One of the basic concepts of probability theory is the concept of an event.

Event refers to any fact that may or may not occur as a result of the test.

Under test (experience, experiment) in this definition is understood the fulfillment of a certain set of conditions in which this or that phenomenon is observed and this or that result is recorded.

For example, a shooter shoots at a target. In this case, a shot is a test, a hit or miss is an event. Another example: from an urn containing balls of different colors, one ball is drawn. In this case, retrieving the ball from the urn is a test. The appearance of a ball of a certain color is an event.

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

The event is called reliable , if as a result of the test it must necessarily occur. The event is called random , if as a result of the test it may or may not occur. The event is called impossible , if as a result of the test it cannot occur at all.

For example, a die is thrown. In this case, the appearance of an integer is a reliable event, the appearance of the number 2 is a random event, and the appearance of the number 8 is an impossible event.

The events are called incompatible , if the occurrence of one of them excludes the occurrence of any other. Otherwise the events are called joint .

For example, a student receiving “excellent”, “good” and “satisfactory” grades on an exam in one discipline are incompatible events, but receiving the same grades in three different disciplines are joint events.

The events are called the only possible , if the occurrence of one and only one of them as a result of the test is a reliable event.

For example, two students came to take a test. One of the following events will definitely happen: both students will pass the test (event A), only one student will pass the test (event IN), none of the students will pass the test (event WITH). Events A, IN, WITH are the only possible ones.

The events are called equally possible , if, according to the conditions of symmetry, there is reason to believe that none of these events is objectively more possible than the others.

For example, the appearance of a coat of arms or heads when tossing a coin are equally possible events. Indeed, it is assumed that the coin is made of a homogeneous material, has a regular cylindrical shape, and the presence of minting does not affect the loss of one side or another of the coin.

Several events form full group , if they are the only possible and incompatible outcomes of the trial. This means that one and only one of these events must occur as a result of the test.

For example, a student answers questions on an exam paper. The ticket contains two questions. The following test outcomes are possible: the student will answer both questions (event A 1), will answer one question (event A 2), will not answer any question (event A 3). Events A 1 , A 2 and A 3 form a complete group.

Opposite name two uniquely possible events that form a complete group.

For example, the event that a student is currently in the classroom and the event that he is outside the classroom are opposites.

If one of two opposing events is indicated by A, then something else is usually designated as .

Classification of events into possible, probable and random. Concepts of simple and complex elementary events. Operations on events. Classic definition of the probability of a random event and its properties. Elements of combinatorics in probability theory. Geometric probability. Axioms of probability theory.

Event classification

One of the basic concepts of probability theory is the concept of an event. Under event understand any fact that may occur as a result of an experience or test. Under experience, or test, refers to the implementation of a certain set of conditions.

Examples of events:

– hitting the target when fired from a gun (experience - making a shot; event - hitting the target);
– the loss of two emblems when throwing a coin three times (experience - throwing a coin three times; event - the loss of two emblems);
– the appearance of a measurement error within specified limits when measuring the range to a target (experience - range measurement; event - measurement error).

Countless similar examples can be given. Events are indicated by capital letters of the Latin alphabet, etc.

Distinguish joint events And incompatible. Events are called joint if the occurrence of one of them does not exclude the occurrence of the other. Otherwise, the events are called incompatible. For example, two dice are tossed. The event is the loss of three points on the first die, the event is the loss of three points on the second die. and - joint events. Let the store receive a batch of shoes of the same style and size, but different colors. Event - a box taken at random will contain black shoes, an event - the box will contain brown shoes, and - incompatible events.

The event is called reliable, if it is sure to occur under the conditions of a given experiment.

An event is called impossible if it cannot occur under the conditions of a given experience. For example, the event that a standard part will be taken from a batch of standard parts is reliable, but a non-standard part is impossible.

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

The events are called equally possible, if, according to the test conditions, none of these events is objectively more possible than the others. For example, let a store be supplied with light bulbs (in equal quantities) by several manufacturing plants. Events involving the purchase of a light bulb from any of these factories are equally possible.

An important concept is full group of events. Several events in a given experiment form a complete group if at least one of them is sure to appear as a result of the experiment. For example, an urn contains ten balls, six of them are red, four are white, and five balls have numbers. - the appearance of a red ball during one draw, - the appearance of a white ball, - the appearance of a ball with a number. Events form a complete group of joint events.

Let us introduce the concept of an opposite, or additional, event. Under opposite An event is understood as an event that must necessarily occur if some event does not occur. Opposite events are incompatible and the only possible ones. They form a complete group of events. For example, if a batch of manufactured products consists of good and defective products, then when one product is removed, it may turn out to be either good - event, or defective - event.

Operations on events

When developing an apparatus and methodology for studying random events in probability theory, the concept of the sum and product of events is very important.

The sum, or union, of several events is an event consisting of the occurrence of at least one of these events.

The sum of events is indicated as follows:

For example, if an event is hitting the target with the first shot, an event - with the second, then the event is hitting the target in general, it does not matter with which shot - the first, second or both.

The product, or intersection, of several events is an event consisting of the joint occurrence of all these events.

The production of events is indicated

For example, if the event is that the target is hit with the first shot, the event is that the target is hit with the second shot, then the event is that the target was hit with both shots.

The concepts of sum and product of events have a clear geometric interpretation. Let the event consist of a point getting into the region , the event consists of getting into the region , then the event consists of the point getting into the region shaded in Fig. 1, and the event is when a point hits the area shaded in Fig. 2.

Classic definition of the probability of a random event

To quantitatively compare events according to the degree of possibility of their occurrence, a numerical measure is introduced, which is called the probability of an event.

The probability of an event is a number that expresses the measure of the objective possibility of the occurrence of an event.

The probability of an event will be denoted by the symbol.

The probability of an event is equal to the ratio of the number of cases favorable to it, out of the total number of uniquely possible, equally possible and incompatible cases, to the number i.e.

This is the classic definition of probability. Thus, to find the probability of an event, it is necessary, having considered the various outcomes of the test, to find a set of uniquely possible, equally possible and incompatible cases, calculate their total number, the number of cases favorable to a given event, and then perform the calculation using formula (1.1).

From formula (1.1) it follows that the probability of an event is a non-negative number and can vary from zero to one depending on the proportion of the favorable number of cases from the total number of cases:

Properties of Probability

Property 1. If all cases are favorable for a given event, then this event is sure to occur. Consequently, the event in question is reliable, and the probability of its occurrence is , since in this case

Property 2. If there is not a single case favorable for a given event, then this event cannot occur as a result of experience. Consequently, the event in question is impossible, and the probability of its occurrence is , since in this case:

Property 3. The probability of the occurrence of events that form a complete group is equal to one.

Property 4. The probability of the occurrence of the opposite event is determined in the same way as the probability of the occurrence of the event:

where is the number of cases favorable to the occurrence of the opposite event. Hence the probability of the opposite event occurring is equal to the difference between unity and the probability of the event occurring:

An important advantage of the classical definition of the probability of an event is that with its help the probability of an event can be determined without resorting to experience, but based on logical reasoning.

Example 1. While dialing a phone number, the subscriber forgot one digit and dialed it at random. Find the probability that the correct number is dialed.

Solution. Let us denote the event that the required number is dialed. The subscriber could dial any of the 10 digits, so the total number of possible outcomes is 10. These outcomes are the only possible (one of the digits must be dialed) and equally possible (the digit is dialed at random). Only one outcome favors the event (there is only one required number). The required probability is equal to the ratio of the number of outcomes favorable to the event to the number of all outcomes:

Elements of combinatorics

In probability theory, placements, permutations and combinations are often used. If a set is given, then placement (combination) of the elements by is any ordered (unordered) subset of the elements of the set. When placed is called rearrangement from elements.

Let, for example, be given a set. The placements of three elements of this set of two are , , , , , ; combinations - , , .

Two combinations differ in at least one element, and placements differ either in the elements themselves or in the order in which they appear. The number of combinations of elements by is calculated by the formula

is the number of placements of elements by ; - number of permutations of elements.

Example 2. In a batch of 10 parts there are 7 standard ones. Find the probability that among 6 parts taken at random there are exactly 4 standard ones.

Solution. The total number of possible test outcomes is equal to the number of ways in which 6 parts can be extracted from 10, i.e., equal to the number of combinations of 10 elements of 6. The number of outcomes favorable to the event (among the 6 taken parts there are exactly 4 standard ones) is determined as follows: 4 standard parts can be taken from 7 standard parts in different ways; in this case, the remaining parts must be non-standard; There are ways to take 2 non-standard parts from non-standard parts. Therefore, the number of favorable outcomes is equal to . The initial probability is equal to the ratio of the number of outcomes favorable to the event to the number of all outcomes:

Statistical definition of probability

Formula (1.1) is used to directly calculate the probabilities of events only when experience is reduced to a pattern of cases. In practice, the classical definition of probability is often not applicable for two reasons: first, the classical definition of probability assumes that the total number of cases must be finite. In fact, it is often not limited. Secondly, it is often impossible to present the outcomes of an experiment in the form of equally possible and incompatible events.

The frequency of occurrence of events during repeated Experiments tends to stabilize around some constant value. Thus, a certain constant value can be associated with the event under consideration, around which frequencies are grouped and which is a characteristic of the objective connection between the set of conditions under which experiments are carried out and the event.

The probability of a random event is the number around which the frequencies of this event are grouped as the number of trials increases.

This definition of probability is called statistical.

The advantage of the statistical method of determining probability is that it is based on a real experiment. However, its significant drawback is that to determine the probability it is necessary to perform a large number of experiments, which are very often associated with material costs. The statistical definition of the probability of an event, although it quite fully reveals the content of this concept, does not make it possible to actually calculate the probability.

The classical definition of probability considers the complete group of a finite number of equally possible events. In practice, very often the number of possible test outcomes is infinite. In such cases, the classical definition of probability is not applicable. However, sometimes in such cases you can use another method of calculating probability. For definiteness, we restrict ourselves to the two-dimensional case.

Let a certain region of area , which contains another region of area, be given on the plane (Fig. 3). A dot is thrown into the area at random. What is the probability that a point will fall into the region? It is assumed that a point thrown at random can hit any point in the region, and the probability of hitting any part of the region is proportional to the area of ​​the part and does not depend on its location and shape. In this case, the probability of hitting the area when throwing a point at random into the area is

Thus, in the general case, if the possibility of a random appearance of a point inside a certain area on a line, plane or in space is determined not by the position of this area and its boundaries, but only by its size, i.e. length, area or volume, then the probability of a random point falling inside a certain region is defined as the ratio of the size of this region to the size of the entire region in which a given point can appear. This is the geometric definition of probability.

Example 3. A round target rotates at a constant angular velocity. One fifth of the target is painted green, and the rest is white (Fig. 4). A shot is fired at the target in such a way that hitting the target is a reliable event. You need to determine the probability of hitting the target sector colored green.

Solution. Let’s denote “the shot hit the sector colored green.” Then . The probability is obtained as the ratio of the area of ​​the part of the target painted green to the entire area of ​​the target, since hits on any part of the target are equally possible.

Axioms of probability theory

From the statistical definition of the probability of a random event it follows that the probability of an event is the number around which the frequencies of this event observed experimentally are grouped. Therefore, the axioms of probability theory are introduced so that the probability of an event has the basic properties of frequency.

Axiom 1. Each event corresponds to a certain number that satisfies the condition and is called its probability.

Probability theory – a mathematical science that studies the patterns of random phenomena. Random phenomena are understood as phenomena with an uncertain outcome that occur when a certain set of conditions are repeatedly reproduced.

For example, when throwing a coin, you cannot predict which side it will land on. The result of tossing a coin is random. But with a sufficiently large number of coin tosses, there is a certain pattern (the coat of arms and the hash mark will fall out approximately the same number of times).

Basic concepts of probability theory

Test (experience, experiment) - the implementation of a certain set of conditions in which this or that phenomenon is observed and this or that result is recorded.

For example: tossing a dice and getting a number of points; air temperature difference; method of treating the disease; some period of a person's life.

Random event (or just an event) – test outcome.

Examples of random events:

getting one point when throwing a die;

exacerbation of coronary heart disease with a sharp increase in air temperature in summer;

development of complications of the disease due to the wrong choice of treatment method;

admission to a university after successful studies at school.

Events are designated in capital letters of the Latin alphabet: A , B , C , …

The event is called reliable , if as a result of the test it must necessarily occur.

The event is called impossible , if as a result of the test it cannot occur at all.

For example, if all products in a batch are standard, then extracting a standard product from it is a reliable event, but extracting a defective product under the same conditions is an impossible event.

CLASSICAL DEFINITION OF PROBABILITY

Probability is one of the basic concepts of probability theory.

Classic event probability is called the ratio of the number of cases favoring an event , to the total number of cases, i.e.

, (5.1)

Where
- probability of event ,

- number of cases favorable to the event ,

- total number of cases.

Properties of event probability

The probability of any event lies between zero and one, i.e.

The probability of a reliable event is equal to one, i.e.

.

The probability of an impossible event is zero, i.e.

.

(Offer to solve several simple problems orally).

STATISTICAL DETERMINATION OF PROBABILITY

In practice, estimating the probabilities of events is often based on how often a given event will occur in the tests performed. In this case, the statistical definition of probability is used.

Statistical probability of an event called the relative frequency limit (the ratio of the number of cases m, favorable for the occurrence of an event , to the total number tests performed), when the number of tests tends to infinity, i.e.

Where
- statistical probability of an event ,
- number of trials in which the event appeared , - total number of tests.

Unlike classical probability, statistical probability is a characteristic of experimental probability. Classical probability serves to theoretically calculate the probability of an event under given conditions and does not require that tests be carried out in reality. The statistical probability formula is used to experimentally determine the probability of an event, i.e. it is assumed that the tests were actually carried out.

Statistical probability is approximately equal to the relative frequency of a random event, therefore, in practice, the relative frequency is taken as the statistical probability, because statistical probability is practically impossible to find.

The statistical definition of probability is applicable to random events that have the following properties:

Probability addition and multiplication theorems

Basic Concepts

a) The only possible events

Events
They are called the only possible ones if, as a result of each test, at least one of them will certainly occur.

These events form a complete group of events.

For example, when tossing a die, the only possible events are the sides with one, two, three, four, five and six points. They form a complete group of events.

b) Events are called incompatible, if the occurrence of one of them excludes the occurrence of other events in the same trial. Otherwise they are called joint.

c) Opposite name two uniquely possible events that form a complete group. Designate And .

G) Events are called independent, if the probability of the occurrence of one of them does not depend on the commission or non-completion of others.

Actions on events

The sum of several events is an event consisting of the occurrence of at least one of these events.

If And – joint events, then their sum
or
denotes the occurrence of either event A, or event B, or both events together.

If And – incompatible events, then their sum
means occurrence or events , or events .

Amount events mean:

The product (intersection) of several events is an event consisting of the joint occurrence of all these events.

The product of two events is denoted by
or
.

Work events represent

Theorem for adding probabilities of incompatible events

The probability of the sum of two or more incompatible events is equal to the sum of the probabilities of these events:

For two events;

- For events.

Consequences:

a) Sum of probabilities of opposite events And equal to one:

The probability of the opposite event is denoted by :
.

b) Sum of probabilities of events forming a complete group of events is equal to one: or
.

Theorem for adding probabilities of joint events

The probability of the sum of two joint events is equal to the sum of the probabilities of these events without the probabilities of their intersection, i.e.

Probability multiplication theorem

a) For two independent events:

b) For two dependent events

Where
– conditional probability of an event , i.e. probability of an event , calculated under the condition that the event happened.

c) For independent events:

.

d) Probability of at least one of the events occurring , forming a complete group of independent events:

Conditional probability

Probability of event , calculated assuming that the event occurred , is called the conditional probability of the event and is designated
or
.

When calculating conditional probability using the classical probability formula, the number of outcomes And
calculated taking into account the fact that before the event occurs an event occurred .

Events and their classification

Basic concepts of probability theory

When constructing any mathematical theory, first of all, the simplest concepts are identified, which are accepted as the initial facts. Such basic concepts in probability theory are the concept random experiment, random event, probability of a random event.

Random experiment– this is the process of recording an observation of an event of interest to us, which is carried out under the condition of a given stationary (not changing over time) a real set of conditions, including the inevitability of the influence of a large number of random (not amenable to strict accounting and control) factors.

These factors, in turn, do not allow us to draw completely reliable conclusions about whether the event of interest to us will occur or not. In this case, it is assumed that we have a fundamental possibility (at least mentally realizable) of repeating our experiment or observation many times within the framework of the same set of conditions.

Here are some examples of random experiments.

1. A random experiment consisting of tossing a perfectly symmetrical coin involves random factors such as the force with which the coin is thrown, the trajectory of the coin, the initial speed, the moment of rotation, etc. These random factors make it impossible to accurately determine the outcome of each individual trial: “when tossing a coin, a coat of arms will appear” or “tossing a coin, tails will appear.”

2. The Stalkanat plant tests manufactured cables for the maximum permissible load. The load varies within certain limits from one experiment to another. This is due to such random factors as micro defects in the material from which the cables are made, various interferences in the operation of equipment that occur during the production of cables, storage conditions, experimental conditions, etc.

3. A series of shots are fired from the same gun at a specific target. Hitting the target depends on many random factors, which include the condition of the gun and the projectile, the installation of the gun, the skill of the gunner, weather conditions (wind, light, etc.).

Definition. The implementation of a certain set of conditions is called test. The test result is called event.

Random events are indicated in capital letters of the Latin alphabet: A, B, C...or capital letter with index: .

For example, passing an exam under a given set of conditions (a written exam, including a rating system, etc.) is a test for the student, and receiving a certain grade is an event;

firing a gun under a given set of conditions (weather conditions, condition of the gun, etc.) is a test, and hitting or missing the target is an event.

We can repeat the same experiment many times under the same conditions. The presence of a large number of random factors characterizing the conditions of each such experiment makes it impossible to make a completely definite conclusion about whether the event of interest to us will occur or not in a separate test. Note that in probability theory such a problem is not posed.

Event classification

Events happen reliable, impossible And random.

Definition. The event is called reliable, if under a given set of conditions it necessarily occurs.

All reliable events are indicated by a letter (the first letter of the English word universal- general)

Examples of reliable events are: the emergence of a white ball from an urn containing only white balls; winning a win-win lottery.

Definition. The event is called impossible, if under a given set of conditions it cannot occur.

All impossible events are indicated by the letter .

For example, in Euclidean geometry, the sum of the angles of a triangle cannot be greater than , and you cannot get a grade of “6” on an exam with a five-point grading system.

Definition. The event is called random, if it may or may not appear under a given set of conditions.

For example, random events are: the event of the appearance of an ace from a deck of cards; event winning a football team match; event of winning a cash and clothing lottery; event purchase of a defective TV, etc.

Definition. Events are called incompatible, if the occurrence of one of these events excludes the occurrence of any other.

Example 1. If we consider the test, which consists of tossing a coin, then the events - the appearance of a coat of arms and the appearance of a number - are incompatible events.

Definition. Events are called joint, if the occurrence of one of these events does not exclude the occurrence of other events.

Example 2. If a shot is fired from three guns, then the following events are combined: a hit from the first gun; hit from the second gun; hit from the third gun.

Definition. Events are called the only possible, if when a given set of conditions is realized, at least one of the specified events must occur.

Example 3. When throwing a die, the following are the only possible events:

A 1 – appearance of one point,

A 2 – appearance of two points,

A 3 – appearance of three points,

A 4 – appearance of four points,

A 5 – appearance of five points,

A 6 – appearance of six points.

Definition. They say that events form full group of events, if these events are the only possible and incompatible.

The events that were considered in examples 1, 3 form a complete group, since they are incompatible and the only possible ones.

Definition. Two events that form a complete group are called opposite.

If is some event, then the opposite event is denoted by .

Example 4. If the event is a coat of arms, then the event is a tails.

Opposite events are also: “the student passed the exam” and “the student did not pass the exam,” “the plant fulfilled the plan” and “the plant did not fulfill the plan.”

Definition. Events are called equally probable or equally possible, if during the test they all objectively have the same possibility of appearing.

Note that equally possible events can only appear in experiments with symmetry of outcomes, which is ensured by special methods (for example, making absolutely symmetrical coins, dice, careful shuffling of cards, dominoes, mixing balls in an urn, etc.).

Definition. If the outcomes of some test are the only possible, incompatible and equally possible, then they are called elementary outcomes, cases or chances, and the test itself is called case diagram or "urn scheme"(since any probability problem for the test in question can be replaced by an equivalent problem with urns and balls of different colors) .

Example 5. If there are 3 white and 3 black balls in the urn, identical to the touch, then the event A 1 – appearance of a white ball and event A 2 – the appearance of a black ball are equally probable events.

Definition. They say that the event favors event or event entails event , if upon appearance event definitely comes.

If an event entails an event, then this is indicated by the symbols equivalent or equivalent and denote

Thus, equivalent events and at each test, either both occur or both do not occur.

To build a probability theory, in addition to the basic concepts already introduced (random experiment, random event), it is necessary to introduce one more basic concept - probability of a random event.

Note that ideas about the probability of an event changed during the development of probability theory. Let us trace the history of the development of this concept.

Under probability random event understand the measure of the objective possibility of the occurrence of an event.

This definition reflects the concept of probability from a qualitative point of view. It was known in the ancient world.

A quantitative definition of the probability of an event was first given in the works of the founders of probability theory, who considered random experiments with symmetry or objective equipossibility of outcomes. Such random experiments, as noted above, most often include artificially organized experiments in which special methods are taken to ensure equal outcomes (shuffling cards or dominoes, making perfectly symmetrical dice, coins, etc.). In relation to such random experiments in the seventeenth century. The French mathematician Laplace formulated the classical definition of probability.

Many, when faced with the concept of “probability theory,” get scared, thinking that it is something overwhelming, very complex. But everything is actually not so tragic. Today we will look at the basic concept of probability theory and 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. Scientists first 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 established by experience or observation. But what is experience? Another basic concept of probability theory. It means that this set of circumstances was created not by chance, but for a specific purpose. As for observation, here the researcher himself does not participate in the experiment, but is simply a witness to these events; he does not influence what is happening in any way.

Events

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

• Reliable.
• Impossible.
• Random.

Regardless of what kind of events they are, observed or created during the experience, they are all subject to this classification. We invite you to get acquainted with each type separately.

Reliable event

This is a circumstance for 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. The theory of probability includes such an important concept as a reliable event. Here are some examples:

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

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

Impossible events

Now we are considering elements of probability theory. We propose to move on to an explanation of the next type of event, namely the impossible. First, let's 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 those described above very clearly reflect the essence of this category. An impossible event will never occur during an experiment under any circumstances.

Random Events

When studying the elements, special attention should be paid to this particular type of event. This is what science studies. As a result of the experience, something may or may not happen. In addition, the test can be carried out an unlimited number of times. Vivid examples include:

• The toss of a coin is an experience or test, the landing of heads is an event.
• Pulling a ball out of a bag blindly is a test; getting a red ball 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 the events, a table is provided. Probability theory studies only the last type of all presented.

Name	definition
Reliable	Events that occur with a 100% guarantee if certain conditions are met.	Admission to an educational institution upon passing the entrance exam well.
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.	A hit or miss when throwing a basketball into a hoop.

Laws

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

• Convergence of sequences of random variables.
• Law of large numbers.

When calculating the possibility of something 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 proven using certain 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:

• The sequence of random variables converges in probability.
• Almost impossible.
• Mean square convergence.
• Distribution convergence.

So, right off the bat, it’s very difficult to understand the essence. Here are definitions that will help you understand this topic. Let's start with the first view. 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 view, 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 mean square 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 look at it briefly so that we can move directly to solving problems. Convergence in distribution has another name - “weak”, and we will explain why later. Weak convergence is the convergence of 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 in the probability space. This is possible because the condition is formed exclusively using distribution functions.

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 may drag on for several dozen sheets. Our main task is to apply probability theory in practice. We suggest you do this right now. But before that, let’s look at 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 an 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 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 ranges from zero to one. The closer the value is to one, the greater the chances; if the value approaches zero, the probability is very low. Let's write this in mathematical language: 0<Р(С)<1.

Let's 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 it in mathematical language: P(A+B)=P(A)+P(B).

The axioms of probability theory are the simplest rules that are not difficult to remember. Let's try to solve some problems based on the knowledge we have already acquired.

Lottery ticket

First, let's look at the simplest example - a 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 are participating in the circulation, one of which has a prize of five hundred rubles, ten of them have a hundred rubles each, fifty have a prize of twenty rubles, and one hundred have a prize of five. Probability problems are based on finding the possibility of luck. Now together we will analyze the solution to the above task.

If we use the letter A to denote a win of five hundred rubles, then the probability of getting A will be equal to 0.001. How did we get this? 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)

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

We are not interested in the remaining tickets, 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 a given event; we have already found them in previous actions. All that remains is to add up the necessary data, and the answer we get is 0.061. This number will be the answer to the task question.

Card deck

Problems in probability theory can be more complex; for example, let’s take the following task. In front of you is a deck of thirty-six cards. Your task is to draw two cards in a row without shuffling the stack, 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 probability of the second event depends on which card we drew first, we wonder whether 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: we multiply the probability of one event by the conditional probability of another, which we calculate, assuming that the first event occurred, that is, we drew an ace with the first card.

To make everything clear, let’s give a designation to such an element as events. It is calculated assuming that event A has occurred. It is 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 equal to (4/36) * ((3/35)/(4/36). We calculate by 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 nine hundredths. The value is very small, which means that the probability of the event occurring is extremely small.

Forgotten number

We propose to analyze several more variants of 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 one by one. We need to calculate the probability that he will call no more than three times. The solution to the problem is simplest if the rules, laws and axioms of probability theory are known.

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

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

Cards with numbers

There are nine cards in front of you, on each of which a number from one to nine is 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 appear;
• 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. Let's find the probability that the number will be even. It won’t be difficult to calculate that there are four even numbers, this will be our m, there are nine possible options in total, that is, m=9. Then the probability is 0.44 or 4/9.

Let's 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.