Total probability formula. Definition of probability

Whether we like it or not, our life is full of all kinds of accidents, both pleasant and not so pleasant. Therefore, it would not hurt each of us to know how to find the probability of a particular event. This will help you make the right decisions under any circumstances that involve uncertainty. For example, such knowledge will be very useful when choosing investment options, assessing the possibility of winning a stock or lottery, determining the reality of achieving personal goals, etc., etc.

Probability theory formula

In principle, studying this topic does not take too much time. In order to get an answer to the question: “How to find the probability of a phenomenon?”, you need to understand the key concepts and remember the basic principles on which the calculation is based. So, according to statistics, the events under study are denoted by A1, A2,..., An. Each of them has both favorable outcomes (m) and total elementary outcomes. For example, we are interested in how to find the probability that there will be an even number of points on the top side of the cube. Then A is a roll of m - rolling out 2, 4 or 6 points (three favorable options), and n is all six possible options.

The calculation formula itself is as follows:

With one outcome everything is extremely easy. But how to find the probability if events happen one after another? Consider this example: one card is shown from a card deck (36 pieces), then it is hidden back into the deck, and after shuffling, the next one is pulled out. How to find the probability that at least in one case the queen of spades was drawn? Exists next rule: If you are considering a complex event that can be divided into several incompatible simple events, you can first calculate the result for each of them and then add them together. In our case it will look like this: 1/36 + 1/36 = 1/18. But what happens when several occur simultaneously? Then we multiply the results! For example, the probability that when two coins are tossed simultaneously, two heads will appear will be equal to: ½ * ½ = 0.25.

Now let's take an even more complex example. Suppose we entered a book lottery in which ten out of thirty tickets are winning. You need to determine:

1. The probability that both will be winners.
2. At least one of them will bring a prize.
3. Both will be losers.

So, let's consider the first case. It can be broken down into two events: the first ticket will be lucky, and the second will also be lucky. Let's take into account that the events are dependent, since after each pull out the total number of options decreases. We get:

10 / 30 * 9 / 29 = 0,1034.

In the second case, you will need to determine the probability of a losing ticket and take into account that it can be either the first or the second: 10/30 * 20/29 + 20/29 * 10/30 = 0.4598.

Finally, the third case, when you won’t be able to get even one book from the lottery: 20 / 30 * 19 / 29 = 0.4368.

as an ontological category reflects the extent of the possibility of the emergence of any entity under any conditions. In contrast to the mathematical and logical interpretation of this concept, ontological mathematics does not associate itself with the obligation of quantitative expression. The meaning of V. is revealed in the context of understanding determinism and the nature of development in general.

Great definition

Incomplete definition ↓

PROBABILITY

concept characterizing quantities. the measure of the possibility of the occurrence of a certain event at a certain conditions. In scientific knowledge there are three interpretations of V. The classical concept of V., which arose from mathematical. analysis gambling and most fully developed by B. Pascal, J. Bernoulli and P. Laplace, considers victory as the ratio of the number of favorable cases to the total number of all equally possible ones. For example, when throwing a dice that has 6 sides, each of them can be expected to land with a value of 1/6, since no one side has advantages over another. Such symmetry of experimental outcomes is specially taken into account when organizing games, but is relatively rare in the study of objective events in science and practice. Classic V.'s interpretation gave way to statistics. V.'s concepts, which are based on the actual observing the occurrence of a certain event over a long period of time. experience under precisely fixed conditions. Practice confirms that the more often an event occurs, the more degree objective possibility of its occurrence, or B. Therefore, statistical. V.'s interpretation is based on the concept of relates. frequency, which can be determined experimentally. V. as a theoretical the concept never coincides with the empirically determined frequency, however, in plural. In cases, it differs practically little from the relative one. frequency found as a result of duration. observations. Many statisticians consider V. as a “double” refers. frequencies, edges are determined statistically. study of observational results

or experiments. Less realistic was the definition of V. as the limit relates. frequencies of mass events, or groups, proposed by R. Mises. As further development The frequency approach to V. puts forward a dispositional, or propensitive, interpretation of V. (K. Popper, J. Hacking, M. Bunge, T. Settle). According to this interpretation, V. characterizes the property of generating conditions, for example. experiment. installations to obtain a sequence of massive random events. It is precisely this attitude that gives rise to physical dispositions, or predispositions, V. which can be checked using relatives. frequency

Statistical V.'s interpretation dominates scientific research. cognition, because it reflects specific. the nature of the patterns inherent in mass phenomena of a random nature. In many physical, biological, economic, demographic. and etc. social processes it is necessary to take into account the effect of many random factors, which are characterized by a stable frequency. Identifying these stable frequencies and quantities. its assessment with the help of V. makes it possible to reveal the necessity that makes its way through the cumulative action of many accidents. This is where the dialectic of transforming chance into necessity finds its manifestation (see F. Engels, in the book: K. Marx and F. Engels, Works, vol. 20, pp. 535-36).

Logical, or inductive, reasoning characterizes the relationship between the premises and the conclusion of non-demonstrative and, in particular, inductive reasoning. Unlike deduction, the premises of induction do not guarantee the truth of the conclusion, but only make it more or less plausible. This plausibility, with precisely formulated premises, can sometimes be assessed using V. The value of this V. is most often determined by comparison. concepts (more than, less than or equal to), and sometimes in a numerical way. Logical interpretation is often used to analyze inductive reasoning and construct various systems of probabilistic logic (R. Carnap, R. Jeffrey). In semantics logical concepts V. is often defined as the degree to which one statement is confirmed by others (for example, a hypothesis by its empirical data).

In connection with the development of theories of decision making and games, the so-called personalistic interpretation of V. Although V. at the same time expresses the degree of faith of the subject and the occurrence of a certain event, V. themselves must be chosen in such a way that the axioms of the calculus of V. are satisfied. Therefore, V. with such an interpretation expresses not so much the degree of subjective, but rather reasonable faith . Consequently, decisions made on the basis of such V. will be rational, because they do not take into account the psychological. characteristics and inclinations of the subject.

With epistemological t.zr. difference between statistical, logical. and personalistic interpretations of V. is that if the first characterizes the objective properties and relationships of mass phenomena of a random nature, then the last two analyze the features of the subjective, cognizant. human activities under conditions of uncertainty.

PROBABILITY

one of the most important concepts of science, characterizing a special systemic vision of the world, its structure, evolution and knowledge. The specificity of the probabilistic view of the world is revealed through the inclusion of the concepts of randomness, independence and hierarchy (the idea of ​​levels in the structure and determination of systems) among the basic concepts of existence.

Ideas about probability originated in ancient times and related to the characteristics of our knowledge, while the existence of probabilistic knowledge was recognized, which differed from reliable knowledge and from false knowledge. The impact of the idea of ​​probability on scientific thinking and on the development of knowledge is directly related to the development of probability theory as a mathematical discipline. The origin of the mathematical doctrine of probability dates back to the 17th century, when the development of a core of concepts allowing. quantitative (numerical) characteristics and expressing a probabilistic idea.

Intensive applications of probability to the development of cognition occur in the 2nd half. 19- 1st floor. 20th century Probability has entered the structures of such fundamental sciences of nature as classical statistical physics, genetics, quantum theory, and cybernetics (information theory). Accordingly, probability personifies that stage in the development of science, which is now defined as non-classical science. To reveal the novelty and features of the probabilistic way of thinking, it is necessary to proceed from an analysis of the subject of probability theory and the foundations of its numerous applications. Probability theory is usually defined as a mathematical discipline that studies the patterns of mass random phenomena under certain conditions. Randomness means that within the framework of mass character, the existence of each elementary phenomenon does not depend on and is not determined by the existence of other phenomena. At the same time, the mass nature of phenomena itself has a stable structure and contains certain regularities. A mass phenomenon is quite strictly divided into subsystems, and the relative number of elementary phenomena in each of the subsystems (relative frequency) is very stable. This stability is compared with probability. A mass phenomenon as a whole is characterized by a probability distribution, that is, by specifying subsystems and their corresponding probabilities. The language of probability theory is the language of probability distributions. Accordingly, probability theory is defined as the abstract science of operating with distributions.

Probability gave rise in science to ideas about statistical patterns and statistical systems. The last essence systems formed from independent or quasi-independent entities, their structure is characterized by probability distributions. But how is it possible to form systems from independent entities? It is usually assumed that for the formation of systems with integral characteristics, it is necessary that sufficiently stable connections exist between their elements that cement the systems. Stability of statistical systems is given by the presence of external conditions, external environment, external rather than internal forces. The very definition of probability is always based on setting the conditions for the formation of the initial mass phenomenon. Another important idea characterizing the probabilistic paradigm is the idea of ​​hierarchy (subordination). This idea expresses the relationship between the characteristics of individual elements and the integral characteristics of systems: the latter, as it were, are built on top of the former.

The importance of probabilistic methods in cognition lies in the fact that they make it possible to study and theoretically express the patterns of structure and behavior of objects and systems that have a hierarchical, “two-level” structure.

Analysis of the nature of probability is based on its frequency, statistical interpretation. At the same time, very long time In science, such an understanding of probability prevailed, which was called logical, or inductive, probability. Logical probability is interested in questions of the validity of a separate, individual judgment under certain conditions. Is it possible to evaluate the degree of confirmation (reliability, truth) of an inductive conclusion (hypothetical conclusion) in quantitative form? During the development of probability theory, such questions were repeatedly discussed, and they began to talk about the degrees of confirmation of hypothetical conclusions. This measure of probability is determined by the available this person information, his experience, views on the world and psychological mindset. In all similar cases the magnitude of probability is not amenable to strict measurements and practically lies outside the competence of probability theory as a consistent mathematical discipline.

The objective, frequentist interpretation of probability was established in science with significant difficulties. Initially, the understanding of the nature of probability was strongly influenced by those philosophical and methodological views that were characteristic of classical science. Historically, the development of probabilistic methods in physics occurred under the determining influence of the ideas of mechanics: statistical systems were interpreted simply as mechanical. Since the corresponding problems were not solved by strict methods of mechanics, assertions arose that turning to probabilistic methods and statistical laws is the result of the incompleteness of our knowledge. In the history of the development of classical statistical physics, numerous attempts were made to substantiate it on the basis of classical mechanics, but they all failed. The basis of probability is that it expresses the structural features of a certain class of systems, other than mechanical systems: the state of the elements of these systems is characterized by instability and a special (not reducible to mechanics) nature of interactions.

The entry of probability into knowledge leads to the denial of the concept of hard determinism, to the denial of the basic model of being and knowledge developed in the process of the formation of classical science. The basic models represented by statistical theories are of a different, more general nature: they include the ideas of randomness and independence. The idea of ​​probability is associated with the disclosure of the internal dynamics of objects and systems, which cannot be entirely determined by external conditions and circumstances.

The concept of a probabilistic vision of the world, based on the absolutization of ideas about independence (as before the paradigm of rigid determination), has now revealed its limitations, which most strongly affects the transition modern science To analytical methods research into complex systems and the physical and mathematical foundations of self-organization phenomena.

Great definition

Incomplete definition ↓

In fact, formulas (1) and (2) are a short record of conditional probability based on a contingency table of characteristics. Let's return to the example discussed (Fig. 1). Suppose we learn that a family is planning to buy a wide-screen television. What is the probability that this family will actually buy such a TV?

Rice. 1. Widescreen TV Buying Behavior

In this case, we need to calculate the conditional probability P (purchase completed | purchase planned). Since we know that the family is planning to buy, the sample space does not consist of all 1000 families, but only those planning to buy a wide-screen TV. Of the 250 such families, 200 actually bought this TV. Therefore, the probability that a family will actually buy a wide-screen TV if they have planned to do so can be calculated using the following formula:

P (purchase completed | purchase planned) = number of families who planned and bought a wide-screen TV / number of families planning to buy a wide-screen TV = 200 / 250 = 0.8

Formula (2) gives the same result:

where is the event A is that the family is planning to purchase a widescreen TV, and the event IN- that she will actually buy it. Substituting real data into the formula, we get:

Decision tree

In Fig. 1 families are divided into four categories: those who planned to buy a wide-screen TV and those who did not, as well as those who bought such a TV and those who did not. A similar classification can be performed using a decision tree (Fig. 2). The tree shown in Fig. 2 has two branches corresponding to families who planned to purchase a widescreen TV and families who did not. Each of these branches splits into two additional branches corresponding to households that did and did not purchase a widescreen TV. The probabilities written at the ends of the two main branches are the unconditional probabilities of events A And A'. The probabilities written at the ends of the four additional branches are the conditional probabilities of each combination of events A And IN. Conditional probabilities are calculated by dividing the joint probability of events by the corresponding unconditional probability of each of them.

Rice. 2. Decision tree

For example, to calculate the probability that a family will buy a wide-screen television if it has planned to do so, one must determine the probability of the event purchase planned and completed, and then divide it by the probability of the event purchase planned. Moving along the decision tree shown in Fig. 2, we get the following (similar to the previous) answer:

Statistical independence

In the example of buying a wide-screen TV, the probability that a randomly selected family purchased a wide-screen TV given that they planned to do so is 200/250 = 0.8. Recall that the unconditional probability that a randomly selected family purchased a wide-screen TV is 300/1000 = 0.3. This leads to a very important conclusion. Prior information that the family was planning a purchase influences the likelihood of the purchase itself. In other words, these two events depend on each other. In contrast to this example, there are statistically independent events whose probabilities do not depend on each other. Statistical independence is expressed by the identity: P(A|B) = P(A), Where P(A|B)- probability of event A provided that the event occurred IN, P(A)- unconditional probability of event A.

Please note that events A And IN P(A|B) = P(A). If in a contingency table of characteristics having a size of 2×2, this condition is satisfied for at least one combination of events A And IN, it will be valid for any other combination. In our example events purchase planned And purchase completed are not statistically independent because information about one event affects the probability of another.

Let's look at an example that shows how to test the statistical independence of two events. Let's ask 300 families who bought a widescreen TV if they were satisfied with their purchase (Fig. 3). Determine whether the degree of satisfaction with the purchase and the type of TV are related.

Rice. 3. Data characterizing the degree of satisfaction of buyers of widescreen TVs

Judging by these data,

In the same time,

P (customer satisfied) = 240 / 300 = 0.80

Therefore, the probability that the customer is satisfied with the purchase and that the family purchased an HDTV are equal, and these events are statistically independent because they are not related in any way.

Probability multiplication rule

The formula for calculating conditional probability allows you to determine the probability of a joint event A and B. Having resolved formula (1)

relative to joint probability P(A and B), we obtain a general rule for multiplying probabilities. Probability of event A and B equal to the probability of the event A provided that the event occurs IN IN:

(3) P(A and B) = P(A|B) * P(B)

Let's take as an example 80 families who bought a widescreen HDTV television (Fig. 3). The table shows that 64 families are satisfied with the purchase and 16 are not. Let us assume that two families are randomly selected from among them. Determine the probability that both customers will be satisfied. Using formula (3), we obtain:

P(A and B) = P(A|B) * P(B)

where is the event A is that the second family is satisfied with their purchase, and the event IN- that the first family is satisfied with their purchase. The probability that the first family is satisfied with their purchase is 64/80. However, the likelihood that the second family is also satisfied with their purchase depends on the first family's response. If the first family does not return to the sample after the survey (selection without return), the number of respondents is reduced to 79. If the first family is satisfied with their purchase, the probability that the second family will also be satisfied is 63/79, since there are only 63 left in the sample families satisfied with their purchase. Thus, substituting specific data into formula (3), we get the following answer:

P(A and B) = (63/79)(64/80) = 0.638.

Therefore, the probability that both families are satisfied with their purchases is 63.8%.

Suppose that after the survey the first family returns to the sample. Determine the probability that both families will be satisfied with their purchase. In this case, the probability that both families are satisfied with their purchase is the same, equal to 64/80. Therefore, P(A and B) = (64/80)(64/80) = 0.64. Thus, the probability that both families are satisfied with their purchases is 64.0%. This example shows that the choice of the second family does not depend on the choice of the first. Thus, replacing the conditional probability in formula (3) P(A|B) probability P(A), we obtain a formula for multiplying the probabilities of independent events.

The rule for multiplying the probabilities of independent events. If events A And IN are statistically independent, the probability of an event A and B equal to the probability of the event A, multiplied by the probability of the event IN.

(4) P(A and B) = P(A)P(B)

If this rule is true for events A And IN, which means they are statistically independent. Thus, there are two ways to determine the statistical independence of two events:

1. Events A And IN are statistically independent of each other if and only if P(A|B) = P(A).
2. Events A And B are statistically independent of each other if and only if P(A and B) = P(A)P(B).

If in a contingency table of characteristics having a size of 2×2, one of these conditions is met for at least one combination of events A And B, it will be valid for any other combination.

Unconditional probability of an elementary event

(5) P(A) = P(A|B 1)P(B 1) + P(A|B 2)P(B 2) + … + P(A|B k)P(B k)

where events B 1, B 2, ... B k are mutually exclusive and exhaustive.

Let us illustrate the application of this formula using the example of Fig. 1. Using formula (5), we obtain:

P(A) = P(A|B 1)P(B 1) + P(A|B 2)P(B 2)

Where P(A)- the likelihood that the purchase was planned, P(B 1)- the probability that the purchase is made, P(B 2)- the probability that the purchase is not completed.

BAYES' THEOREM

The conditional probability of an event takes into account information that some other event has occurred. This approach can be used both to refine the probability taking into account newly received information, and to calculate the probability that the observed effect is a consequence of some specific reason. The procedure for refining these probabilities is called Bayes' theorem. It was first developed by Thomas Bayes in the 18th century.

Let's assume that the company mentioned above is researching the market for a new TV model. In the past, 40% of the TVs created by the company were successful, while 60% of the models were not recognized. Before announcing the release of a new model, marketing specialists carefully research the market and record demand. In the past, 80% of successful models were predicted to be successful, while 30% of successful predictions turned out to be wrong. The marketing department gave a favorable forecast for the new model. What is the likelihood that a new TV model will be in demand?

Bayes' theorem can be derived from the definitions of conditional probability (1) and (2). To calculate the probability P(B|A), take formula (2):

and substitute instead of P(A and B) the value from formula (3):

P(A and B) = P(A|B) * P(B)

Substituting formula (5) instead of P(A), we obtain Bayes’ theorem:

where events B 1, B 2, ... B k are mutually exclusive and exhaustive.

Let us introduce the following notation: event S - TV is in demand, event S’ - TV is not in demand, event F - favorable prognosis, event F’ - poor prognosis. Let’s assume that P(S) = 0.4, P(S’) = 0.6, P(F|S) = 0.8, P(F|S’) = 0.3. Applying Bayes' theorem we get:

The probability of demand for a new TV model, given a favorable forecast, is 0.64. Thus, the probability of lack of demand given a favorable forecast is 1–0.64=0.36. The calculation process is shown in Fig. 4.

Rice. 4. (a) Calculations using the Bayes formula to estimate the probability of demand for televisions; (b) Decision tree when studying demand for a new TV model

Let's look at an example of using Bayes' theorem for medical diagnostics. The probability that a person suffers from a particular disease is 0.03. A medical test can check if this is true. If a person is truly sick, the probability of an accurate diagnosis (saying that the person is sick when he really is sick) is 0.9. If a person is healthy, the probability of a false positive diagnosis (saying that a person is sick when he is healthy) is 0.02. Let's assume that medical test gave positive result. What is the probability that a person is actually sick? What is the likelihood of an accurate diagnosis?

Let us introduce the following notation: event D - the person is sick, event D’ - the person is healthy, event T - diagnosis is positive, event T’ - diagnosis negative. From the conditions of the problem it follows that P(D) = 0.03, P(D’) = 0.97, P(T|D) = 0.90, P(T|D’) = 0.02. Applying formula (6), we obtain:

The probability that with a positive diagnosis a person is really sick is 0.582 (see also Fig. 5). Please note that the denominator of the Bayes formula is equal to the probability of a positive diagnosis, i.e. 0.0464.

Brief theory

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 a random event is a number that expresses the measure of the objective possibility of an event occurring.

The quantities that determine how significant the objective reasons are to expect the occurrence of an event are characterized by the probability of the event. It must be emphasized that probability is an objective quantity that exists independently of the knower and is conditioned by the entire set of conditions that contribute to the occurrence of an event.

The explanations we have given for the concept of probability are not a mathematical definition, since they do not quantify the concept. There are several definitions of the probability of a random event, which are widely used in solving specific problems (classical, axiomatic, statistical, etc.).

Classic definition of event probability reduces this concept to the more elementary concept of equally possible events, which is no longer subject to definition and is assumed to be intuitively clear. For example, if a die is a homogeneous cube, then the loss of any of the faces of this cube will be equally possible events.

Let a reliable event be divided into equally possible cases, the sum of which gives the event. That is, the cases into which it breaks down are called favorable for the event, since the appearance of one of them ensures the occurrence.

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 total number of the only 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 n, the number of cases m favorable for a given event, and then perform the calculation using the above formula.

The probability of an event equal to the ratio of the number of experimental outcomes favorable to the event to the total number of experimental outcomes is called classical probability random event.

The following properties of probability follow from the definition:

Property 1. The probability of a reliable event is equal to one.

Property 2. The probability of an impossible event is zero.

Property 3. The probability of a random event is a positive number between zero and one.

Property 4. The probability of the occurrence of events that form a complete group is equal to one.

Property 5. Probability of occurrence opposite event is determined in the same way as the probability of the occurrence of event A.

The number of cases favoring the occurrence of an opposite event. Hence, the probability of the occurrence of the opposite event is equal to the difference between unity and the probability of the occurrence of event A:

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.

When a set of conditions is met, a reliable event will definitely happen, but an impossible event will definitely not happen. Among the events that may or may not occur when a set of conditions is created, the occurrence of some can be counted on with good reason, and the occurrence of others with less reason. If, for example, there are more white balls in an urn than black balls, then there is more reason to hope for the appearance of a white ball when drawn from the urn at random than for the appearance of a black ball.

Example of problem solution

Example 1

A box contains 8 white, 4 black and 7 red balls. 3 balls are drawn at random. Find the probabilities of the following events: – at least 1 red ball is drawn, – there are at least 2 balls of the same color, – there are at least 1 red and 1 white ball.

The solution of the problem

We find the total number of test outcomes as the number of combinations of 19 (8+4+7) elements of 3:

Let's find the probability of the event– at least 1 red ball is drawn (1,2 or 3 red balls)

Required probability:

Let the event– there are at least 2 balls of the same color (2 or 3 white balls, 2 or 3 black balls and 2 or 3 red balls)

Number of outcomes favorable to the event:

Required probability:

Let the event– there is at least one red and 1 white ball

(1 red, 1 white, 1 black or 1 red, 2 white or 2 red, 1 white)

Number of outcomes favorable to the event:

Required probability:

Answer: P(A)=0.773;P(C)=0.7688; P(D)=0.6068

Example 2

Two dice are thrown. Find the probability that the sum of points is at least 5.

Solution

Let the event be a score of at least 5

Let's use the classic definition of probability:

Total number of possible test outcomes

Number of trials favoring the event of interest

On the fallen edge of the first dice one point, two points..., six points may appear. similarly, six outcomes are possible when rolling the second die. Each of the outcomes of throwing the first die can be combined with each of the outcomes of the second. Thus, the total number of possible elementary test outcomes is equal to the number of placements with repetitions (choice with placements of 2 elements from a set of volume 6):

Let's find the probability of the opposite event - the sum of points is less than 5

The following combinations of dropped points will favor the event:

Explained geometric definition probability and a solution to the well-known meeting problem is given.

I understand that everyone wants to know in advance how the sporting event will end, who will win and who will lose. With this information, you can bet on sporting events without fear. But is it even possible, and if so, how to calculate the probability of an event?

Probability is a relative quantity, therefore it cannot speak with certainty about any event. This value allows you to analyze and evaluate the need to place a bet on a particular competition. Determining probabilities is a whole science, requiring careful study and understanding.

Probability coefficient in probability theory

In sports betting, there are several options for the outcome of the competition:

• first team victory;
• victory of the second team;
• draw;
• total

Each outcome of the competition has its own probability and frequency with which this event will occur, subject to conservation initial characteristics. As we said earlier, it is impossible to accurately calculate the probability of any event - it may or may not coincide. Thus, your bet can either win or lose.

There cannot be a 100% accurate prediction of the results of the competition, since many factors influence the outcome of the match. Naturally, bookmakers do not know the outcome of the match in advance and only assume the result, making decisions using their analysis system and offering certain odds for betting.

How to calculate the probability of an event?

Let’s assume that the bookmaker’s odds are 2.1/2 – we get 50%. It turns out that coefficient 2 is equal to the probability of 50%. Using the same principle, you can get a break-even probability ratio - 1/probability.

Many players think that after several repeated defeats, a win will definitely happen - this is a mistaken opinion. The probability of winning a bet does not depend on the number of losses. Even if you flip several heads in a row in a coin game, the probability of flipping tails remains the same - 50%.