Life stories about the law of probability. Research work "probability theory"

Mathematics, the queen of all sciences, is often put on trial by young people. We put forward the thesis “Mathematics is useless.” And we refute it using 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 field of mathematics that studies random events and, naturally, their probability. This kind of mathematics originated not in boring gray offices, but... in gambling halls. The first approaches to assessing the probability of a particular event were popular back in the Middle Ages among the “Hamlers” of that time. However, then they had only empirical research (that is, evaluation in practice, by experiment). It is impossible to attribute the authorship of the theory of probability to a specific person, since many famous people worked on it, each of whom contributed their own share.

The first of these people were Pascal and Fermat. They studied probability theory using dice statistics. She discovered the first laws. H. Huygens had done similar work 20 years earlier, but the theorems were not formulated precisely. Important contributions to probability theory were made by Jacob Bernoulli, Laplace, Poisson and many others.

Pierre Fermat

The theory of probability in life

I will surprise you: we all, to one degree or another, use the theory of probability, based on the analysis of events that have happened in our lives. We know that death from a car accident is more likely than from a lightning strike because the former, unfortunately, happens so often. One way or another, we pay attention to the likelihood of things in order to predict our behavior. But unfortunately, a person cannot always 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 about), that this is not at all the case. The fact is that our life “eye” sometimes fails, because air transport seems much more frightening to people who are accustomed to walking firmly on the ground. And most people do not use this type of transport very often. Even if we can estimate the probability of an event correctly, it is most likely extremely inaccurate, which will not make any sense, say, in space engineering, where parts per million 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 releasing a certain product to the market, a competent entrepreneur will certainly take into account the risks, as well as the likelihood of purchase in a particular market, country, etc. Brokers on world markets practically cannot imagine their life without probability theory. Predicting the money exchange rate (which definitely cannot be done without the theory of probability) on money options or the famous Forex market makes it possible to earn serious money from this theory.

The theory of probability is important at the beginning of almost any activity, as well as its regulation. By assessing 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 possibilities of a terrorist attack in the metro, an economic crisis or a nuclear war - all this can be expressed as a percentage. And most importantly, take appropriate counteractions based on the data received.

I was lucky enough to attend a mathematical scientific conference in my city, where one of the winning papers spoke about the practical significance theories of probability in life. You probably, like all people, don’t like standing in lines for a long time. This work proved how the purchasing process can be accelerated if we use the probability theory of calculating people in line and regulating activities (opening cash registers, increasing the number of salespeople, etc.). Unfortunately, now the majority of even large networks ignore this fact and rely only on their own visual calculations.

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

Real life turns out to be not so simple and unambiguous. The outcomes of many phenomena cannot be predicted in advance, no matter how complete information we have about them. It is impossible, for example, to say for sure which side a coin thrown up will fall, when the first snow will fall next year, or how many people in the city will want to make a phone call within the next hour. Such unpredictable phenomena are called random. However, chance also has its own laws, which begin to manifest themselves when random phenomena are repeated many times. It is these patterns that are studied in a special section of mathematics - Probability Theory.

As a science, probability theory originated in the 17th century. The emergence of the concept of probability was associated both with the needs of insurance, which became widespread in that era when trade relations and sea travel grew noticeably, and in connection with the demands of gambling. The word “excitement”, which usually means strong passion, fervor, is a transcription of the French word hazard, literally meaning “case”, “risk”.

Gambling games are those games in which the winnings depend mainly not on the player’s skill, but on chance. The gambling scheme was very simple and could be subjected to comprehensive logical analysis. The first attempts of this kind are associated with the names of famous scientists, algebraist Gerolamo Cardan () and Galileo Galilei (). However, the honor of discovering this theory, which not only makes it possible to compare random variables, but also to perform certain mathematical operations with them, belongs to two outstanding scientists Blaise Pascal () and Pierre Fermat.

Even in ancient times, it was noticed that there are phenomena that have a peculiarity: with a small number of observations, no correctness is observed over them, but as the number of observations increases, a certain pattern becomes more and more clear. It all started with a game of dice.

The emergence of probability theory as a science dates back to the Middle Ages and the first attempts at mathematical analysis of gambling (flake, dice, roulette). Initially, its basic concepts did not have a strictly mathematical form; they could be treated as some empirical facts, as properties of real events, and they were formulated in visual representations. The earliest works of scientists in the field of probability theory date back to the 17th century. While studying the prediction of winnings in gambling, Blaise Pascal and Pierre Fermat discovered the first probabilistic patterns that arise when throwing dice.

Jacob Bernoulli made an important contribution to probability theory: he gave a proof of the law of large numbers in the simplest case of independent trials. In the first half of the 19th century, probability theory began to be applied to the analysis of observational errors; Laplace and Poisson proved the first limit theorems. In the second half of the 19th century, the main contribution was made by Russian scientists P. L. Chebyshev, A. A. Markov and A. M. Lyapunov. At this time, the law of large numbers and the central limit theorem were proven, and the theory of Markov chains was developed. Probability theory received its modern form thanks to the axiomatization proposed by Andrei Nikolaevich Kolmogorov. As a result, probability theory acquired a strict mathematical form and finally began to be perceived as one of the branches of mathematics. Jacob Bernoulli's law of large numbers of the 19th century Laplace-Poisson of the 19th century P. L. Chebysheva A. A. Markov A. M. Lyapunovlaw of large numbers central limit theorem Markov chain axiomatization by Andrei Nikolaevich Kolmogorov sections of mathematics

Many people ask what is theory of probability, cognition and everything, what it affects and what its functions are. As you know, there are many theories and few of them work in practice. Of course, the theory of probability, knowledge and everything has long been proven by scientists, so we will consider it in this article in order to use it to our advantage.

In the article you will learn what the theory of probability, knowledge and everything is, what its functions are, how it manifests itself and how to use it to your advantage. After all, probability and knowledge are very important in our lives and therefore we need to use what has already been tested by scientists and proven by science.

Certainly Probability theory is a mathematical and physical science that studies this or that phenomenon and what is the probability that everything will happen exactly the way you want. For example, how likely is it that the end of the world will happen in 27 years, and so on.

Also, the theory of probability is applicable in our lives, when we strive for our goals and do not know how to calculate the probability of whether we will achieve our goal or not. Of course, this will be based on your hard work, a clear plan and real actions, which can be calculated for many years.

Theory of knowledge

The theory of knowledge is also important in life, as it determines our subconscious and consciousness. Because we are learning about this world and developing every day. The best way to learn something new is by reading interesting books written by successful authors who have achieved something in life. Knowledge also allows us to feel God within ourselves and create reality for ourselves the way we want, or trust God and become a puppet in his hands.

Theory of everything

But here theory of everything tells us that the world came into existence precisely because of the big bang, which separated energy into several cells in a matter of seconds and as we see large populations, this is actually the division of energy. When there are fewer people, this will mean that the World is returning to its original point again, and when the world is restored, there is a high probability of another explosion.

Denisova Ekaterina

Report at the scientific and practical conference

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International

Scientific research

conference

High school students and students

"Education. The science. Profession"

Section "Mathematics"

"The theory of probability in our lives"

Completed by: Ekaterina Denisova, 11th grade student

Municipal educational institution Kabanovskaya secondary school

Head: Zolotareva Valentina Viktorovna,

Mathematic teacher

Otradny

year 2012

1. Main part

1. Basic concepts of the theory
2. Problems and examples
3. Forecasting the results of the Unified State Examination in mathematics in 2012

1. Conclusion. Practical application of probability theory

1. Introduction. The world is ruled by chance

"Probability theory is essentially

Nothing more than common sense reduced to calculus"

Laplace

At first glance, it may seem that there are and cannot be any laws governing the phenomena in our lives. However, if you look at it, random phenomena do not occur so chaotically. In many cases, patterns emerge. These patterns are not similar to the ordinary laws of physical phenomena; they are very diverse.So, each of us every day has to make many decisions under conditions of uncertainty. However, this uncertainty can be “transformed” into some certainty. And then this knowledge can provide significant assistance in making a decision.

Every “random” event has a clear probability of its occurrence.

In a stable system, the probability of events occurring is maintained from year to year. That is, from a person’s point of view, a random event happened to him. And from the point of view of the system, it was predetermined.

A reasonable person should strive to think based on the laws of probability (statistics). But in life, few people think about probability. Decisions are made emotionally.

People are afraid to fly by plane. Meanwhile, the most dangerous thing about flying on an airplane is the road to the airport by car. But try to explain to someone that a car is more dangerous than an airplane.

According to research: in the United States, in the first 3 months after the terrorist attacks of September 11, 2001, another thousand people died... indirectly. Out of fear, they stopped flying by plane and began moving around the country in cars. And since it is more dangerous, the number of deaths has increased.

The world is ruled by probability and we need to remember this.

II. Main part

1. History of the emergence of probability theory

The word "probability" “, a synonym for which is, for example, the word “chance”, which is often used in everyday life. I think everyone is familiar with the phrases: “Tomorrow it will probably snow,” or “I’ll probably go outdoors this weekend,” or “this is simply incredible,” or “there’s a chance to get an automatic test.” These kinds of phrases intuitively assess the likelihood that some random event will occur.

The outcomes of many phenomena cannot be predicted in advance, no matter how complete information we have about them. It is impossible, for example, to say for sure which side a coin thrown up will fall, when the first snow will fall next year, or how many people in the city will want to make a phone call within the next hour.

Such unpredictable events are called random.

Probability theory took shape as an independent science relatively recently, althoughhistory of probability theorybegan in antiquity. Thus, Lucretius, Democritus, Carus and some other scientists of ancient Greece in their reasoning spoke about equally probable outcomes of such an event, such as the possibility that all matter consists of molecules. Thus, the concept of probability was used on an intuitive level, but it was not separated into a new category. However, ancient scientists laid an excellent foundation for the emergence of this scientific concept. In the Middle Ages, one might say, the theory of probability was born, when the first attempts at mathematical analysis and such gambling games as dice, toss, and roulette were made. In archaeological excavations, animal bones specially processed for playing have been found since the 5th century BC. The oldest dice was found in Northern Iraq and dates back to the 4th millennium BC. People who have repeatedly watched the throwing of dice have noticed certain patterns that govern this game.

The results of these observations were formulated as the “Golden Rules” and were known to many players.

One of the most famous problems that contributed to the development of probability theory was the problem of dividing the bet, placed in the book of Luca Paccioli (1445 - ca. 1514).

The book was called "The Summa of Arithmetic, Geometry, Ratio and Proportion" and was published in Venice in 1494.

The next person who made a significant contribution to the understanding of the laws governing chance was Galileo Galilei (1564 -1642).

It was he who noticed thatmeasurement resultsare random.

The first scientific works on probability theory appeared in the 17th century. When scientists such as Blaise Pascal and Pierre Fermat discovered certain patterns that occur when throwing dice. At the same time, another scientist, Christian Huygens, showed interest in this issue. In 1657, in his work, he introduced the following concepts of probability theory: the concept of probability as the value of chance or possibility; mathematical expectation for discrete cases, in the form of the price of chance, as well as theorems of addition and multiplication of probabilities, which, however, were not formulated explicitly. At the same time, probability theory began to find areas of application - demography, insurance, and assessment of observation errors.

But as a mathematical science, probability theory begins with the work of the outstanding Swiss mathematician Jacob Bernoulli (1654 -1705) “The Art of Conjecture.”

This treatise proves a number of theorems, including the most famous theorem, “The Law of Large Numbers”

The most significant contribution to laying the foundations of the theory was made by A.N. Kolmogorov.

To dateprobability theoryThis is an independent science with a huge scope of application.

1. Basic concepts of the theory

Take the coin game for example. When tossing, there can be two equally probable outcomes: the coin can land up head or tail. When you toss a coin once, you cannot predict which side will end up on top. However, after tossing a coin 100 times, you can draw conclusions. You can say in advance that the coat of arms will appear not 1 or 2 times, but more, but not 99 or 98 times, but less. The number of drops of the coat of arms will be close to 50. In fact, and from experience one can be convinced of this that this number will be between 40 and 60.

It is also statistically established that per 1000 children there are 511 boys and 489 girls (i.e. 48.9% and 51.1%, respectively). This information allows us to predict with great accuracy the likelihood of the number of boys or girls in a given year (these calculations, for example, are used by the draft board).

• The subject of research in probability theory is events , appearing under certain conditions, which can be played an unlimited number of times.
• Each realization of these conditions is called test

Test examples:throwing a dice, weighing a body on an analytical balance

Examples of events:rolling a six or If an even number of points is rolled, the measurement error will not exceed a predetermined number

Degree of objective possibilitya random event can be measured by a number.

This number is calledthe probability of a random event.

The relative frequencies of a given random event are grouped around this number

The event is called reliable, if it always occurs, under any test.

The probability of a certain event is always equal to 1.

Examples of reliable events

1. The dice will roll less than seven;
2. After summer, autumn will come.

The event is called impossible , if it never occurs, then there are 0 favorable outcomes for it.

The probability of an impossible event is 0.

Examples of impossible events

1. A coin falling on its edge

1. Rolling a seven on the dice

The event is called random , if under the same conditions it may or may not happen.

Examples of random events

1. An even number of points appears on the dice;
2. Landing heads when tossing a coin;
3. Winning combination of numbers on Russian lotto cards.

The union of events A and B is an event that consists in the fact that at least one of these events occurred as a result of an experiment (i.e.).

The intersection of events A and B is an event such that both of these events occur as a result of an experiment (i.e.).

Events A and B are called incompatible , if they cannot occur simultaneously, or, in the language of sets, A ∩ B = ∅ .

Examples of incompatible events

1. When rolling two dice, an odd number of points is obtained and equal numbers on both dice;
2. Take 2 balls out of the box with multi-colored balls. The following events will be incompatible: both balls are red and both balls are blue.

Events A and B are called independent , if the probability of their product is equal to the product of their probabilities: P(AB) = P(A)⋅ P(B).

Examples of independent events

1. Both dice will roll a six;
2. When flipping two coins, two heads will appear;
3. When two balls are drawn from an urn, both balls will be red.

With every event A connected opposite event, consisting in the fact that the event A is not implemented.

Opposite events are obviously incompatible.

The sum of the probabilities of opposite events is 1

Examples of opposite events

1. The die will roll an even number and the die will roll an odd number;
2. The coin landed heads up and the coin landed tails up;
3. The lamp is on and the lamp is not on.

Event A favors event B if event B follows from the fact that event A occurs (i.e.)

Conditional probability of event B given condition Acalled attitude

Law of large numbers.

Let us carry out the tests K times, and N times as a result of the experiment, event A occurs. Then the numberwill be called the frequency of occurrence of event A.

You can always choose N large enough to satisfy the following relation:

Where (upsilon) - an arbitrarily small positive number that is not equal to zero.

This means that with a sufficiently large number of tests, the frequency of occurrence of a particular event will differ as little as desired from zero.

This relationship makes it possible to establish experimentally with a fairly good approximation the probability of an event unknown to us.

3. Problems and examples.

The first calculations of the probabilities of events began in the 17th century with the calculation of the chances of players in gambling. First of all, it was a game of dice.

Task 1.

They threw a die. What is the probability that the number rolled is 5?

Solution.

There are 6 types of bone loss in total (n = 6). All these options are equally probable, because the die is made so that all sides have the same chance of being on top, hence m = 1; Means

Where P(5) is the probability of rolling a five.

Task 2.

What is the probability that when throwing an even number of points?

Solution.

There are three favorable opportunities here: 2; 4; 6. Therefore m = 3, there are 6 outcomes in total (n = 6), therefore

Where P(even) is the probability of getting an even number.

Task 3.

We threw 2 dice and counted the total points. What is more likely - to get a total of 7 or 8?

Solution.

We are interested in the events A = “7 points are rolled” and B = “8 points are rolled.” Number of all possible outcomes n = 6 2 = 36 (each of the 6 points on the white die can be combined with any of the 6 points on the black die). Of these 36 outcomes, event A will be favored by the following outcomes: (1; 6); (2; 5); (3; 4); (4; 3); (5; 2); (6; 1), i.e. total 6 (m = 6). According to the formula we have:

Event B will be favored by the following outcomes: (2;6); (3;5); (4;4); (5;3); (6;2), i.e. only 5. According to the formula, we have:

Therefore, getting a total of 7 points is a more likely event than getting 8.

This problem was first solved by dice players, and only then solved mathematically. She became one of the first, during the discussion of which the Theory began to take shape.

Definition: Two events A and B are called independent if the equality holds:

Task 4.

Two hunters, independently of each other, simultaneously shoot at a hare. The hare will be killed if both are hit. What are the hare's chances of surviving if the first hunter hits with a probability of 0.8, and the second with a probability of 0.75?

Solution.

Let's consider two events: A = “the 1st hunter hit the hare” and B = “the 2nd hunter hit the hare.” We are interested in the event(i.e. both event A and event B occurred). Due to the independence of events, we have:

This means that in 6 out of 10 cases the hare will be shot.

Task 5.

One French knight, de Mere, was a passionate dice player. He tried in every possible way to get rich and came up with various complicated rules for this.

In particular, he came up with the following rules: they throw 4 dice and he bets that at least one of them will get a 6. He believed that in most cases he would win. To confirm this, he turned to his old acquaintance, Blaise Pascal, with a request to calculate what the probability of winning in this game was.

Let us present Pascal's calculation.

For each individual roll, the probability of the event A = “a six is ​​thrown” =. Probability of event B = “missing a six” =. The cubes do not depend on each other, therefore, according to the formula

The probability of not rolling a six twice in a row is

In the same way, it is shown that when rolled three times, the probability of not getting a 6 is

And with four times -

A , therefore, the probability of winning. This means that in each game more than half the chances were that De Mere would win; if the game was repeated many times, he would certainly win.

It is reasonable to ask the question, what should be the probability of an event in order for it to be considered reliable? It is known that approximately 5% of scheduled concerts are cancelled, but this does not stop us from buying tickets. But if 5% of planes crashed, then hardly anyone would use air transport.

III.Conclusion. Practical application of probability theory

However, already at the end of the 17th century. began to use the Theory when insuring ships, i.e. they began to calculate how many chances there were that the ship would return to port unharmed, that it would not be sunk by a storm, that the cargo would not get wet, that it would not be captured by pirates, etc. This calculation made it possible to determine what amount of insurance should be paid and what insurance premium to take so that it would be profitable for the company.

In the first half of the 18th century. Jacob Bernoulli, a member of the Russian Academy of Sciences, did a lot for the theory. The works of S. Laplace, S. Poisson, and C. Gauss should be noted.

With all this, during the second half of the 18th century. The theory, in a certain sense, “was marking time.” At that time, the connection between various phenomena in life and the science of mass phenomena was not yet clear. In the middle of the 19th century. A big shift in the development of the theory was made by the Russian mathematician P. Chebyshev. Markov, Lyapunov, Bernstein, Kolmogorov made a great contribution.

The theory played a large practical role in the Second World War. Let us give an example from the military field. It is clear that it is very difficult to shoot down an airplane with one shot from a rifle. After all, the shooter must not only hit the plane, but also hit the most vulnerable spot, such as the fuel tank. Therefore, the likelihood that one shooter will shoot down an airplane with a rifle is negligible. Mass shelling is a completely different matter. Assuming that the probability of shooting down a plane with one rifle is 0.004; accordingly, the probability of a miss is 0.996. Now suppose there are 500 shooters shooting; as we proved above, the probability of a miss is

Thus, the probability of shooting down a plane in one salvo is 0.86. And if it is possible to fire 2–3 salvos, then the aircraft’s chances of surviving are close to zero.

The Theory also made it possible to determine areas in which it made sense to search for aircraft and submarines or to indicate routes to avoid meeting them. A typical problem here is how to more profitably lead caravans of merchant ships across an ocean in which enemy submarines operate. If you organize caravans of a large number of ships, then you can get by with fewer raids, but the possible losses when meeting an enemy fleet will be greater. The theory helped to calculate the optimal sizes of caravans and the frequency of their departure. Many problems of this kind arose, so special groups were organized at headquarters to calculate probabilities. After the war, similar calculations began to be applied to economic issues in peacetime. They constituted the content of a new large area called operations research, which is being formalized into a whole science.

Many people, when starting to play roulette, remember that they once heard about the theory of probability.

Unfortunately, all this “probability theory” will not help when playing roulette, but will only cause harm.

What follows from this is only that probabilities can be used with an unlimited increase in the number of repetitions of the experiment. When we play roulette, we have a fairly limited number of repetitions of experience (roulette wheel rotations). For an unlimited increase in the number of experiments, we do not have an unlimited amount of money and time in stock.

Probability theory is one of the most interesting sections of the Science of Higher Mathematics. This theory is a complex discipline and has application in real life. It is of undoubted value for general education. This science allows not only to obtain knowledge that helps to understand the patterns of the world around us, but also to find practical application in everyday life.

So each of us has to make many decisions every day under conditions of uncertainty. However, this uncertainty can be “transformed” into some certainty. And then this knowledge can provide significant assistance in making a decision.

Probability theory is a mathematical science that studies the patterns of mass randomness of phenomena (events).

A random event (or simply an event) is any phenomenon that may or may not occur under the implementation of a certain set of conditions. The theory of probability deals with such events, which are of a massive nature. This means that this set of conditions can be reproduced an unlimited number of times. Each such implementation of a given set of conditions is called a test (or experience).

Let event A occur m times during n trials.

The ratio m/n is called the frequency of event A and is denoted:

Experience shows that when tests are repeated many times, the frequency P(A) of a random event is stable.

An event is called reliable if it must necessarily occur in a given experience; on the contrary, an event is called impossible if it cannot happen in a given experience.

If the event is reliable, then it will occur on every trial (m=n).

Therefore, the frequency of event reliability is always equal to one or 100%. On the contrary, if an event is impossible, then it will not occur in any trial (m=0). Therefore, the frequency of an impossible event in any series of trials is 0.

The combination of two (AB) or more (ABC) events is an event consisting of the joint occurrence of events. D=AB; D= ABC

The union of two events A and B is called a C event, which means that at least one of the events either A or B will occur. This event is denoted C=A+B

The union of several events is an event consisting of the occurrence of at least one of them. The notation D=A+B+C means that event D is a union of events A, B, and C.

Two events A and B are said to be incompatible if the occurrence of event A excludes event B.

It follows that if events A and B are incompatible, then event AB is impossible.

Let's look at an example: I want to have a great figure! In order to be physically healthy I need to do a number of exercises. Daily training will lead me to physical success. If I do 2 workouts in 7 days, then it turns out P(A) = 2/7 = 0.29 (or 29% of 100% possible). It's a low probability that my body will get the right shape at the right time. For this, the best option is to practice daily, i.e. 7 workouts in 7 days m=n; 7=7; P(A)=7/7=1 (100%) Therefore, this event takes on a reliable form. If we are not training and m=0, then what kind of figure can we talk about, with m=0 the event is not reliable.