Mom washed the frame

Towards the end of a long summer vacation, it's time to slowly return to higher mathematics and solemnly open an empty Verd file in order to start creating a new section - . I confess that the first lines are not easy, but the first step is half the way, so I suggest everyone to carefully study the introductory article, after which it will be 2 times easier to master the topic! I'm not exaggerating at all. ... On the eve of the next September 1, I remember the first grade and primer .... Letters form syllables, syllables into words, words into short sentences - Mom washed the frame. Cope with terver and mathematical statistics as easy as learning to read! However, for this it is necessary to know the key terms, concepts and designations, as well as some specific rules, to which this lesson is devoted.

But first, please accept my congratulations on the beginning (continuation, completion, note as necessary) school year and accept the gift. The best gift is a book, and for independent work I recommend the following literature:

1) Gmurman V.E. Theory of Probability and Mathematical Statistics

legendary tutorial over ten editions. It differs in intelligibility and the ultimate simple presentation of the material, and the first chapters are completely accessible, I think, already for students in grades 6-7.

2) Gmurman V.E. Guide to Problem Solving in Probability and Mathematical Statistics

Reshebnik of the same Vladimir Efimovich with detailed examples and tasks.

NECESSARILY download both books from the Internet or get their paper originals! A 60s-70s version will do, which is even better for dummies. Although the phrase "probability theory for dummies" sounds rather ridiculous, since almost everything is limited to elementary arithmetic operations. They slip, however, in places derivatives And integrals, but this is only in places.

I will try to achieve the same clarity of presentation, but I must warn you that my course is focused on problem solving and theoretical calculations are kept to a minimum. Thus, if you need a detailed theory, proofs of theorems (yes, theorems!), please refer to the textbook.

For those who want learn to solve problems in a matter of days, created crash course in pdf format (according to the site). Well, right now, without postponing the matter in a long folder, we are starting to study terver and matstat - follow me!

Enough to get started =)

As you read the articles, it is useful to get acquainted (at least briefly) with additional problems of the types considered. On the page Ready-made solutions for higher mathematics the corresponding pdf-ki with examples of solutions are placed. Also, significant assistance will be provided IDZ 18.1-18.2 Ryabushko(easier) and solved IDZ according to the collection of Chudesenko(more difficult).

1) sum two events and is called the event which consists in the fact that or event or event or both events at the same time. In case the events incompatible, the last option disappears, that is, it can occur or event or event .

The rule also applies to more terms, for example, an event is what will happen at least one from events , but if the events are incompatible – that one and only one event from this sum: or event , or event , or event , or event , or event .

Plenty of examples:

The event (when throwing a die does not drop 5 points) is that or 1, or 2, or 3, or 4, or 6 points.

Event (will drop no more two points) is that 1 or 2points.

Event (there will be an even number of points) is that the or 2 or 4 or 6 points.

The event is that a card of red suit (heart) will be drawn from the deck or tambourine), and the event - that the “picture” will be extracted (jack or lady or King or ace).

A little more interesting is the case with joint events:

The event is that a club will be drawn from the deck or seven or seven of clubs According to the above definition, at least something- or any club or any seven or their "crossing" - seven clubs. It is easy to calculate that this event corresponds to 12 elementary outcomes (9 club cards + 3 remaining sevens).

The event is that tomorrow at 12.00 AT LEAST ONE of the summable joint events, namely:

- or there will be only rain / only thunder / only sun;
- or only some pair of events will come (rain + thunderstorm / rain + sun / thunderstorm + sun);
– or all three events will appear at the same time.

That is, the event includes 7 possible outcomes.

The second pillar of the algebra of events:

2) work two events and call the event, which consists in the joint appearance of these events, in other words, multiplication means that under certain circumstances there will come And event , And event . A similar statement is true for a larger number of events, for example, the work implies that under certain conditions, there will be And event , And event , And event , …, And event .

Consider a trial in which two coins are tossed and the following events:

- heads will fall on the 1st coin;
- the 1st coin will land tails;
- the 2nd coin will land heads;
- the 2nd coin will come up tails.

Then:
And on the 2nd) an eagle will fall out;
- the event consists in the fact that on both coins (on the 1st And on the 2nd) tails will fall out;
– the event is that the 1st coin will land heads And on the 2nd coin tails;
- the event is that the 1st coin will come up tails And on the 2nd coin an eagle.

It is easy to see that the events incompatible (since it cannot, for example, fall out 2 heads and 2 tails at the same time) and form full group (since taken into account all possible outcomes of tossing two coins). Let's summarize these events: . How to interpret this entry? Very simple - multiplication means logical connection AND, and the addition is OR. Thus, the sum is easy to read in understandable human language: “two eagles will fall or two tails or heads on the 1st coin And on the 2nd tail or heads on the 1st coin And eagle on the 2nd coin »

This was an example when in one test several objects are involved, in this case two coins. Another scheme commonly used in practice is repeated tests when, for example, the same dice is thrown 3 times in a row. As a demonstration, consider the following events:

- in the 1st throw, 4 points will fall out;
- in the 2nd roll, 5 points will fall out;
- in the 3rd throw, 6 points will fall out.

Then the event consists in the fact that in the 1st roll 4 points will fall out And in the 2nd roll will drop 5 points And in the 3rd roll, 6 points will fall. Obviously, in the case of a die, there will be significantly more combinations (outcomes) than if we were tossing a coin.

... I understand that, perhaps, they don’t understand very well interesting examples, but these are things that are often encountered in tasks and cannot be avoided. In addition to a coin, a die and a deck of cards, there are urns with colorful balls, several anonymous people shooting at targets, and a tireless worker who constantly grinds out some details =)

Event Probability

Event Probability is a central concept in probability theory. ...A deadly logical thing, but you had to start somewhere =) There are several approaches to its definition:

;
Geometric definition of probability ;
Statistical definition of probability .

In this article, I will focus on the classical definition of probabilities, which is most widely used in educational tasks.

Notation. The probability of some event is denoted by a capital Latin letter , and the event itself is taken in brackets, acting as a kind of argument. For example:

Also, a small letter is widely used to represent probability. In particular, one can abandon the cumbersome designations of events and their probabilities in favor of the following style:

is the probability that the toss of a coin will result in heads;
- the probability that 5 points will fall out as a result of throwing a dice;
is the probability that a card of the club suit will be drawn from the deck.

This option is popular in solving practical problems, since it allows you to significantly reduce the solution entry. As in the first case, it is convenient to use “talking” subscripts/superscripts here.

Everyone has long guessed about the numbers that I just wrote down above, and now we will find out how they turned out:

The classical definition of probability:

The probability of an event occurring in some test is the ratio , where:

is the total number of all equally possible, elementary outcomes of this test, which form full group of events;

- number elementary outcomes favorable event .

When a coin is tossed, either heads or tails can fall out - these events form full group, thus, the total number of outcomes ; while each of them elementary And equally possible. The event is favored by the outcome (heads). According to the classical definition of probabilities: .

Similarly, as a result of a roll of a die, elementary equally possible outcomes may appear, forming a complete group, and the event is favored by a single outcome (rolling a five). That's why: .THIS IS NOT ACCEPTED TO DO (although it is not forbidden to figure out the percentages in your mind).

It is customary to use fractions of a unit, and, obviously, the probability can vary within . Moreover, if , then the event is impossible, if - authentic, and if , then we are talking about random event.

! If in the course of solving any problem you get some other probability value - look for an error!

In the classical approach to the definition of probability, the extreme values ​​(zero and one) are obtained by exactly the same reasoning. Let 1 ball be drawn at random from an urn containing 10 red balls. Consider the following events:

in a single trial, an unlikely event will not occur.

That is why you will not hit the Jackpot in the lottery if the probability of this event is, say, 0.00000001. Yes, yes, it is you - with the only ticket in a particular circulation. However, more tickets and more draws will not help you much. ... When I tell others about this, I almost always hear in response: "but someone wins." Okay, then let's do the following experiment: please buy any lottery ticket today or tomorrow (don't delay!). And if you win ... well, at least more than 10 kilo rubles, be sure to unsubscribe - I will explain why this happened. For a percentage, of course =) =)

But there is no need to be sad, because there is an opposite principle: if the probability of some event is very close to unity, then in a single test it almost certain will happen. Therefore, before a parachute jump, do not be afraid, on the contrary - smile! After all, absolutely unthinkable and fantastic circumstances must arise for both parachutes to fail.

Although all this is poetry, since, depending on the content of the event, the first principle may turn out to be cheerful, and the second - sad; or even both are parallel.

Probably enough for now, in class Tasks for the classical definition of probability we will squeeze the maximum out of the formula. In the final part of this article, we consider one important theorem:

The sum of the probabilities of events that form a complete group is equal to one. Roughly speaking, if events form a complete group, then with 100% probability one of them will happen. In the simplest case, opposite events form a complete group, for example:

- as a result of a coin toss, an eagle will fall out;
- as a result of tossing a coin, tails will fall out.

According to the theorem:

It is clear that these events are equally likely and their probabilities are the same. .

Because of the equality of probabilities, equally probable events are often called equiprobable . And here is the tongue twister for determining the degree of intoxication turned out =)

Dice example: events are opposite, so .

The theorem under consideration is convenient in that it allows you to quickly find the probability of the opposite event. So, if you know the probability that a five will fall out, it is easy to calculate the probability that it will not fall out:

This is much easier than summing up the probabilities of five elementary outcomes. For elementary outcomes, by the way, this theorem is also valid:
. For example, if is the probability that the shooter will hit the target, then is the probability that he will miss.

! In probability theory, it is undesirable to use the letters and for any other purpose.

In honor of Knowledge Day, I will not ask homework=), but it is very important that you can answer the following questions:

What types of events are there?
– What is chance and equal possibility of an event?
– How do you understand the terms compatibility / incompatibility of events?
– What is a complete group of events, opposite events?
What does the addition and multiplication of events mean?
– What is the essence of the classical definition of probability?
– Why is the addition theorem for the probabilities of events forming a complete group useful?

No, you don’t need to cram anything, these are just the basics of probability theory - a kind of primer that will fit in your head pretty quickly. And so that this happens as soon as possible, I suggest that you read the lessons

INTRODUCTION

Many things are incomprehensible to us, not because our concepts are weak;
but because these things do not enter the circle of our concepts.
Kozma Prutkov

The main goal of studying mathematics in secondary specialized educational institutions is to give students a set of mathematical knowledge and skills necessary for studying other program disciplines that use mathematics to one degree or another, for the ability to perform practical calculations, for the formation and development of logical thinking.

In this paper, all the basic concepts of the section of mathematics "Fundamentals of Probability Theory and Mathematical Statistics", provided for by the program and the State Educational Standards of Secondary Vocational Education (Ministry of Education of the Russian Federation. M., 2002), are consistently introduced, the main theorems are formulated, most of which are not proved . The main tasks and methods for their solution and technologies for applying these methods to solving practical problems are considered. The presentation is accompanied by detailed comments and numerous examples.

Methodical instructions can be used for initial acquaintance with the studied material, when taking notes of lectures, for preparing for practical exercises, for consolidating the acquired knowledge, skills and abilities. In addition, the manual will be useful for undergraduate students as a reference tool that allows you to quickly restore in memory what was previously studied.

At the end of the work, examples and tasks are given that students can perform in self-control mode.

Methodical instructions are intended for students of correspondence and daily form learning.

BASIC CONCEPTS

Probability theory studies the objective regularities of mass random events. It is a theoretical basis for mathematical statistics, dealing with the development of methods for collecting, describing and processing the results of observations. Through observations (tests, experiments), i.e. experience in the broad sense of the word, there is a knowledge of the phenomena of the real world.

In his practical activities we often encounter phenomena whose outcome cannot be predicted, the outcome of which depends on chance.

A random phenomenon can be characterized by the ratio of the number of its occurrences to the number of trials, in each of which, under the same conditions of all trials, it could occur or not occur.

Probability theory is a branch of mathematics in which random phenomena (events) are studied and regularities are revealed when they are massively repeated.

Mathematical statistics is a branch of mathematics that has as its subject the study of methods for collecting, systematizing, processing and using statistical data to obtain scientifically sound conclusions and make decisions.

At the same time, statistical data is understood as a set of numbers that represent the quantitative characteristics of the features of the studied objects that are of interest to us. Statistical data are obtained as a result of specially designed experiments and observations.

Statistical data in its essence depend on many random factors, so mathematical statistics is closely related to probability theory, which is its theoretical basis.

I. PROBABILITY. THEOREMS OF ADDITION AND PROBABILITY MULTIPLICATION

1.1. Basic concepts of combinatorics

In the section of mathematics called combinatorics, some problems are solved related to the consideration of sets and the compilation of various combinations of elements of these sets. For example, if we take 10 different numbers 0, 1, 2, 3,:, 9 and make combinations of them, we will get different numbers, for example 143, 431, 5671, 1207, 43, etc.

We see that some of these combinations differ only in the order of the digits (for example, 143 and 431), others in the numbers included in them (for example, 5671 and 1207), and others also differ in the number of digits (for example, 143 and 43).

Thus, the obtained combinations satisfy various conditions.

Depending on the compilation rules, three types of combinations can be distinguished: permutations, placements, combinations.

Let's first get acquainted with the concept factorial.

product of all natural numbers from 1 to n inclusive are called n-factorial and write.

Calculate: a) ; b) ; in) .

Solution. but) .

b) as well as , then you can take it out of brackets

Then we get

in) .

Permutations.

A combination of n elements that differ from each other only in the order of the elements is called a permutation.

Permutations are denoted by the symbol P n , where n is the number of elements in each permutation. ( R- the first letter of the French word permutation- permutation).

The number of permutations can be calculated using the formula

or with factorial:

Let's remember that 0!=1 and 1!=1.

Example 2. In how many ways can six different books be arranged on one shelf?

Solution. The desired number of ways is equal to the number of permutations of 6 elements, i.e.

Accommodations.

Placements from m elements in n in each, such compounds are called that differ from each other either by the elements themselves (at least one), or by the order from the location.

Locations are denoted by the symbol , where m is the number of all available elements, n is the number of elements in each combination. ( BUT- first letter of the French word arrangement, which means "placement, putting in order").

At the same time, it is assumed that nm.

The number of placements can be calculated using the formula

,

those. the number of all possible placements from m elements by n is equal to the product n consecutive integers, of which the greater is m.

We write this formula in factorial form:

Example 3. How many options for the distribution of three vouchers to a sanatorium of various profiles can be made for five applicants?

Solution. The desired number of options is equal to the number of placements of 5 elements by 3 elements, i.e.

.

Combinations.

Combinations are all possible combinations of m elements by n, which differ from each other by at least one element (here m And n- natural numbers, and nm).

Number of combinations from m elements by n are denoted ( FROM- the first letter of the French word combination- combination).

In general, the number of m elements by n equal to the number of placements from m elements by n divided by the number of permutations from n elements:

Using factorial formulas for placement and permutation numbers, we get:

Example 4. In a team of 25 people, you need to allocate four to work in a certain area. In how many ways can this be done?

Solution. Since the order of the chosen four people does not matter, this can be done in ways.

We find by the first formula

.

In addition, when solving problems, the following formulas are used that express the main properties of combinations:

(by definition, and are assumed);

.

1.2. Solving combinatorial problems

Task 1. 16 subjects are studied at the faculty. On Monday, you need to put 3 subjects in the schedule. In how many ways can this be done?

Solution. There are as many ways to schedule three items out of 16 as there are placements of 16 elements of 3 each.

Task 2. Out of 15 objects, 10 objects must be selected. In how many ways can this be done?

Task 3. Four teams participated in the competition. How many options for the distribution of seats between them are possible?

.

Problem 4. In how many ways can a patrol of three soldiers and one officer be formed if there are 80 soldiers and 3 officers?

Solution. Soldier on patrol can be selected

ways, and officers ways. Since any officer can go with each team of soldiers, there are only ways.

Task 5. Find if it is known that .

Since , we get

,

,

By definition of combination it follows that , . That. .

1.3. The concept of a random event. Event types. Event Probability

Any action, phenomenon, observation with several different outcomes, realized under a given set of conditions, will be called test.

The result of this action or observation is called event .

If an event under given conditions can occur or not occur, then it is called random . In the event that an event must certainly occur, it is called authentic , and in the case when it certainly cannot happen, - impossible.

The events are called incompatible if only one of them can appear each time.

The events are called joint if, under the given conditions, the occurrence of one of these events does not exclude the occurrence of the other in the same test.

The events are called opposite , if under the test conditions they, being its only outcomes, are incompatible.

Events are usually denoted by capital letters of the Latin alphabet: A, B, C, D, : .

A complete system of events A 1 , A 2 , A 3 , : , A n is a set of incompatible events, the occurrence of at least one of which is mandatory for a given test.

If a complete system consists of two incompatible events, then such events are called opposite and are denoted by A and .

Example. There are 30 numbered balls in a box. Determine which of the following events are impossible, certain, opposite:

got a numbered ball (BUT);

draw an even numbered ball (IN);

drawn a ball with an odd number (FROM);

got a ball without a number (D).

Which of them form a complete group?

Solution . BUT- certain event; D- impossible event;

In and FROM- opposite events.

The complete group of events is BUT And D, V And FROM.

The probability of an event is considered as a measure of the objective possibility of the occurrence of a random event.

1.4. The classical definition of probability

The number, which is an expression of the measure of the objective possibility of the occurrence of an event, is called probability this event and is denoted by the symbol P(A).

Definition. Probability of an event BUT is the ratio of the number of outcomes m that favor the occurrence of a given event BUT, to the number n all outcomes (incompatible, unique and equally possible), i.e. .

Therefore, in order to find the probability of an event, it is necessary, after considering the various outcomes of the test, to calculate all possible incompatible outcomes n, choose the number of outcomes we are interested in m and calculate the ratio m to n.

The following properties follow from this definition:

The probability of any trial is a non-negative number not exceeding one.

Indeed, the number m of the desired events lies within . Dividing both parts into n, we get

2. The probability of a certain event is equal to one, because .

3. The probability of an impossible event is zero because .

Problem 1. There are 200 winners out of 1000 tickets in the lottery. One ticket is drawn at random. What is the probability that this ticket wins?

Solution. The total number of different outcomes is n=1000. The number of outcomes favoring the winning is m=200. According to the formula, we get

.

Task 2. In a batch of 18 parts, there are 4 defective ones. 5 pieces are chosen at random. Find the probability that two out of these 5 parts are defective.

Solution. Number of all equally possible independent outcomes n is equal to the number of combinations from 18 to 5 i.e.

Let's calculate the number m that favor event A. Among the 5 randomly selected parts, there should be 3 high-quality and 2 defective ones. The number of ways to select two defective parts from 4 available defective parts is equal to the number of combinations from 4 to 2:

The number of ways to select three quality parts from 14 available quality parts is equal to

.

Any group of quality parts can be combined with any group of defective parts, so the total number of combinations m is

The desired probability of the event A is equal to the ratio of the number of outcomes m that favor this event to the number n of all equally possible independent outcomes:

.

The sum of a finite number of events is an event consisting in the occurrence of at least one of them.

The sum of two events is denoted by the symbol A + B, and the sum n events symbol A 1 +A 2 + : +A n .

The theorem of addition of probabilities.

The probability of the sum of two incompatible events is equal to the sum of the probabilities of these events.

Corollary 1. If the event А 1 , А 2 , : , А n form a complete system, then the sum of the probabilities of these events is equal to one.

Corollary 2. The sum of the probabilities of opposite events and is equal to one.

.

Problem 1. There are 100 lottery tickets. It is known that 5 tickets get a win of 20,000 rubles, 10 - 15,000 rubles, 15 - 10,000 rubles, 25 - 2,000 rubles. and nothing for the rest. Find the probability that the purchased ticket will win at least 10,000 rubles.

Solution. Let A, B, and C be events consisting in the fact that a prize equal to 20,000, 15,000 and 10,000 rubles falls on the purchased ticket. since the events A, B and C are incompatible, then

Task 2. On extramural technical school receives tests in mathematics from the cities A, B And FROM. The probability of receipt of control work from the city BUT equal to 0.6, from the city IN- 0.1. Find the probability that the next test will come from the city FROM.

The simplest example of a connection between two events is a causal relationship, when the occurrence of one of the events necessarily leads to the occurrence of the other, or vice versa, when the occurrence of one excludes the possibility of the occurrence of the other.

To characterize the dependence of some events on others, the concept is introduced conditional probability.

Definition. Let be BUT And IN- two random events of the same test. Then the conditional probability of the event BUT or the probability of event A, provided that event B has occurred, is called the number.

Denoting the conditional probability , we obtain the formula

, .

Task 1. Calculate the probability that a second boy will be born in a family with one boy child.

Solution. Let the event BUT consists in the fact that there are two boys in the family, and the event IN- that one boy.

Consider all possible outcomes: boy and boy; boy and girl; girl and boy; girl and girl.

Then , and by the formula we find

.

Event BUT called independent from the event IN if the occurrence of the event IN has no effect on the probability of an event occurring BUT.

Probability multiplication theorem

The probability of the simultaneous occurrence of two independent events is equal to the product of the probabilities of these events:

The probability of the occurrence of several events that are independent in the aggregate is calculated by the formula

Problem 2. The first urn contains 6 black and 4 white balls, the second urn contains 5 black and 7 white balls. One ball is drawn from each urn. What is the probability that both balls are white.

A and IN there is an event AB. Consequently,

b) If the first element works, then an event occurs (the opposite of the event BUT- the failure of this element); if the second element works - event IN. Find the probabilities of events and :

Then the event consisting in the fact that both elements will work is, and, therefore,

Many, faced with the concept of "probability theory", are frightened, thinking that this is something overwhelming, very complex. But it's really not all that tragic. Today we will consider the basic concept of probability theory, learn how to solve problems using specific examples.

The science

What does such a branch of mathematics as “probability theory” study? She notes patterns and magnitudes. For the first time, scientists became interested in this issue back in the eighteenth century, when they studied gambling. The basic concept of probability theory is an event. It is any fact that is ascertained by experience or observation. But what is experience? Another basic concept of probability theory. It means that this composition of circumstances was not created by chance, but for a specific purpose. As for observation, here the researcher himself does not participate in the experiment, but simply is a witness to these events, he does not influence what is happening in any way.

Developments

We learned that the basic concept of probability theory is an event, but did not consider the classification. All of them fall into the following categories:

• Reliable.
• Impossible.
• Random.

No matter what kind of events are observed or created in the course of experience, they are all subject to this classification. We offer to get acquainted with each of the species separately.

Credible Event

This is a circumstance before which the necessary set of measures has been taken. In order to better understand the essence, it is better to give a few examples. Physics, chemistry, economics, and higher mathematics are subject to this law. Probability theory includes such an important concept as a certain event. Here are some examples:

• We work and receive remuneration in the form of wages.
• We passed the exams well, passed the competition, for this we receive a reward in the form of admission to educational institution.
• We invested money in the bank, if necessary, we will get it back.

Such events are reliable. If we have fulfilled all the necessary conditions, then we will definitely get the expected result.

Impossible events

We now consider elements of probability theory. We propose to move on to an explanation of the next type of event, namely, the impossible. To begin with, we will stipulate the most important rule - the probability of an impossible event is zero.

It is impossible to deviate from this formulation when solving problems. To clarify, here are examples of such events:

• The water froze at a temperature of plus ten (this is impossible).
• The lack of electricity does not affect production in any way (just as impossible as in the previous example).

More examples should not be given, since the ones described above very clearly reflect the essence of this category. The impossible event will never happen during the experience under any circumstances.

random events

When studying the elements, special attention should be paid to this particular type of event. That is what science is studying. As a result of experience, something may or may not happen. In addition, the test can be repeated an unlimited number of times. Prominent examples are:

• Tossing a coin is an experience, or a test, heading is an event.
• Pulling the ball out of the bag blindly is a test, a red ball is caught is an event, and so on.

There can be an unlimited number of such examples, but, in general, the essence should be clear. To summarize and systematize the knowledge gained about events, a table is given. Probability theory studies only the last type of all presented.

title	definition
Credible	Events that occur with a 100% guarantee, subject to certain conditions.	Admission to an educational institution with a good passing of the entrance exam.
Impossible	Events that will never happen under any circumstances.	It is snowing at an air temperature of plus thirty degrees Celsius.
Random	An event that may or may not occur during an experiment/test.	Hit or miss when throwing a basketball into the hoop.

Laws

Probability theory is a science that studies the possibility of an event occurring. Like the others, it has some rules. There are the following laws of probability theory:

• Convergence of sequences of random variables.
• The law of large numbers.

When calculating the possibility of the complex, a complex of simple events can be used to achieve the result in an easier and faster way. Note that the laws of probability theory are easily proved with the help of some theorems. Let's start with the first law.

Convergence of sequences of random variables

Note that there are several types of convergence:

• The sequence of random variables is convergent in probability.
• Almost impossible.
• RMS convergence.
• Distribution Convergence.

So, on the fly, it's very hard to get to the bottom of it. Here are some definitions to help you understand this topic. Let's start with the first look. The sequence is called convergent in probability, if the following condition is met: n tends to infinity, the number to which the sequence tends is greater than zero and close to one.

Let's move on to the next one, almost certainly. The sequence is said to converge almost certainly to a random variable with n tending to infinity, and P tending to a value close to unity.

The next type is RMS convergence. When using SC convergence, the study of vector random processes is reduced to the study of their coordinate random processes.

The last type remains, let's briefly analyze it in order to proceed directly to solving problems. Distribution convergence has another name - “weak”, we will explain why below. Weak convergence is the convergence of distribution functions at all points of continuity of the limiting distribution function.

We will definitely fulfill the promise: weak convergence differs from all of the above in that random value is not defined on the probability space. This is possible because the condition is formed exclusively using distribution functions.

Law of Large Numbers

Excellent assistants in proving this law will be theorems of probability theory, such as:

• Chebyshev's inequality.
• Chebyshev's theorem.
• Generalized Chebyshev's theorem.
• Markov's theorem.

If we consider all these theorems, then this question can drag on for several tens of sheets. Our main task is to apply the theory of probability in practice. We invite you to do this right now. But before that, let's consider the axioms of probability theory, they will be the main assistants in solving problems.

Axioms

We already met the first one when we talked about the impossible event. Let's remember: the probability of an impossible event is zero. We gave a very vivid and memorable example: snow fell at an air temperature of thirty degrees Celsius.

The second is as follows: a certain event occurs with a probability equal to one. Now let's show how to write it down using the mathematical language: P(B)=1.

Third: A random event may or may not occur, but the possibility always ranges from zero to one. The closer the value is to one, the greater the chance; if the value approaches zero, the probability is very low. Let's write it in mathematical language: 0<Р(С)<1.

Consider the last, fourth axiom, which sounds like this: the probability of the sum of two events is equal to the sum of their probabilities. We write in mathematical language: P (A + B) \u003d P (A) + P (B).

The axioms of probability theory are the simplest rules that are easy to remember. Let's try to solve some problems, based on the knowledge already gained.

Lottery ticket

To begin with, consider the simplest example - the lottery. Imagine that you bought one lottery ticket for good luck. What is the probability that you will win at least twenty rubles? In total, a thousand tickets participate in the circulation, one of which has a prize of five hundred rubles, ten of one hundred rubles, fifty of twenty rubles, and one hundred of five. Problems in probability theory are based on finding the possibility of luck. Let's take a look at the solution to the above problem together.

If we denote by the letter A a win of five hundred rubles, then the probability of getting A will be 0.001. How did we get it? You just need to divide the number of "happy" tickets by their total number (in this case: 1/1000).

B is a win of one hundred rubles, the probability will be equal to 0.01. Now we acted on the same principle as in the previous action (10/1000)

C - the winnings are equal to twenty rubles. We find the probability, it is equal to 0.05.

The remaining tickets are of no interest to us, since their prize fund is less than that specified in the condition. Let's apply the fourth axiom: The probability of winning at least twenty rubles is P(A)+P(B)+P(C). The letter P denotes the probability of the occurrence of this event, we have already found them in the previous steps. It remains only to add the necessary data, in the answer we get 0.061. This number will be the answer to the question of the assignment.

card deck

Problems in the theory of probability are also more complex, for example, take the following task. Before you is a deck of thirty-six cards. Your task is to draw two cards in a row without mixing the pile, the first and second cards must be aces, the suit does not matter.

To begin with, we find the probability that the first card will be an ace, for this we divide four by thirty-six. They put it aside. We take out the second card, it will be an ace with a probability of three thirty-fifths. The probability of the second event depends on which card we drew first, we are interested in whether it was an ace or not. It follows that event B depends on event A.

The next step is to find the probability of simultaneous implementation, that is, we multiply A and B. Their product is found as follows: we multiply the probability of one event by the conditional probability of another, which we calculate, assuming that the first event happened, that is, we drew an ace with the first card.

In order to make everything clear, let's give a designation to such an element as events. It is calculated assuming that event A has occurred. Calculated as follows: P(B/A).

Let's continue the solution of our problem: P (A * B) \u003d P (A) * P (B / A) or P (A * B) \u003d P (B) * P (A / B). The probability is (4/36) * ((3/35)/(4/36). Calculate by rounding to hundredths. We have: 0.11 * (0.09/0.11)=0.11 * 0, 82 = 0.09 The probability that we will draw two aces in a row is nine hundredths.The value is very small, it follows that the probability of the occurrence of the event is extremely small.

Forgotten number

We propose to analyze a few more options for tasks that are studied by probability theory. You have already seen examples of solving some of them in this article, let's try to solve the following problem: the boy forgot the last digit of his friend's phone number, but since the call was very important, he began to dial everything in turn. We need to calculate the probability that he will call no more than three times. The solution of the problem is the simplest if the rules, laws and axioms of probability theory are known.

Before looking at the solution, try to solve it yourself. We know that the last digit can be from zero to nine, that is, there are ten values ​​in total. The probability of getting the right one is 1/10.

Next, we need to consider options for the origin of the event, suppose that the boy guessed right and immediately scored the right one, the probability of such an event is 1/10. The second option: the first call is a miss, and the second is on target. We calculate the probability of such an event: multiply 9/10 by 1/9, as a result we also get 1/10. The third option: the first and second calls turned out to be at the wrong address, only from the third the boy got where he wanted. We calculate the probability of such an event: we multiply 9/10 by 8/9 and by 1/8, we get 1/10 as a result. According to the condition of the problem, we are not interested in other options, so it remains for us to add up the results, as a result we have 3/10. Answer: The probability that the boy calls no more than three times is 0.3.

Cards with numbers

There are nine cards in front of you, each of which contains a number from one to nine, the numbers are not repeated. They were placed in a box and mixed thoroughly. You need to calculate the probability that

• an even number will come up;
• two-digit.

Before moving on to the solution, let's stipulate that m is the number of successful cases, and n is the total number of options. Find the probability that the number is even. It will not be difficult to calculate that there are four even numbers, this will be our m, there are nine options in total, that is, m = 9. Then the probability is 0.44 or 4/9.

We consider the second case: the number of options is nine, and there can be no successful outcomes at all, that is, m equals zero. The probability that the drawn card will contain a two-digit number is also zero.

The classical definition of probability is based on the concept probabilistic experience, or probabilistic experiment. Its result is one of several possible outcomes, called elementary outcomes, and there is no reason to expect that any elementary outcome will appear more often than others when repeating a probabilistic experiment. For example, consider a probabilistic experiment on throwing a dice (dice). The result of this experience is the loss of one of the 6 points drawn on the faces of the die.

Thus, in this experiment there are 6 elementary outcomes:

and each of them is equally expected.

event in a classical probabilistic experiment is an arbitrary subset of the set of elementary outcomes. In the considered example of throwing a dice, the event is, for example, the loss of an even number of points, which consists of elementary outcomes.

The probability of an event is a number:

where is the number of elementary outcomes that make up the event (sometimes they say that this is the number of elementary outcomes that favor the appearance of the event), and is the number of all elementary outcomes.

In our example:

Elements of combinatorics.

When describing many probabilistic experiments, elementary outcomes can be identified with one of the following objects of combinatorics (the science of finite sets).

permutation from numbers is called an arbitrary ordered record of these numbers without repetitions. For example, for a set of three numbers, there are 6 different permutations:

, , , , , .

For an arbitrary number of permutations is

(the product of consecutive numbers of the natural series, starting from 1).

A combination of is an arbitrary unordered set of any elements of the set . For example, for a set of three numbers, there are 3 different combinations of 3 to 2:

For an arbitrary pair , , the number of combinations of by is

For example,

Hypergeometric distribution.

Consider the following probabilistic experiment. There is a black box containing white and black balls. The balls are the same size and indistinguishable by touch. The experiment is that we randomly pull out the balls. The event , the probability of which is to be found, is that of these balls are white and the rest are black.

Renumber all the balls with numbers from 1 to . Let the numbers 1, ¼, correspond to white balls, and the numbers , ¼, to black balls. The elementary outcome in this experiment is an unordered set of elements from the set , that is, a combination of by . Therefore, there are all elementary outcomes.

Let us find the number of elementary outcomes favoring the appearance of the event . The corresponding sets consist of "white" and "black" numbers. You can choose numbers from “white” numbers in ways, and numbers from “black” numbers in ¾ ways. White and black sets can be connected arbitrarily, so there are only elementary outcomes that favor the event.

The probability of an event is

The resulting formula is called the hypergeometric distribution.

Problem 5.1. The box contains 55 standard and 6 defective parts of the same type. What is the probability that among three randomly selected parts there will be at least one defective?

Solution. There are 61 parts in total, we take 3. An elementary outcome is a combination of 61 by 3. The number of all elementary outcomes is . Favorable outcomes are divided into three groups: 1) these are those outcomes in which 1 part is defective, and 2 are good; 2) 2 parts are defective, and 1 is good; 3) all 3 parts are defective. The number of sets of the first kind is equal to , the number of sets of the second kind is equal to , the number of sets of the third kind is equal to . Therefore, the occurrence of an event is favored by elementary outcomes. The probability of an event is

Algebra of events

Space of elementary events is the set of all elementary outcomes related to a given experience.

sum of two events is called an event, which consists of elementary outcomes belonging to the event or event .

work two events is called an event consisting of elementary outcomes belonging simultaneously to the events and .

Events and are called incompatible if .

The event is called opposite event , if the event is favored by all those elementary outcomes that do not belong to the event . In particular, , .

THEOREM about the sum.

In particular, .

Conditional Probability event, provided that the event occurred, is called the ratio of the number of elementary outcomes belonging to the intersection to the number of elementary outcomes belonging to . In other words, the conditional probability of an event is determined by the classical probability formula, in which the new probability space is . The conditional probability of an event is denoted by .

THEOREM about the product. .

The events are called independent, if . For independent events, the product theorem gives the relation .

A consequence of the sum and product theorems is the following two formulas.

Total Probability Formula. A complete group of hypotheses is an arbitrary set of incompatible events , , ¼, , in the sum of the components of the entire probability space:

In this situation, for an arbitrary event, a formula is valid, called the total probability formula,

where is the Laplace function , , . The Laplace function is tabulated, and its values ​​for a given value can be found in any textbook on probability theory and mathematical statistics.

Problem 5.3. It is known that in a large batch of parts there are 11% of defective ones. 100 parts are selected for verification. What is the probability that there are at most 14 defective ones among them? Evaluate the answer using the Moivre-Laplace theorem.

Solution. We are dealing with the Bernoulli test, where , , . Finding a defective part is considered a success, and the number of successes satisfies the inequality . Consequently,

Direct counting gives:

, , , , , , , , , , , , , , .

Consequently, . Now we apply the Moivre-Laplace integral theorem. We get:

Using the table of function values, taking into account the oddness of the function, we obtain

The approximate calculation error does not exceed .

random variables

A random variable is a numerical characteristic of a probabilistic experience, which is a function of elementary outcomes. If , , ¼, is a set of elementary outcomes, then the random variable is a function of . It is more convenient, however, to characterize the random variable by listing all its possible values ​​and the probabilities with which it takes this value.

Such a table is called the law of distribution of a random variable. Since the events form a complete group, the probabilistic normalization law holds

The mathematical expectation, or average value, of a random variable is a number equal to the sum of the products of the values ​​of the random variable by the corresponding probabilities.

The variance (the degree of spread of values ​​around the mathematical expectation) of a random variable is the mathematical expectation of a random variable,

It can be shown that

Value

is called the mean square deviation of the random variable.

The distribution function for a random variable is the probability to fall on the set , that is

It is a non-negative, non-decreasing function that takes values ​​from 0 to 1. For a random variable that has a finite set of values, it is a piecewise constant function with discontinuities of the second kind at the state points. Moreover, is continuous on the left and .

Problem 5.4. Two dice are thrown consecutively. If one, three or five points fall out on one dice, the player loses 5 rubles. If two or four points fall out, the player receives 7 rubles. If six points fall out, the player loses 12 rubles. Random value x is the player's payoff for two throws of the dice. Find the distribution law x, plot the distribution function, find the mathematical expectation and variance x.

Solution. Let us first consider what the player's payoff is when one roll of the die is equal. Let the event be that 1, 3 or 5 points fell out. Then , and the winnings will be Rs. Let the event be that 2 or 4 points fell out. Then , and the winnings will be Rs. Finally, let the event mean a roll of 6 points. Then the payoff is equal to Rs.

Now consider all possible combinations of events , and for two throws of the die, and determine the payoff values ​​for each such combination.

If an event occurs, then , at the same time .

If an event occurs, then , at the same time .

Similarly, for , we obtain , .

All found states and the total probabilities of these states are written in the table:

We check the fulfillment of the law of probabilistic normalization: on the real line, you need to be able to determine the probability of a random variable falling into this interval 1) and rapidly decreasing at, ¼,

Math for Programmers: Probability Theory

Ivan Kamyshan

Some programmers, after working in the development of conventional commercial applications, are thinking about mastering machine learning and becoming a data analyst. Often they do not understand why certain methods work, and most machine learning methods seem like magic. In fact, machine learning is based on mathematical statistics, and that, in turn, is based on probability theory. Therefore, in this article we will pay attention to the basic concepts of probability theory: we will touch on the definitions of probability, distribution, and analyze a few simple examples.

You may know that probability theory is conditionally divided into 2 parts. Discrete probability theory studies phenomena that can be described by a distribution with a finite (or countable) number of possible behaviors (throws of dice, coins). Continuous probability theory studies phenomena distributed on some dense set, for example, on a segment or in a circle.

It is possible to consider the subject of probability theory with a simple example. Imagine yourself as a shooter developer. An integral part of the development of games in this genre is the mechanics of shooting. It is clear that a shooter in which all weapons shoot absolutely accurately will be of little interest to players. Therefore, it is necessary to add spread to the weapon. But simply randomizing weapon hitpoints won't allow for fine-tuning, so adjusting the game balance will be difficult. At the same time, using random variables and their distributions, you can analyze how the weapon will work with a given spread, and help make the necessary adjustments.

Space of elementary outcomes

Suppose, from some random experiment that we can repeat many times (for example, tossing a coin), we can extract some formalizable information (heads or tails). This information is called an elementary outcome, and it is advisable to consider the set of all elementary outcomes, often denoted by the letter Ω (Omega).

The structure of this space depends entirely on the nature of the experiment. For example, if we consider shooting at a sufficiently large circular target, the space of elementary outcomes will be a circle, for convenience, placed with the center at zero, and the outcome will be a point in this circle.

In addition, they consider sets of elementary outcomes - events (for example, hitting the "top ten" is a concentric circle of small radius with a target). In the discrete case, everything is quite simple: we can get any event, including or excluding elementary outcomes in a finite time. In the continuous case, however, everything is much more complicated: we need some good enough family of sets to consider, called an algebra, by analogy with simple real numbers that can be added, subtracted, divided and multiplied. Sets in an algebra can be intersected and combined, and the result of the operation will be in the algebra. This is a very important property for the mathematics behind all these concepts. The minimal family consists of only two sets - the empty set and the space of elementary outcomes.

Measure and Probability

Probability is a way of making inferences about the behavior of very complex objects without understanding how they work. Thus, the probability is defined as a function of an event (from that very good family of sets), which returns a number - some characteristic of how often such an event can occur in reality. For definiteness, mathematicians agreed that this number should lie between zero and one. In addition, requirements are imposed on this function: the probability of an impossible event is zero, the probability of the entire set of outcomes is unity, and the probability of combining two independent events (disjoint sets) is equal to the sum of the probabilities. Another name for probability is a probability measure. The most commonly used Lebesgue measure, which generalizes the concepts of length, area, volume to any dimensions (n-dimensional volume), and thus it is applicable to a wide class of sets.

Together, the set of a set of elementary outcomes, a family of sets, and a probability measure is called probability space. Let's look at how we can construct a probability space for the target shooting example.

Consider shooting at a large round target of radius R that cannot be missed. As a set of elementary events, we put a circle centered at the origin of coordinates of radius R . Since we are going to use the area (the Lebesgue measure for two-dimensional sets) to describe the probability of an event, we will use the family of measurable (for which this measure exists) sets.

Note Actually, this is a technical point and in simple problems the process of determining the measure and the family of sets does not play a special role. But it is necessary to understand that these two objects exist, because in many books on probability theory, theorems begin with the words: “ Let (Ω,Σ,P) be a probability space…».

As mentioned above, the probability of the entire space of elementary outcomes must be equal to one. The area (the two-dimensional Lebesgue measure, which we will denote by λ 2 (A), where A is the event) of the circle, according to the well-known formula from school, is π * R 2. Then we can introduce the probability P(A) = λ 2 (A) / (π *R 2) , and this value will already lie between 0 and 1 for any event A.

If we assume that hitting any point of the target is equally probable, the search for the probability of hitting by the shooter in some area of ​​the target is reduced to finding the area of ​​this set (hence we can conclude that the probability of hitting a specific point is zero, because the area of ​​the point is zero).

For example, we want to know what is the probability that the shooter will hit the "ten" (event A - the shooter hit the right set). In our model, "ten" is represented by a circle centered at zero and with radius r. Then the probability of falling into this circle is P(A) = λ 2 /(A)π *R 2 = π * r 2 /(π R 2)= (r/R) 2 .

This is one of the simplest varieties of "geometric probability" problems - most of these problems require finding an area.

random variables

A random variable is a function that converts elementary outcomes into real numbers. For example, in the considered problem, we can introduce a random variable ρ(ω) - the distance from the point of impact to the center of the target. The simplicity of our model allows us to explicitly specify the space of elementary outcomes: Ω = (ω = (x,y) numbers such that x 2 +y 2 ≤ R 2 ) . Then the random variable ρ(ω) = ρ(x,y) = x 2 +y 2 .

Means of abstraction from the probability space. Distribution function and density

It is good when the structure of space is well known, but in reality this is not always the case. Even if the structure of space is known, it can be complex. To describe random variables, if their expression is unknown, there is the concept of a distribution function, which is denoted by F ξ (x) = P(ξ< x) (нижний индекс ξ здесь означает случайную величину). Т.е. это вероятность множества всех таких элементарных исходов, для которых значение случайной величины ξ на этом событии меньше, чем заданный параметр x .

The distribution function has several properties:

1. First, it is between 0 and 1 .
2. Second, it does not decrease when its argument x increases.
3. Third, when the number -x is very large, the distribution function is close to 0, and when x itself is large, the distribution function is close to 1.

Probably, the meaning of this construction is not very clear on the first reading. One of the useful properties - the distribution function allows you to look for the probability that the value takes a value from the interval. So, P (random variable ξ takes values ​​from the interval ) = F ξ (b)-F ξ (a) . Based on this equality, we can investigate how this value changes if the boundaries a and b of the interval are close.

Let d = b-a , then b = a+d . And therefore, F ξ (b)-F ξ (a) = F ξ (a+d) - F ξ (a) . For small values ​​of d, the above difference is also small (if the distribution is continuous). It makes sense to consider the relation p ξ (a,d)= (F ξ (a+d) - F ξ (a))/d . If for sufficiently small values ​​of d this ratio differs little from some constant p ξ (a) , independent of d, then at this point the random variable has a density equal to p ξ (a) .

Note Readers who have previously encountered the concept of a derivative may notice that p ξ (a) is the derivative of the function F ξ (x) at the point a . In any case, you can study the concept of a derivative in an article dedicated to this topic on the Mathprofi website.

Now the meaning of the distribution function can be defined as follows: its derivative (density p ξ , which we defined above) at point a describes how often a random variable will fall into a small interval centered at point a (neighborhood of point a) compared to neighborhoods of other points . In other words, the faster the distribution function grows, the more likely it is that such a value will appear in a random experiment.

Let's go back to the example. We can calculate the distribution function for a random variable, ρ(ω) = ρ(x,y) = x 2 +y 2 , which denotes the distance from the center to the point of a random hit on the target. By definition, F ρ (t) = P(ρ(x,y)< t) . т.е. множество {ρ(x,y) < t)} – состоит из таких точек (x,y) , расстояние от которых до нуля меньше, чем t . Мы уже считали вероятность такого события, когда вычисляли вероятность попадания в «десятку» - она равна t 2 /R 2 . Таким образом, Fρ(t) = P(ρ(x,y) < t) = t 2 /R 2 , для 0

We can find the density p ρ of this random variable. We note right away that it is zero outside the interval, since the distribution function on this interval is unchanged. At the ends of this interval, the density is not determined. Inside the interval, it can be found using a table of derivatives (for example, from the Mathprofi website) and elementary differentiation rules. The derivative of t 2 /R 2 is 2t/R 2 . This means that we found the density on the entire axis of real numbers.

Another useful property of density is the probability that a function takes a value from an interval is calculated using the integral of the density over this interval (you can find out what it is in the articles on proper, improper, indefinite integrals on the Mathprofi website).

On the first reading, the span integral of the function f(x) can be thought of as the area of ​​a curvilinear trapezoid. Its sides are a fragment of the Ox axis, a gap (of the horizontal coordinate axis), vertical segments connecting the points (a,f(a)), (b,f(b)) on a curve with points (a,0), (b,0 ) on the x-axis. The last side is a fragment of the graph of the function f from (a,f(a)) to (b,f(b)) . We can talk about the integral over the interval (-∞; b] when, for sufficiently large negative values, a, the value of the integral over the interval will change negligibly small compared to the change in the number a. The integral over the intervals is defined in a similar way)