capital letter or by enumerating its elements given in curly brackets, or by indicating (in the same curly brackets) the rule by which an element belongs to a set. For example, a finite set M natural numbers from 1 to 100 can be written as

M = (1, 2, ..., 100) = (i - integer; 1 i 100).

Suppose that some experience (experiment, test) is being made, the result of which is unknown in advance, random. Then the set of all possible outcomes of experience represents the space of elementary events, and each of its elements (one separate outcome of experience) is an elementary event. Any set of elementary events (any combination of them) is considered subset (part of) the set and is random event, i.e. any event BUT is a subset of the set : BUT . For example, the space of elementary events when throwing a dice is six possible outcomes = (1, 2, 3, 4, 5, 6). Taking into account empty sets , which contains no elements at all, in the space a total of 2 6 = 64 subsets can be distinguished:

; {1}; … ; {6}; {1, 2}; … ; {5, 6}; {1, 2, 3}; … ; .

In general, if the set contains n elements, then 2 n subsets (events) can be distinguished in it.

Considering an event (because every set is its own subset), it can be noted that it is certain event, i.e., it is carried out with any experience. Empty set how the event is impossible, i.e., in any experiment, it certainly cannot happen. For the previous example: certain event = (1, 2, 3, 4, 5, 6) = (drop one of the six points); impossible event = (7) = (loss of 7 points with one throw of a dice).

Joint (non-joint) events - such events, the occurrence of one of which does not exclude (excludes) the possibility of the occurrence of another.

Dependent (independent) events - such events, the occurrence of one of which affects (does not affect) the occurrence of another event.

Opposite event relative to some selected event BUT– an event consisting in the non-occurrence of this selected event (denoted ).

A complete group of events is such a set of events in which at least one of the events of this set must occur as a result of the experiment. Obviously the events BUT and constitute a complete group of events.

One of the reasons for the use of set theory in probability theory is that important transformations are defined for sets, which have a simple geometric representation and make it easier to understand the meaning of these transformations. It is called the Euler-Venn diagram, and on it the space is depicted in the form of a rectangle, and various sets - in the form of flat figures bounded by closed lines. Example diagram illustrating set inclusion CB A, shown in fig. one.

It's clear that B is a subset BUT, a C- subset B(and at the same time a subset BUT).

II.2. 2. Algebra of events.

In applied problems, the main methods are not direct, but indirect methods for calculating the probabilities of events of interest to us through the probabilities of others associated with them. To do this, we need to be able to express the events of interest to us through others, that is, to use the algebra of events.

Note that all the concepts introduced below are valid when the events in question are subsets of the same space of elementary events .

Sum or event aggregation BUT1 , BUT2 , …, An- such an event BUT, the appearance of which in an experiment is equivalent to the appearance in the same experiment of at least one of the events BUT1 , BUT2 , …, An. The amount is indicated:

where - sign of logical addition of events, - sign of the logical sum of events.

Work or event intersectionBUT1 , BUT2 , …, An- such an event BUT, the appearance of which in the experiment is equivalent to the appearance in the same experience of all events BUT1 , BUT2 , …, An simultaneously. The work is denoted

where - sign of logical multiplication of events, - a sign of the logical product of events.

The operations of addition and multiplication of events have a number of properties inherent in ordinary addition and multiplication, namely: commutative, associative and distributive properties, which are obvious and do not need explanation.

Euler-Venn diagrams for the sum (a) and product (b) of two events BUT1 and BUT2 shown in fig. 2.

The sum (combination) of events BUT1 and BUT2 is an event consisting in the appearance of at least one of these events (shaded area in Fig. 2, a). Production of events BUT1 and BUT2 is an event consisting in the joint execution of both events (shaded intersection of events BUT1 and BUT2 - rice. 2b).

From the definition of the sum and the product of events it follows that

BUT = BUT BUT; BUT = BUT ; = BUT ;
BUT = AA; = BUT ; BUT = BUT .

If events AI(i=1, …, n) or ( AI) n i=1 make up a complete group of events, then their sum is a reliable event

Depiction of opposite event shown in fig. 3. Region complements BUT to full space . From the definition of the opposite event, it follows that

illustrated in Fig. four.

II.2. 3. Axioms of probability theory

Match each event BUT a number, called, as before, its probability and denoted by P(A) or P(A). The probability is chosen so that it satisfies the following conditions or axioms:

If a AI and Aj incompatible events, i.e. Ai Aj= , then

Using the axioms, one can calculate the probabilities of any events (subsets of the space ) using the probabilities of elementary events. The question of how to determine the probabilities of elementary events is rhetorical. In practice, they are determined either from considerations related to the possible outcomes of the experiment (for example, in the case of tossing a coin, it is natural to consider the probabilities of heads or tails to be the same), or on the basis of experimental data (frequencies).

The latter approach is widely used in applied engineering problems, since it allows one to correlate the results of analysis indirectly with physical reality.

Let us assume that in experience the space can be represented as a complete group of incompatible and equally probable events BUT1 , BUT2 , …, An. According to (3), their sum represents a reliable event:

as the events BUT1 , BUT2 , …, An are inconsistent, then according to axioms (6) and (9):

Because the events BUT1 , BUT2 , …, An are equally likely, then the probability of each of them is the same and is equal to

From this it follows directly frequency definition of probability any event A:

as the ratio of the number of cases ( m A) favorable to the occurrence of the event BUT, to the total number of cases (possible number of outcomes of the experiment) n.

It is quite obvious that the frequency estimate of probability is nothing but a consequence of the axiom of addition of probabilities. Imagine that the number n increases without limit, one can observe a phenomenon called statistical ordering, when the frequency of an event BUT changes less and less and approaches some constant value, which represents the probability of an event BUT.

II.2. 4. Basic rules of probability theory

The probabilities of complex events can be calculated using the probabilities of simpler ones, using the basic rules (theorems): addition and multiplication of probabilities.

II.2.4.1.The theorem of addition of probabilities.

If a BUT1 , BUT2 , …, An- incompatible events and BUT is the sum of these events, then the probability of the event BUT is equal to the sum of the probabilities of events BUT1 , BUT2 , …, An:

In order to formulate the probabilities multiplication theorem in the general case, we introduce the notion of conditional probability.

Conditional Probability developments BUT1 when an event occurs BUT2 - the probability of an event BUT1 , calculated under the assumption that the event BUT2 happened:

For any finite number of events, the multiplication theorem has the form

and for a finite number n of independent events

A consequence of the rules of addition and multiplication of probabilities is experiment repetition theorem (Bernoulli scheme): Experiments are considered independent if the probability of one or another outcome of each of them does not depend on what outcomes the other experiments had.

Let in some experience the probability of an event BUT is equal to P(A) = p, and the probability that it will not happen is P( ) = q, and, according to (13),

P(A) + P() = p + q = 1

If carried out n independent experiments, in each of which an event BUT appears with probability p, then the probability that in a given series of experiments an event BUT appears exactly m times, is determined by the expression

where - binomial coefficient.

For example, the probability of a single error when reading a 32-bit word in computer format, representing a combination of 0 and 1, with the probability of an error reading a binary number p = 10 -3, is according to (19)

where q=1-p=0.999; n = 32; m = 1.

Probability of no read error at m = 0, C 0 32 = 1

Often there are problems of determining the probabilities that some event BUT will happen at least m once or more m once. Such probabilities are determined by adding the probabilities of all outcomes that make up the event in question.

The calculation expressions for this type of situation are:

where P n (i) is determined by (19).

At large m calculation of binomial coefficients C n m and raising p and q to large powers is associated with significant difficulties, so it is advisable to use simplified calculation methods. The approximation called de Moivre-Laplace theorem, is used if npq>>1, and |m-np|<(npq) 0,5 , в таком случае выражение (19) записывается:

II.2. 5. Total probability formula and Bayes formula (formula of hypotheses probabilities)

In the practice of solving a large number of problems, the total probability formula (PPF) and the Bayes formula, which are a consequence of the main theorems, are widely used.

II.2.5.1. Total probability formula.

If the results of the experiment can be done n mutually exclusive assumptions (hypotheses) H1 , H2 , … Hn, representing the complete group of incompatible events (for which ), then the probability of the event BUT, which can only appear with one of these hypotheses, is determined by:

where P(Hi)- the probability of the hypothesis Hi;

P(A|Hi) is the conditional probability of the event BUT under the hypothesis Hi.

Since the event BUT may appear with one of the hypotheses H1 , H2 , … Hn, then A = A H1 H2 … BUT Hn, but H1 , H2 , … Hn inconsistent, so

In view of the dependence of the event BUT from the occurrence of an event (hypothesis) Hi

P(AHi) = P(Hi) P(А| Hi), whence expression (21) follows.

II.2.5.2.Bayes formula (formula of probabilities of hypotheses).

If prior to experience the probabilities of hypotheses H1 , H2 , … Hn were equal P(H1 ), P(H2 ), …, P(Hn), and as a result of the experiment, an event occurred BUT, then the new (conditional) probabilities of the hypotheses are calculated:

Pre-experimental (initial) probabilities of hypotheses P(H1 ), P(H2 ), …, P(Hn) called a priori , and post-experimental - P(H1 | A), … P(Hn| A) – a posteriori .

The Bayes formula allows you to "revise" the possibilities of hypotheses, taking into account the result of the experiment.

The proof of the Bayes formula follows from the previous material. Because the P(Hi A) = P(Hi) P(A| Hi) = P(Hi) P(Hi| A): . More detailed material fromtheoriesprobabilities the reader can get in the Appendix: " Mainconcepts and briefintelligencefromtheoriesprobabilities". 2. Mainintelligence about mathematical models of calculation in theoriesprobabilities ...

Mathematics is the queen of all sciences, often put on trial by young people. We put forward the thesis “Mathematics is useless”. And we refute on the example of one of the most interesting mysterious and interesting theories. How probability theory helps in life, saves the world, what technologies and achievements are based on these seemingly intangible and far from life formulas and complex calculations.

History of probability theory

Probability theory- a branch of mathematics that studies random events, and, of course, their probability. This kind of mathematics was born not at all in boring gray offices, but ... gambling halls. The first approaches to assessing the probability of an event were popular back in the Middle Ages among the “hamlers” of that time. However, then they had only an empirical study (that is, an assessment in practice, by the method of experiment). It is impossible to attribute the authorship of the theory of probability to a certain person, since many famous people worked on it, each of whom invested his share.

The first of these people were Pascal and Fermat. They studied probability theory on the statistics of dice. She discovered the first regularities. H. Huygens did similar work 20 years earlier, but the theorems were not formulated exactly. An important contribution to the theory of probability was made by Jacob Bernoulli, Laplace, Poisson and many others.

Pierre Fermat

Probability theory in life

I will surprise you: we all, to one degree or another, use the theory of probability, based on an analysis of the events that have occurred in our lives. We know that death from a car accident is more likely than from a lightning strike, because the former, unfortunately, happens very often. One way or another, we pay attention to the likelihood of things in order to predict our behavior. But here's an insult, unfortunately, not always a person can accurately determine the likelihood of certain events.

For example, without knowing the statistics, most people tend to think that the chance of dying in a plane crash is greater than in a car accident. Now we know, having studied the facts (which, I think, many have heard of), that this is not at all the case. The fact is that our vital "eye" sometimes fails, because air transport seems much more terrible to people who are accustomed to walking firmly on the ground. And most people do not often use this mode of transport. Even if we can estimate the probability of an event correctly, it is most likely extremely inaccurate, which would not make any sense, say, in space engineering, where millionths decide a lot. And when we need accuracy, who do we turn to? Of course, to mathematics.

There are many examples of the real use of probability theory in life. Almost the entire modern economy is based on it. When launching a certain product on the market, a competent entrepreneur will certainly take into account the risks, as well as the likelihood of buying in a particular market, country, etc. Practically do not imagine their life without the theory of probability brokers in the world markets. Predicting the money rate (in which probability theory is definitely indispensable) on money options or the famous Forex market makes it possible to earn serious money on this theory.

The theory of probability is important at the beginning of almost any activity, as well as its regulation. Thanks to the assessment of the chances of a particular malfunction (for example, a spacecraft), we know what efforts we need to make, what exactly to check, what to expect in general thousands of kilometers from Earth. The possibility of a terrorist attack in the subway, an economic crisis or a nuclear war - all this can be expressed as a percentage. And most importantly, take appropriate counter-actions based on the data received.

I was lucky enough to get to the mathematical scientific conference of my city, where one of the winning papers spoke about the practical significance probability theory in life. You probably, like all people, do not like to stand in lines for a long time. This work proved how the buying process can be accelerated if we use the probability theory of counting people in the queue and the regulation of activities (opening cash desks, increasing sellers, etc.). Unfortunately, now most even large networks ignore this fact and rely only on their own visual calculations.

Any activity in any field can be analyzed using statistics, calculated using probability theory and significantly improved.

We should rightfully start with statistical physics. Modern natural science proceeds from the idea that all natural phenomena are of a statistical nature and that laws can be formulated precisely only in terms of probability theory. Statistical physics has become the basis of all modern physics, and probability theory - its mathematical apparatus. In statistical physics, problems are considered that describe phenomena that are determined by the behavior of a large number of particles. Statistical physics is very successfully applied in various branches of physics. In molecular physics, with its help, thermal phenomena are explained; in electromagnetism, the dielectric, conductive and magnetic properties of bodies; in optics, it made it possible to create a theory of thermal radiation, molecular scattering of light. In recent years, the range of applications of statistical physics has continued to expand.

Statistical representations made it possible to quickly formalize the mathematical study of the phenomena of nuclear physics. The advent of radio physics and the study of the transmission of radio signals not only increased the significance of statistical concepts, but also led to the progress of mathematical science itself - the emergence of information theory.

Understanding the nature of chemical reactions, dynamic equilibrium is also impossible without statistical concepts. All physical chemistry, its mathematical apparatus and the models it proposes are statistical.

The processing of observational results, which are always accompanied by both random observational errors and random changes for the observer in the conditions of the experiment, led researchers back in the 19th century to create a theory of observational errors, and this theory is completely based on statistical concepts.

Astronomy in a number of its sections uses the statistical apparatus. Stellar astronomy, the study of the distribution of matter in space, the study of cosmic particle fluxes, the distribution of sunspots (centers of solar activity) on the surface of the sun, and much more require the use of statistical representations.

Biologists have noticed that the spread in the sizes of the organs of living beings of the same species fits perfectly into the general theoretical and probabilistic laws. The famous laws of Mendel, which marked the beginning of modern genetics, require probabilistic-statistical reasoning. The study of such significant problems of biology as the transfer of excitation, the structure of memory, the transfer of hereditary properties, questions of the distribution of animals in the territory, the relationship between predator and prey requires a good knowledge of probability theory and mathematical statistics.

The humanities unite very diverse disciplines, from linguistics and literature to psychology and economics. Statistical methods are increasingly being used in historical research, especially in archaeology. The statistical approach is used to decipher inscriptions in the language of ancient peoples. The ideas that guided J. Champollion in deciphering ancient hieroglyphic writing are basically statistical. The art of encryption and decryption is based on the use of the statistical patterns of language. Other areas are related to the study of the frequency of words and letters, the distribution of stress in words, the calculation of the informativeness of the language of specific writers and poets. Statistical methods are used to establish authorship and expose literary forgeries. For example, the authorship of M.A. Sholokhov based on the novel "Quiet Don" was established using probabilistic-statistical methods. Revealing the frequency of the appearance of sounds of a language in oral and written speech allows us to raise the question of the optimal coding of the letters of a given language for transmitting information. The frequency of use of letters determines the ratio of the number of characters in the typesetting box office. The arrangement of letters on the carriage of a typewriter and on a computer keyboard is determined by a statistical study of the frequency of letter combinations in a given language.

Many problems of pedagogy and psychology also require the involvement of a probabilistic-statistical apparatus. Economic issues cannot but interest the society, since all aspects of its development are connected with it. Without statistical analysis, it is impossible to foresee changes in the size of the population, its needs, the nature of employment, changes in mass demand, and without this it is impossible to plan economic activity.

Directly related to probabilistic-statistical methods are the issues of checking the quality of products. Often, the manufacture of a product takes incomparably less time than checking its quality. For this reason, it is not possible to check the quality of each product. Therefore, one has to judge the quality of a batch by a relatively small part of the sample. Statistical methods are also used when testing the quality of products leads to their damage or death.

Questions related to agriculture have long been resolved with the extensive use of statistical methods. Breeding of new breeds of animals, new varieties of plants, comparison of yields - this is not a complete list of tasks solved by statistical methods.

It can be said without exaggeration that our whole life is permeated with statistical methods today. In the well-known work of the materialist poet Lucretius Cara "On the Nature of Things" there is a vivid and poetic description of the phenomenon of Brownian motion of dust particles:

"Look: every time the sunlight penetrates into our dwellings and cuts through the darkness with its rays, You will see many small bodies in the void, flickering, Rushing back and forth in a radiant radiance of light; As if in an eternal struggle they fight in battles and "Suddenly they rush into battles in groups, not knowing peace. Either converging, or apart, constantly scattering again. From this you can understand for yourself how tirelessly the Origin of things in the vast void is restless. Thus, small things help to comprehend great things, outlining the paths for achievement, In addition, therefore you need to pay attention to the turmoil in the bodies flickering in the sunlight, that from it you will recognize the movement of matter"

The first opportunity for an experimental study of the relationship between the random motion of individual particles and the regular motion of their large aggregates appeared when, in 1827, the botanist R. Brown discovered a phenomenon that was named "Brownian motion" after his name. Brown observed flower pollen suspended in water under a microscope. To his surprise, he discovered that the particles suspended in water were in continuous random movement, which could not be stopped even with the most careful effort to eliminate any external influences. It was soon discovered that this is a general property of any sufficiently small particles suspended in a liquid. Brownian motion is a classic example of a random process.

Nevolina Ekaterina Nikolaevna Ekaterinburg USUE Head - Knysh A. A. Practical application of the theory of probability. Relevance. Probability theory is one of the branches of mathematics that studies random events, random variables, their properties and operations on them. Methods of probability theory are increasingly being used in various fields of science and technology, as well as in everyday life. The peculiarity of this section of science is the consideration of such phenomena in which there is uncertainty. In the article, I would like to consider examples of some problems that demonstrate the practical application of probability theory. Problems with economic content. 1. One of the firms is going to conclude a contract for the supply of goods with a chain of stores. Provided that a competitor of the firm does not simultaneously apply for a contract, the probability of a contract is estimated at 0.85, otherwise the probability of obtaining a contract is 0.6. According to the company's experts, the probability that a competitor will put forward proposals for concluding a contract is 0.55. What is the probability of a contract for this firm? . This problem is solved using the total probability formula. 2. An economist-analyst conditionally subdivides the economic situation in the country into “good”, “mediocre” and “bad” and estimates their probabilities for a given point in time at 0.2; 0.7 and 0.15, respectively. Some index of economic condition increases with a probability of 0.65 when the situation is "good"; with a probability of 0.35 when the situation is mediocre, and with a probability of 0.1 when the situation is "bad". Suppose that the index of economic condition has increased at the present moment. What is the probability that the country's economy is booming? . The problem is solved using the Bayes formula. 3. The bank issues 9 loans. The probability of loan default is 0.2 for each borrower. What is the probability that three borrowers will not repay the loan? The problem is solved using the Bernoulli formula. 5. The part is considered fit if the deviation X of the linear size in absolute terms is less than 1 mm. Deviation X is a value distributed according to the normal law, with a standard deviation   0.35 . Find the number of defective parts in one batch of manufactured parts (lot size 1000 pieces), the cost of losses from marriage at a batch cost of 15 million rubles, income from the sale of the remaining good parts and economic losses at a market price of 19,000 rubles. per unit of production. Let's consider the solution of this problem. Because X is the deviation of the linear size in absolute terms, then the mathematical expectation is M(X)=a=0. Substituting in the formula  P  X     2      the values ​​   0.35 and   1, we get P X  1  0.9956. Thus, in a batch of 1000 parts, 995 parts will be good. At a party cost of 15 million rubles. the cost of each part will average 15,000 rubles. The cost of losses from marriage will be 75,000 rubles. Income from the sale of suitable parts at the market price will be 995∙19000 = 18.905 million rubles. Due to the inability to sell part of the production, economic losses will amount to 5∙19000=95000 rubles. Probability theory methods are also used in sports betting. With the help of probability theory, it became possible to predict and evaluate the outcomes of various matches, as well as to identify the productivity of a single player. So, for example, if we consider basketball, then as a player's productivity we can consider the probability of his hitting the ring from various points. Let's give examples of tasks. 1. In a basketball competition, the center player of the “N” team throws the ball into the basket. For each goal scored, the team receives 2 points. Find the probability that for a given throw by the post, the team will not receive a single point (0 points are due only for a miss). 2. Two equal basketball teams play basketball. Which is more likely: to score one quarter out of two or two quarters out of four (tied scores are not taken into account)? This problem is solved using the Bernoulli formula. So, finding patterns in random phenomena is the task of probability theories. The theory of probability is a tool for studying the invisible and multivalued relationships of various phenomena in numerous fields of science, technology and economics. Probability theory makes it possible to correctly calculate fluctuations in demand, supply, prices and other economic indicators. Probability theory is a part of basic science like statistics and applied computer science. Since not one application program, and the computer as a whole, cannot work without the theory of probability. And in game theory, it is also the main one. List of sources used: 1. Wentzel E. S. Probability theory [Electron. resource] : Proc. allowance. - Moscow. - Higher School, 1999. - 576 p. – Access mode: http://sernam.ru/book_tp.php 2. Guidelines for students on practical work in the discipline "Mathematics" [Electron. resource]. – Monchegorsk, 2013. – Mode of access: http://www.studfiles.ru/preview/3829108/ 3. Khusnutdinov, R. Sh. Mathematics for Economists in Examples and Tasks [Electron. resource]: textbook. allowance / R. Sh. Khusnutdinov, V. A. Zhikharev. - St. Petersburg: Lan, 2012. - 656 p. - Access mode: https://e.lanbook.com/book/4233