The basic concept of the theory of probability. The laws of the theory of probability. Probability theory and basic concepts of the theory Theory of mathematical probability

The doctrine of the laws, which are subject to the so-called. random phenomena. Dictionary of foreign words included in the Russian language. Chudinov A.N., 1910 ... Dictionary of foreign words of the Russian language

probability theory- - [L.G. Sumenko. The English Russian Dictionary of Information Technology. M .: GP TsNIIS, 2003.] Subjects Information Technology overall EN probability theorytheory of chancesprobability calculation ... Technical translator's guide

Probability theory- there is a part of mathematics that studies the relationship between probabilities (see Probability and Statistics) of various events. We list the most important theorems related to this science. The probability of one of several inconsistent events occurring equals ... ... encyclopedic Dictionary F. Brockhaus and I.A. Efron

THEORY OF PROBABILITIES- mathematical. a science that allows for the probabilities of some random events (see) to find the probabilities of random events associated with c. l. way with the first. Modern TV based on the axiomatics (see. Method axiomatic) A. N. Kolmogorov. On the… … Russian Sociological Encyclopedia

Probability theory- a branch of mathematics, in which, according to the given probabilities of some random events, the probabilities of other events, related in some way to the first, are found. Probability theory also studies random variables and random processes. One of the main ... ... Concepts of modern natural science. Glossary of basic terms

probability theory- tikimybių teorija statusas T sritis fizika atitikmenys: angl. probability theory vok. Wahrscheinlichkeitstheorie, f rus. probability theory, f pranc. théorie des probabilités, f ... Fizikos terminų žodynas

Probability theory- ... Wikipedia

Probability theory- a mathematical discipline that studies the laws of random phenomena ... The beginnings of modern natural science

THEORY OF PROBABILITIES- (probability theory) see Probability ... Comprehensive explanatory sociological dictionary

Probability theory and its applications- ("Theory of probabilities and its applications",) scientific journal of the Department of Mathematics of the Academy of Sciences of the USSR. Publishes original articles and short messages on probability theory, general issues mathematical statistics and their applications in natural science and ... ... Big Soviet encyclopedia

Books

Probability theory. , Wentzel E.S .. The book is a textbook intended for persons familiar with mathematics in the scope of a regular university course and interested in technical applications of the theory of probabilities, in ... Buy for 2056 UAH (only Ukraine)
Probability theory. , Wentzel E.S .. The book is a textbook intended for persons familiar with mathematics within the scope of a regular university course and interested in technical applications of probability theory, in ...

What is probability?

Faced with this term for the first time, I would not understand what it is. Therefore, I will try to explain it in an accessible way.

Probability is the chance that the event we need will occur.

For example, you decided to visit a friend, remember the entrance and even the floor on which he lives. But I forgot the number and location of the apartment. And here you are standing on the staircase, and in front of you are the doors to choose from.

What is the chance (probability) that if you ring the first door, your friend will open for you? The whole apartment, and the friend lives only for one of them. We can choose any door with an equal chance.

But what is this chance?

Doors, the right door. Probability of guessing by ringing the first door:. That is, one time out of three you will guess for sure.

We want to know by calling once, how often will we guess the door? Let's consider all the options:

You called in 1st Door
You called in 2nd Door
You called in 3rd Door

Now let's look at all the options where a friend may be:

a. Per 1st by the door
b. Per 2nd by the door
v. Per 3rd by the door

Let's compare all the options in the form of a table. A tick marks the options when your choice coincides with the location of a friend, a cross - when it does not match.

How do you see everything maybe options the friend's location and your choice of which door to ring.

A favorable outcomes of all . That is, you will guess from time to time by ringing the doorbell. ...

This is probability - the ratio of a favorable outcome (when your choice coincided with the location of a friend) to the number of possible events.

Definition is a formula. The probability is usually denoted p, therefore:

It is not very convenient to write such a formula, therefore we will take for - the number of favorable outcomes, and for - the total number of outcomes.

The probability can be written as a percentage, for this you need to multiply the resulting result by:

Probably the word "outcomes" caught your eye. Since mathematicians call various actions (in our case, such an action is a doorbell ringing) experiments, it is customary to call the outcome of such experiments.

Well, the outcomes are favorable and unfavorable.

Let's go back to our example. Let's say we rang one of the doors, but a stranger opened it to us. We didn't guess. What is the likelihood that if we ring one of the remaining doors, our friend will open for us?

If you thought that, then this is a mistake. Let's figure it out.

We have two doors left. Thus, we have possible steps:

1) Call in 1st Door
2) Call in 2nd Door

A friend, with all this, is definitely behind one of them (after all, he was not behind the one we called):

a) Friend for 1st by the door
b) Friend for 2nd by the door

Let's draw the table again:

As you can see, there are all options, of which are favorable. That is, the probability is equal.

Why not?

The situation we have considered - example of dependent events. The first event is the first doorbell, the second event is the second doorbell.

And they are called dependent because they affect the following actions. After all, if a friend opened the door after the first ring, then what would be the probability that he is behind one of the other two? Right, .

But if there are dependent events, then there must be independent? True, there are.

A textbook example is tossing a coin.

Throw a coin once. What is the probability that, for example, heads will come out? That's right - because the options for everything (either heads or tails, we neglect the probability of a coin to stand on an edge), but only suits us.
But it came up tails. Okay, let's throw it one more time. What is the current probability of getting heads? Nothing has changed, everything is the same. How many options? Two. How much does it suit us? One.

And let it come up tails a thousand times in a row. The probability of getting heads at one time will be the same. There are always options, but favorable ones.

It is easy to distinguish dependent events from independent ones:

If the experiment is carried out once (once they throw a coin, ring the doorbell once, etc.), then the events are always independent.
If the experiment is carried out several times (the coin is thrown once, the doorbell rings several times), then the first event is always independent. And then, if the number of favorable or the number of all outcomes changes, then the events are dependent, and if not, they are independent.

Let's practice determining the probability a little.

Example 1.

The coin is thrown twice. What is the probability of hitting heads twice in a row?

Solution:

Let's consider all possible options:

Eagle-eagle
Heads-tails
Heads-tails
Tails-tails

As you can see, the whole option. Of these, only suits us. That is, the probability:

If the condition is asked to simply find the probability, then the answer must be given in the form of a decimal fraction. If it were indicated that the answer should be given as a percentage, then we would multiply by.

Answer:

Example 2.

In a box of chocolates, all chocolates are packed in the same wrapper. However, from sweets - with nuts, cognac, cherries, caramel and nougat.

What is the probability, taking one candy, to get a candy with nuts. Give your answer as a percentage.

Solution:

How many possible outcomes are there? ...

That is, taking one candy, it will be one of the ones in the box.

How many favorable outcomes?

Because the box contains only chocolates with nuts.

Answer:

Example 3.

In a box of balls. of them white, - black.

What is the probability of pulling out the white ball?
We've added more black balls to the box. What is now the probability of pulling out the white ball?

Solution:

a) There are all balls in the box. Of these, white.

The probability is equal to:

b) Now there are balls in the box. And the same number of whites remained -.

Answer:

Full probability

The probability of all possible events is ().

Let's say in a box of red and green balls. What is the probability of pulling out the red ball? Green ball? Red or green ball?

Possibility of pulling a red ball

Green ball:

Red or green ball:

As you can see, the sum of all possible events is (). Understanding this moment will help you solve many problems.

Example 4.

The box contains markers: green, red, blue, yellow, black.

What is the chance of pulling out a NOT red felt-tip pen?

Solution:

Let's count the amount favorable outcomes.

NOT a red marker, it means green, blue, yellow or black.

The probability of all events. And the probability of events that we consider unfavorable (when we pull out the red felt-tip pen) -.

Thus, the probability of pulling out a NOT red felt-tip pen is.

Answer:

The probability that the event will not occur is equal to minus the probability that the event will occur.

The rule for multiplying the probabilities of independent events

You already know what independent events are.

But what if you need to find the probability that two (or more) independent events will occur in a row?

Let's say we want to know what is the probability that when we flip a coin once, we will see an eagle twice?

We have already counted -.

And if we flip a coin once? What is the probability of seeing an eagle in a row?

All possible options:

Eagle-eagle-eagle
Heads-heads-tails
Heads-tails-heads
Heads-tails-tails
Tails-heads-heads
Tails-heads-tails
Tails-tails-heads
Tails-Tails-Tails

I don’t know about you, but I made a mistake once when making this list. Wow! And only option (first) suits us.

For 5 throws, you can make a list of possible outcomes yourself. But mathematicians are not as hardworking as you are.

Therefore, they first noticed and then proved that the probability of a certain sequence of independent events decreases each time by the probability of one event.

In other words,

Consider the example of the same unfortunate coin.

The likelihood of getting heads in a challenge? ... Now we flip a coin once.

What is the probability of hitting heads once in a row?

This rule works not only if we are asked to find the probability that the same event will occur several times in a row.

If we wanted to find the GRIP-EAGLE-GRILLE sequence for throws in a row, we would do the same.

The probability of getting tails -, heads -.

Probability of falling out of the sequence GRILLE-EAGLE-GRILLE-GRILLE:

You can check it yourself by making a table.

The rule for adding the probabilities of inconsistent events.

So stop! New definition.

Let's figure it out. Take our worn out coin and toss it once.
Possible options:

Eagle-eagle-eagle
Heads-heads-tails
Heads-tails-heads
Heads-tails-tails
Tails-heads-heads
Tails-heads-tails
Tails-tails-heads
Tails-Tails-Tails

So, incompatible events are a definite, predetermined sequence of events. are incompatible events.

If we want to determine what the probability of two (or more) incompatible events is, then we add the probabilities of these events.

You need to understand that falling heads or tails are two independent events.

If we want to determine what is the probability of a sequence) (or any other), then we use the rule of multiplication of probabilities.
What is the probability of getting heads on the first throw, and on the second and third tails?

But if we want to know what is the probability of getting one of several sequences, for example, when heads fall out exactly once, i.e. options and, then we have to add the probabilities of these sequences.

All the options are suitable for us.

We can get the same thing by adding the probabilities of each sequence:

Thus, we add probabilities when we want to determine the probabilities of some inconsistent sequences of events.

There is a great rule of thumb to help you avoid confusion when to multiply and when to add:

Let's go back to the example when we flipped a coin once, and we want to know the probability of seeing heads once.
What is going to happen?

Should drop:
(heads AND tails AND tails) OR (tails AND heads AND tails) OR (tails AND tails AND heads).
So it turns out:

Let's look at a few examples.

Example 5.

The box contains pencils. reds, greens, oranges and yellows and blacks. What is the likelihood of pulling out red or green pencils?

Solution:

What is going to happen? We have to pull out (red OR green).

Now it is clear, we add the probabilities of these events:

Answer:

Example 6.

The dice are rolled twice, what is the chance of a total of 8 points?

Solution.

How can we get points?

(and) or (and) or (and) or (and) or (and).

The probability of falling out of one (any) face -.

We calculate the probability:

Answer:

Workout.

I think now it became clear to you when to count the probabilities, when to add them, and when to multiply them. Is not it? Let's practice a little.

Tasks:

Let's take a card deck, in which cards, including spades, hearts, 13 clubs and 13 diamonds. From to ace of each suit.

What is the probability of drawing clubs in a row (we put the first drawn card back into the deck and shuffle it)?
What is the probability of drawing a black card (spades or clubs)?
What is the probability of pulling a picture (jack, queen, king or ace)?
What is the probability of drawing two pictures in a row (we remove the first drawn card from the deck)?
What is the probability, having taken two cards, to collect a combination - (jack, queen or king) and an ace The sequence in which the cards will be drawn does not matter.

Answers:

In the deck, cards of each rank means:
Events are dependent, since after the first card is drawn, the number of cards in the deck has decreased (as well as the number of "pictures"). Total jacks, queens, kings and aces in the deck initially, which means the probability of the first card to pull out the "picture":
Since we are removing the first card from the deck, it means that there is already a card in the deck, of which there are pictures. The probability of pulling a picture with the second card:
Since we are interested in the situation when we get from the deck: "picture" AND "picture", then we need to multiply the probabilities:
Answer:
After the first card is drawn, the number of cards in the deck will decrease, so we have two options:
1) With the first card we take out the Ace, the second - the jack, queen or king
2) With the first card we take out a jack, queen or king, the second - an ace. (ace and (jack or queen or king)) or ((jack or queen or king) and ace). Don't forget about reducing the number of cards in the deck!

If you were able to solve all the problems yourself, then you are a great fellow! Now you will be clicking on problems on the theory of probability in the exam!

THEORY OF PROBABILITIES. AVERAGE LEVEL

Let's look at an example. Let's say we roll a die. What kind of bone is this, you know? This is the name of a cube with numbers on the edges. How many faces, so many numbers: from to how many? Before.

So, we roll the die and want to roll or. And it falls to us.

Probability says what happened favorable event(not to be confused with the prosperous).

If it fell, the event would also be favorable. In total, only two favorable events can occur.

And how many are unfavorable? Since there are all possible events, it means that unfavorable events are among them (this is if it falls out or).

Definition:

Probability is the ratio of the number of favorable events to the number of all possible events... That is, the probability shows what proportion of all possible events are favorable.

They denote the probability by a Latin letter (apparently, from english word probability - probability).

It is customary to measure the probability as a percentage (see topics and). To do this, the probability value must be multiplied by. In the dice example, the probability.

And as a percentage:.

Examples (decide for yourself):

What is the probability of getting heads when flipping a coin? How likely is it to come up tails?
What is the probability of an even number being rolled on a die? And with which - odd?
In a box of pencils, blue and red pencils. Draw one pencil at random. What is the probability of pulling out a simple one?

Solutions:

How many options are there? Heads and tails are just two. How many of them are favorable? Only one is an eagle. So the probability
It's the same with tails:.
Total options: (how many sides the cube has, so many different options). Favorable ones: (these are all even numbers :).
Probability. With odd, of course, the same thing.
Total: . Favorable:. Probability: .

Full probability

All pencils in the drawer are green. What is the probability of pulling out a red pencil? There is no chance: probability (after all, favorable events -).

Such an event is called impossible.

What is the probability of pulling out a green pencil? There are exactly the same number of favorable events as there are total events (all events are favorable). Hence, the probability is equal to or.

Such an event is called reliable.

If there are green and red pencils in the box, what is the chance of pulling out the green or red? Yet again. Note this thing: the probability of pulling green is equal, and red is.

In sum, these probabilities are exactly equal. That is, the sum of the probabilities of all possible events is equal to or.

Example:

In a box of pencils, among them blue, red, green, plain, yellow, and the rest are orange. What is the probability of not pulling green?

Solution:

Remember that all probabilities add up. And the probability of pulling green is equal to. This means that the probability of not pulling green is equal to.

Remember this trick: the probability that the event will not occur is equal to minus the probability that the event will occur.

Independent events and the multiplication rule

You flip a coin once, and you want heads to fall both times. What is the likelihood of this happening?

Let's go over all the possible options and determine how many there are:

Heads-Heads, Heads-Heads, Heads-Heads, Heads-Heads. What else?

The whole option. Of these, only one is suitable for us: Eagle-Eagle. Total, the probability is.

Okay. And now we throw a coin once. Count it yourself. Happened? (answer).

You may have noticed that with the addition of each next throw, the probability decreases in times. General rule called multiplication rule:

The probabilities of independent events change.

What are independent events? Everything is logical: these are those that do not depend on each other. For example, when we toss a coin several times, each time a new toss is made, the result of which does not depend on all previous tosses. We can just as well flip two different coins at the same time.

More examples:

The dice are rolled twice. What is the probability that both times will be rolled?
The coin is thrown once. What is the likelihood that it will land heads first and then tails twice?
The player rolls two dice. What is the probability that the sum of the numbers on them will be equal?

Answers:

The events are independent, which means that the multiplication rule works:.
The probability of an eagle is. The likelihood of tails is also. We multiply:
12 can only be obtained if two -ki are rolled:.

Incompatible events and the addition rule

Incompatible events are called events that complement each other to full likelihood. As the name suggests, they cannot happen at the same time. For example, if we flip a coin, it can come up either heads or tails.

Example.

In a box of pencils, among them blue, red, green, plain, yellow, and the rest are orange. What is the probability of pulling green or red?

Solution .

The probability of pulling out a green pencil is. Red - .

Auspicious events in all: green + red. This means that the probability of pulling out green or red is equal to.

The same probability can be represented as follows:.

This is the addition rule: the probabilities of inconsistent events add up.

Mixed problems

Example.

The coin is thrown twice. What is the likelihood that the result of the throws will be different?

Solution .

This means that if the first hit is heads, the second should be tails, and vice versa. It turns out that there are two pairs of independent events, and these pairs are incompatible with each other. How not to get confused, where to multiply, and where to add.

There is a simple rule of thumb for these situations. Try to describe what is going to happen by connecting the events with AND or OR. For example, in this case:

Should come up (heads and tails) or (tails and heads).

Where there is a conjunction "and", there will be multiplication, and where "or" - addition:

Try it yourself:

What is the likelihood that the same side will land on two tosses of a coin both times?
The dice are rolled twice. What is the probability that the total will be points?

Solutions:

(Heads fell and heads fell) or (tails fell and tails fell):.
What are the options? and. Then:
Dropped out (and) or (and) or (and):.

Another example:

We toss a coin once. What is the probability that heads will come out at least once?

Solution:

Oh, how you don’t want to go through the options ... Heads-tails-tails, Heads-heads-tails, ... And don't! We recall the full probability. Remembered? What is the probability that an eagle will not be dropped even once? It's simple: tails are flying all the time, so.

THEORY OF PROBABILITIES. BRIEFLY ABOUT THE MAIN

Probability is the ratio of the number of favorable events to the number of all possible events.

Independent events

Two events are independent if at the occurrence of one the probability of the occurrence of the other does not change.

Full probability

The probability of all possible events is ().

The probability that the event will not occur is equal to minus the probability that the event will occur.

The rule for multiplying the probabilities of independent events

The probability of a certain sequence of independent events is equal to the product of the probabilities of each of the events

Incompatible events

Incompatible events are called events that cannot happen simultaneously as a result of an experiment. A number of inconsistent events form a complete group of events.

The probabilities of inconsistent events add up.

Having described what should happen, using the conjunctions "AND" or "OR", instead of "AND" we put the sign of multiplication, and instead of "OR" - addition.

THE REMAINING 2/3 ARTICLES ARE AVAILABLE ONLY TO YOUCLEVER STUDENTS!

Become a YouClever student,

Prepare for the OGE or USE in mathematics at the price of "a cup of coffee per month",

And also get unlimited access to the "YouClever" textbook, the "100gia" training program (reshebnik), an unlimited trial USE and OGE, 6000 problems with analysis of solutions and to other YouClever and 100gia services.

INTRODUCTION

Many things are incomprehensible to us, not because our concepts are weak;
but because these things are not included in the range of our concepts.
Kozma Prutkov

The main goal of studying mathematics in secondary specialized educational institutions is to provide students with a set of mathematical knowledge and skills necessary to study other program disciplines that use mathematics to some extent, for the ability to perform practical calculations, for the formation and development of logical thinking.

This work consistently introduces all the basic concepts of the section of mathematics "Fundamentals of the theory of probability and mathematical statistics" provided by the program and the State educational standards of secondary vocational education (Ministry of Education of the Russian Federation. M., 2002), formulates the main theorems, most of which are not proven ... 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.

Methodological 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, abilities and skills. In addition, the manual will be useful for senior students as a reference tool, allowing you to quickly recall what was studied earlier.

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

Methodical instructions are intended for students of part-time and full-time forms of education.

BASIC CONCEPTS

Probability theory studies the objective laws of mass random events. It is a theoretical basis for mathematical statistics, engaged in the development of methods for collecting, describing and processing observation results. Through observations (tests, experiments), i.e. experience in the broad sense of the word, cognition of the phenomena of the real world takes place.

In our practice, we often come across phenomena, the outcome of which cannot be predicted, the result of which depends on the case.

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

Probability theory is a branch of mathematics in which random phenomena (events) are studied and patterns are revealed during their massive repetition.

Mathematical statistics is a branch of mathematics that has as its subject of study the methods of collecting, organizing, processing and using statistical data to obtain scientifically based conclusions and decision-making.

In this case, statistical data is understood as a set of numbers that represent the quantitative characteristics of the features of the objects of interest to us. Statistical data are obtained as a result of specially set experiments and observations.

Statistical data inherently depends on many random factors, therefore, mathematical statistics is closely related to the theory of probability, which is its theoretical basis.

I. PROBABILITY. ADDITION AND MULTIPLICATION OF PROBABILITIES

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 the elements of these sets. For example, if we take 10 different digits 0, 1, 2, 3,:, 9 and make combinations from 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 still others differ in the number of digits (for example, 143 and 43).

Thus, the combinations obtained satisfy various conditions.

Three types of combinations can be distinguished depending on the rules of composition: rearrangement, placement, combination.

Let's first get acquainted with the concept factorial.

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

Calculate: a); b); v) .

Solution. a) .

b) Since and , then you can take out the brackets

Then we get

v) .

Permutations.

A combination of n elements that differ from each other only in the order of the elements are called permutations.

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

The number of permutations can be calculated by the formula

or using factorial:

Remember that 0! = 1 and 1! = 1.

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

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

Accommodation.

Accommodations 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 of the arrangement.

Placements are indicated by the symbol, where m- the number of all available elements, n- the number of elements in each combination. ( A- first letter of a French word arrangement, which means "placement, putting in order").

Moreover, it is believed that nm.

The number of placements can be calculated using the formula

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

Let's write this formula in factorial form:

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

Solution. The required number of variants 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 that differ from each other by at least one element (here m and n- natural numbers, and n m).

Number of combinations of m elements by n are denoted ( WITH-first letter of a French word combination- combination).

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

Using factorial formulas for the numbers of placements and permutations, we get:

Example 4. In a team of 25 people, you need to allocate four to work on a specific site. How many ways can this be done?

Solution. Since the order of the selected four people does not matter, there are several ways to do this.

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, it is assumed and);

1.2. Solving combinatorial problems

Task 1. 16 subjects are studied at the faculty. On Monday, you need to schedule 3 items. How many ways can you do this?

Solution. There are as many ways to schedule three items out of 16 as you can make placements from 16 items of 3 each.

Problem 2. From 15 objects it is necessary to select 10 objects. How many ways can this be done?

Problem 3. Four teams took part in the competition. How many options for the distribution of seats between them are possible?

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

Solution. You can choose a soldier on patrol

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

Problem 5. Find, if it is known that.

Since, we get

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

1.3. The concept of a random event. Types of events. 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 the event at given conditions may or may not happen, then it is called random ... In the event that an event must certainly occur, it is called reliable , and in the case when it obviously cannot happen, - impossible.

Events are called inconsistent if only one of them may appear at a time.

Events are called joint if under the given conditions the occurrence of one of these events does not exclude the occurrence of another during the same test.

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

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

The complete system of events А 1, А 2, А 3,:, А n is a set of incompatible events, the onset of at least one of which is obligatory for a given test.

If the complete system consists of two incompatible events, then such events are called opposite and are designated A and.

Example. The box contains 30 numbered balls. Establish which of the following events are impossible, reliable, opposite:

got a numbered ball (A);

got a ball with an even number (V);

got an odd-numbered ball (WITH);

got a ball without a number (D).

Which ones make up a complete group?

Solution ... A- a reliable event; D- an impossible event;

In and WITH- opposite events.

The full group of events consists of A and D, B and WITH.

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

1.4. Classical definition of probability

A number that is an expression of a measure of the objective possibility of an event occurring is called probability this event and is indicated by the symbol P (A).

Definition. Probability of the event A is the ratio of the number of outcomes m, favorable to the onset of a given event A, to the number n all outcomes (inconsistent, unique and equally possible), i.e. ...

Therefore, to find the probability of an event, it is necessary, after considering the various outcomes of the trial, to calculate all possible inconsistent 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 test is a non-negative number not exceeding one.

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

2. The probability of a reliable event is equal to one, since ...

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

Problem 1. In the lottery of 1000 tickets, there are 200 winning. Take out one ticket at random. What is the probability that this ticket is a winner?

Solution. The total number of different outcomes is n= 1000. The number of outcomes favorable to getting a win is m = 200. According to the formula, we get

Problem 2. There are 4 defective parts in a batch of 18 parts. 5 parts are chosen at random. Find the probability that out of these 5 parts, two will turn out to be defective.

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

Let's count the number m, favorable for event A. Among 5 parts taken at random, there should be 3 high-quality and 2 defective. 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 methods for sampling three high-quality parts from 14 available high-quality parts is

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

The sought probability of event A is equal to the ratio of the number of outcomes m, favorable to 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 by the symbol А 1 + А 2 +: + А n.

The addition theorem for 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 is equal to one.

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

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

Problem 2. On extramural the technical school receives tests in mathematics from cities A, B and WITH... Probability of receipt of test work from the city A equals 0.6, from city V- 0.1. Find the probability that the next test will come from the city WITH.

Many, when faced with the concept of "probability theory", get scared, thinking that this is something overwhelming, very difficult. But everything is actually not so tragic. Today we will consider the basic concept of the theory of probability, we will 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 quantities. For the first time, scientists became interested in this issue back in the eighteenth century, when they studied gambling. The basic concept of the theory of probability is an event. This is any fact that is ascertained by experience or observation. But what is experience? Another basic concept of the theory of probability. It means that this set 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 witnesses these events, he does not in any way affect what is happening.

Events

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

Credible.
Impossible.
Random.

Regardless of what kind of events are observed or created in the course of the experiment, they are all subject to this classification. We invite you to get acquainted with each of the types separately.

Credible event

This is such a circumstance, in front of 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 all subject to this law. Probability theory includes such an important concept as a reliable 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 have invested money in the bank, if necessary, we will get it back.

Such events are credible. If we have done everything the necessary conditions, then we will definitely get the expected result.

Impossible events

We are now looking at the elements of the theory of probability. 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.

One cannot deviate from this formulation when solving problems. For clarification, 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).

It is not worth giving more examples, since the ones described above very clearly reflect the essence of this category. An impossible event will never happen during an experience under any circumstances.

Random events

Studying the elements Special attention it is worth giving to this particular type of event. It is them that this science studies. As a result of the experience, something can happen or not. In addition, the test can be carried out an unlimited number of times. Striking examples can serve:

The toss of a coin is an experience, or a test; the falling of a head is an event.
Pulling the ball out of the bag blindly is a test, a red ball is caught - this 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 species of all presented.

title	definition
Credible	Events occurring 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 the experiment / test.	Hitting or missing when throwing a basketball into the basket.

The laws

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

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

When calculating the possibility of a complex, you can use a set of simple events to achieve a result in an easier and faster way. Note that the laws of probability theory are easily proved using some theorems. We suggest that you first get acquainted with the first law.

Convergence of sequences of random variables

Note that there are several types of convergence:

A sequence of random variables converges in probability.
Almost impossible.
Root-mean-square convergence.
Convergence in distribution.

So, on the fly, it is very difficult to grasp the essence. Here are some definitions that will help you understand this topic. For starters, the first view. The sequence is called converging in probability, if the following condition is met: n tends to infinity, the number to which the sequence tends is greater than zero and is close to one.

Let's move on to the next form, almost surely... The sequence is said to converge almost surely to a random variable as n tends to infinity, and P tends to a value close to unity.

The next type is RMS convergence... When using SK-convergence, the study of vector stochastic processes is reduced to the study of their coordinate stochastic processes.

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

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

The law of large numbers

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 pages. Our main task is to apply the theory of probability in practice. We suggest you do this right now and do it. But before that, consider the axioms of probability theory, they will be the main helpers in solving problems.

Axioms

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

The second is as follows: a reliable event occurs with a probability equal to one. Now we will show how to write this using mathematical language: P (B) = 1.

Third: A random event may or may not happen, but the possibility always varies from zero to one. The closer the value is to one, the greater the chances; if the value approaches zero, the probability is very small. 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) = P (A) + P (B).

The axioms of the theory of probability are the simplest rules that will not be difficult to remember. Let's try to solve some problems, relying on the already acquired knowledge.

Lottery ticket

Let's start by looking at the simplest example - a lottery. Imagine 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 drawing, one of which has a prize of five hundred rubles, ten for one hundred rubles, fifty for twenty rubles, and one hundred for five. Probability problems are based on finding the opportunity for luck. Now we will analyze the solution of the above presented task together.

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

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

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

The rest of the tickets are not of interest to us, since their prize fund is less than the one 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 previous actions. It remains only to add the necessary data, in the answer we get 0.061. This number will be the answer to the task question.

Card deck

Probability theory problems can be more complex, for example, let's take the following task. Here 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.

First, let's 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 likelihood of a second event depends on which card we draw first, we wonder if it was an ace or not. It follows from this that event B depends on event A.

The next step is to find the probability of simultaneous occurrence, that is, we multiply A and B. Their product is found as follows: the probability of one event is multiplied 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, we will 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 solving our problem: P (A * B) = P (A) * P (B / A) or P (A * B) = P (B) * P (A / B). The probability is (4/36) * ((3/35) / (4/36). Calculate, rounding to the nearest hundredth. 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 equal to nine hundredths The value is very small, which means that the probability of the occurrence of the event is extremely small.

Forgotten number

We propose to analyze several more options for tasks that the theory of probability studies. 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 to the problem is the simplest if the rules, laws and axioms of the 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 only ten values. The probability of getting the required one is 1/10.

Next, we need to consider options for the origin of the event, suppose that the boy guessed right and immediately typed the desired 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. Let's calculate the probability of such an event: multiply 9/10 by 1/9, in the end we also get 1/10. The third option: the first and second calls were at the wrong address, only from the third the boy got where he wanted. We calculate the probability of such an event: multiply 9/10 by 8/9 and by 1/8, we get 1/10 as a result. We are not interested in other options according to the condition of the problem, so we have to add up the results obtained, in the end we have 3/10. Answer: The probability that a boy will call no more than three times is 0.3.

Number cards

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

an even number will be dropped;
two-digit.

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

Consider the second case: the number of options is nine, but 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.

Probability theory is a branch of mathematics that studies the laws of random phenomena: random events, random variables, their properties and operations on them.

For a long time, the theory of probability did not have a clear definition. It was only formulated in 1929. The emergence of probability theory as a science is attributed to the Middle Ages and the first attempts at mathematical analysis of gambling (coin, dice, roulette). French mathematicians of the 17th century Blaise Pascal and Pierre Fermat, investigating the prediction of winnings in gambling, discovered the first probability laws arising from throwing dice.

Probability theory arose as a science from the belief that certain patterns lie at the heart of random mass events. Probability theory studies these patterns.

Probability theory deals with the study of events, the occurrence of which is not known for certain. It allows you to judge the degree of probability of the occurrence of some events in comparison with others.

For example: it is impossible to determine unambiguously the result of getting "heads" or "tails" as a result of a coin toss, but with repeated tossing, approximately the same number of "heads" and "tails" falls out, which means that the probability of getting "heads" or "tails" "Is equal to 50%.

Test in this case, the implementation of a certain set of conditions is called, that is, in this case, tossing a coin. The challenge can be played an unlimited number of times. In this case, the complex of conditions includes random factors.

The test result is event... The event happens:

Credible (always happens as a result of a test).
Impossible (never happens).
Accidental (may or may not happen as a result of the test).

For example, when a coin is tossed, an impossible event - the coin will be on the edge, a random event - the falling of "heads" or "tails". The specific test result is called elementary event... As a result of the test, only elementary events occur. The totality of all possible, different, specific test outcomes is called space of elementary events.

Basic concepts of the theory

Probability- the degree of possibility of the origin of the event. When the reasons for some possible event to actually occur outweigh the opposite reasons, then the event is called probable, otherwise - unlikely or improbable.

Random value is a value that, as a result of testing, can take on a particular value, and it is not known in advance which one. For example: the number to the fire station per day, the number of hits with 10 shots, etc.

Random variables can be divided into two categories.

Discrete random variable is a quantity that, as a result of a test, can take on certain values with a certain probability, forming a countable set (a set, the elements of which can be numbered). This set can be both finite and infinite. For example, the number of shots before the first hit on the target is a discrete random variable, since this value can take on an infinite, albeit countable number of values.
Continuous random variable such a quantity is called that can take any values from a certain finite or infinite interval. Obviously, the number of possible values of a continuous random variable is infinite.

Probability space- a concept introduced by A.N. Kolmogorov in the 30s of the XX century to formalize the concept of probability, which gave rise to the rapid development of probability theory as a rigorous mathematical discipline.

Probability space is a triplet (sometimes surrounded by angle brackets:, where

This is an arbitrary set, the elements of which are called elementary events, outcomes, or points;
- sigma-algebra of subsets called (random) events;
- a probabilistic measure or probability, i.e. sigma-additive finite measure such that.

Moivre-Laplace theorem- one of the limit theorems of probability theory, established by Laplace in 1812. She argues that the number of successes with multiple repetitions of the same random experiment with two possible outcomes has an approximately normal distribution. It allows you to find an approximate value of the probability.

If, for each of the independent tests, the probability of the occurrence of some random event is equal to () and is the number of tests in which it actually occurs, then the probability of the inequality is close (for large) to the value of the Laplace integral.

Distribution function in probability theory- a function that characterizes the distribution of a random variable or random vector; the probability that a random variable X will take a value less than or equal to x, where x is an arbitrary real number. If certain conditions are met, it completely determines the random variable.

Expected value- the average value of the random variable (this is the probability distribution of the random variable, considered in the theory of probability). In English-language literature, it is denoted by, in Russian -. In statistics, the notation is often used.

Let a probability space and a random variable defined on it be given. That is, by definition, it is a measurable function. Then, if there is a Lebesgue integral of over space, then it is called the mathematical expectation, or the mean value and is denoted.

Variance of a random variable- a measure of the spread of a given random variable, i.e., its deviation from the mathematical expectation. It is indicated in Russian literature and in foreign literature. In statistics, the designation or is often used. The square root of the variance is called the standard deviation, standard deviation, or standard deviation.

Let be a random variable defined on a certain probability space. Then

where the symbol denotes the mathematical expectation.

In probability theory, two random events are called independent if the occurrence of one of them does not change the probability of the occurrence of the other. Similarly, two random variables are called dependent if the value of one of them affects the probability of the values of the other.

The simplest form of the law of large numbers is Bernoulli's theorem, which states that if the probability of an event is the same in all trials, then with an increase in the number of trials, the frequency of the event tends to the probability of the event and ceases to be random.

The law of large numbers in probability theory states that the arithmetic mean of a finite sample from a fixed distribution is close to the theoretical mean mathematical expectation of that distribution. Depending on the type of convergence, a distinction is made between the weak law of large numbers, when there is convergence in probability, and the strong law of large numbers, when the convergence is almost certain.

The general meaning of the law of large numbers is that the joint action of a large number of identical and independent random factors leads to a result that does not depend on the case in the limit.

Methods for estimating the probability based on the analysis of a finite sample are based on this property. An illustrative example is the forecast of election results based on a poll of a sample of voters.

Central limit theorems- the class of theorems in the theory of probability, asserting that the sum of a sufficiently large number of weakly dependent random variables having approximately the same scales (none of the terms dominates, does not make a determining contribution to the sum) has a distribution close to normal.

Since many random variables in applications are formed under the influence of several weakly dependent random factors, their distribution is considered normal. In this case, the condition must be met that none of the factors is dominant. The central limit theorems in these cases justify the application of the normal distribution.