In fact, formulas (1) and (2) are a short record of the conditional probability based on the contingency table of features. Let's return to the example considered (Fig. 1). Let's say we know that a certain family is going to buy a widescreen TV. What is the probability that this family will actually buy such a TV?
Rice. 1. Widescreen TV Buyer Behavior
In this case, we need to calculate the conditional probability P (the purchase was made | the purchase was planned). Since we know that a family is planning to buy, the sample space does not consist of all 1,000 families, but only those that are planning to buy a widescreen TV. Of the 250 such families, 200 actually bought this TV. Therefore, the probability that a family will actually buy a widescreen TV, if they planned to do so, can be calculated using the following formula:
P (purchase made | purchase planned) = number of families planning and purchasing a widescreen TV / number of families planning to buy a widescreen TV = 200 / 250 = 0.8
The same result is given by formula (2):
where is the event BUT is that the family plans to buy a widescreen TV, and the event AT- that she will actually buy it. Substituting real data into the formula, we get:
decision tree
On fig. 1 families were divided into four categories: those who planned to buy a widescreen TV and those who did not, and those who bought such a TV and those who did not. A similar classification can be done 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 is divided into two additional branches, corresponding to families who bought and did not buy a widescreen TV. The probabilities written at the ends of the two main branches are the unconditional probabilities of events BUT and BUT'. The probabilities written at the ends of the four additional branches are the conditional probabilities of each combination of events BUT and AT. 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 widescreen TV, if they planned to do so, one should 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 one) answer:
Statistical independence
In the example of buying a widescreen TV, the probability that a randomly selected family purchased a widescreen 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 widescreen TV is 300/1000 = 0.3. A very important conclusion follows from this. A priori information that the family was planning a purchase affects the probability 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)- event probability BUT assuming an event has occurred AT, P(A) is the unconditional probability of event A.
Please note that the events BUT and AT P(A|B) = P(A). If in the feature contingency table, which has a size of 2 × 2, this condition is satisfied for at least one combination of events BUT and AT, it will be valid for any other combination. In our example, the 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 whether they are satisfied with their purchase (Fig. 3). Determine if the degree of satisfaction with the purchase and the type of TV are related.
Rice. 3. Customer Satisfaction Data for Widescreen TVs
According to 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 has bought an HDTV is equal, and these events are statistically independent, since they are not related to each other.
Probability multiplication rule
The formula for calculating the conditional probability allows you to determine the probability of a joint event A and B. Resolving formula (1)
with respect to the joint probability P(A and B), we obtain the general rule for multiplication of probabilities. Event Probability A and B is equal to the probability of the event BUT provided that the event AT AT:
(3) P(A and B) = P(A|B) * P(B)
Consider, for example, 80 households who purchased a widescreen HDTV (Figure 3). The table shows that 64 families are satisfied with the purchase and 16 are not. Suppose that two families are randomly selected among them. Determine the probability that both buyers will be satisfied. Using formula (3), we obtain:
P(A and B) = P(A|B) * P(B)
where is the event BUT is that the second family is satisfied with their purchase, and the event AT- 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 probability that the second family is also satisfied with their purchase depends on the response of the first family. If the first family is not returned to the sample after the survey (selection without return), the number of respondents drops to 79. If the first family was satisfied with their purchase, the probability that the second family will also be satisfied is 63/79, since only 63 remained 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 is returned to the sample. Determine the probability that both families will be satisfied with their purchase. In this case, the probabilities that both families are satisfied with their purchase are the same, and 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 in formula (3) the conditional probability P(A|B) probability P(A), we obtain a formula for multiplying the probabilities of independent events.
Rule for multiplying the probabilities of independent events. If events BUT and AT are statistically independent, the probability of an event A and B is equal to the probability of the event BUT multiplied by the probability of the event AT.
(4) P(A and B) = P(A)P(B)
If this rule is true for events BUT and AT, which means they are statistically independent. Thus, there are two ways to determine the statistical independence of two events:
- Events BUT and AT are statistically independent of each other if and only if P(A|B) = P(A).
- Events BUT and B are statistically independent of each other if and only if P(A and B) = P(A)P(B).
If in the feature contingency table, which has a size of 2 × 2, one of these conditions is satisfied for at least one combination of events BUT and B, it will be valid for any other combination.
Unconditional probability of an elementary event
(5) Р(А) = P(A|B 1)Р(B 1) + P(A|B 2)Р(B 2) + … + P(A|B k)Р(B k)
where events B 1 , B 2 , … B k are mutually exclusive and exhaustive.
We illustrate the application of this formula on 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 probability 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 made.
BAYES' THEOREM
The conditional probability of an event takes into account the 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 the result of some specific cause. The procedure for refining these probabilities is called Bayes' theorem. It was first developed by Thomas Bayes in the 18th century.
Suppose 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, and 60% of the models were not recognized. Before announcing the release of a new model, marketers carefully research the market and capture demand. In the past, the success of 80% of models that received recognition was predicted in advance, while 30% of favorable forecasts turned out to be wrong. For the new model, the marketing department gave a favorable forecast. 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 Р(В|А), we take the 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 the Bayes theorem:
where the 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 not in demand, event F - favorable prognosis, event F' - poor prognosis. Let's say 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, subject to a favorable forecast, is 0.64. Thus, the probability of lack of demand under the condition of a favorable forecast is 1–0.64=0.36. The calculation process is shown in fig. 4.
Rice. 4. (a) Bayesian calculations to estimate the probability of TV demand; (b) Decision tree for researching demand for a new TV model
Let's consider an example of application of Bayes' theorem for medical diagnostics. The probability that a person suffers from a certain disease is 0.03. A medical test allows you to check if this is so. If a person is really sick, the probability of an accurate diagnosis (stating that a person is sick when he is really sick) is 0.9. If a person is healthy, the probability of a false positive diagnosis (stating that a person is sick when they are healthy) is 0.02. Let's say a medical test came back positive. What is the probability that the person is actually sick? What is the likelihood of an accurate diagnosis?
Let us introduce the following notation: event D - man is sick, event D' - the person is healthy, event T - positive diagnosis, event T' - the diagnosis is negative. It follows from the conditions of the problem that Р(D) = 0.03, P(D’) = 0.97, Р(T|D) = 0.90, P(T|D’) = 0.02. Applying formula (6), we obtain:
The probability that a person with a positive diagnosis is really sick is 0.582 (see also Fig. 5). Note that the denominator of the Bayes formula is equal to the probability of a positive diagnosis, i.e. 0.0464.
as an ontological category reflects the measure of the possibility of the emergence of any entity in any conditions. In contrast to the mathematical and logical interpretations of this concept, ontological V. does not associate itself with the necessity of a quantitative expression. The value of V. is revealed in the context of understanding determinism and the nature of development in general.
Great Definition
Incomplete definition ↓
PROBABILITY
a concept that characterizes quantities. a measure of the possibility of the appearance 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 the mathematical. analysis of gambling and most fully developed by B. Pascal, J. Bernoulli and P. Laplace, considers V. as the ratio of the number of favorable cases to the total number of all equally possible. For example, when throwing a dice with 6 sides, each of them can be expected to come up with a V equal to 1/6, since neither side has advantages over the other. Such symmetry of the outcomes of experience 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 statistical. V.'s concepts, at the heart of which are valid. observation of the appearance of a certain event during the duration. experience under precisely fixed conditions. Practice confirms that the more often an event occurs, the greater the degree of the objective possibility of its occurrence, or V. Therefore, the statistical. V.'s interpretation is based on the concept of relates. frequencies, a cut can be determined empirically. V. as theoretical. the concept never coincides with an empirically determined frequency, however, in many ways. cases, it practically differs little from the relative. frequency found as a result of the duration. observations. Many statisticians regard V. as a "double" refers. frequency, edge is determined by statistical. study of observational results
or experiments. Less realistic was the definition of V. as the limit relates. frequencies of mass events, or collectives, proposed by R. Mises. As a further development of the frequency approach to V., a dispositional, or propensity, interpretation of V. is put forward (K. Popper, J. Hecking, M. Bunge, T. Setl). According to this interpretation, V. characterizes the property of generating conditions, for example. experiment. installation, to obtain a sequence of massive random events. It is this attitude that gives rise to the physical dispositions, or predispositions, V. to-rykh can be checked by means of relative. frequencies.
Statistical V.'s interpretation dominates the scientific. knowledge, because it reflects the specific. the nature of the patterns inherent in mass phenomena of a random nature. In many physical, biological, economic, demographic and other social processes, it is necessary to take into account the action of many random factors, to-rye are characterized by a stable frequency. Identification of this stable frequency and quantities. its assessment with the help of V. makes it possible to reveal the necessity, which makes its way through the cumulative action of many accidents. This is where the dialectic of the transformation of chance into necessity finds its manifestation (see F. Engels, in the book: K. Marx and F. Engels, Soch., 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 credibility, with precisely formulated premises, can sometimes be estimated with the help of V. The value of this V. is most often determined by comparing. concepts (greater than, less than or equal to), and sometimes in a numerical way. Logic interpretation is often used to analyze inductive reasoning and build various systems of probabilistic logics (R. Carnap, R. Jeffrey). In the semantic logical concepts. V. is often defined as the degree of confirmation of one statement by others (for example, the hypothesis of its empirical data).
In connection with the development of theories of decision-making and games, the so-called. personalistic interpretation of V. Although V. in this case 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 calculation 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.
From epistemological t. sp. difference between statistic., logical. and personalistic interpretations of V. lies in the fact that if the first characterizes the objective properties and relations 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 cognition. The specificity of the probabilistic view of the world is revealed through the inclusion of the concepts of chance, independence and hierarchy (ideas of levels in the structure and determination of systems) among the basic concepts of being.
Ideas about probability originated in antiquity and were related to the characteristics of our knowledge, while the presence of probabilistic knowledge was recognized, which differs from reliable knowledge and from false. The impact of the idea of probability on scientific thinking, on the development of knowledge is directly related to the development of the theory of probability as a mathematical discipline. The origin of the mathematical doctrine of probability dates back to the 17th century, when the development of the core of concepts that allow. quantitative (numerical) characteristics and expressing a probabilistic idea.
Intensive applications of probability to the development of knowledge fall on the 2nd floor. 19- 1st floor. 20th century Probability has entered the structures of such fundamental sciences of nature as classical statistical physics, genetics, quantum theory, 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, features of the probabilistic way of thinking, it is necessary to proceed from the analysis of the subject of probability theory and the foundations of its many applications. Probability theory is usually defined as a mathematical discipline that studies the laws 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 very mass nature of phenomena has a stable structure, 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 distribution of probabilities, i.e., the assignment of subsystems and their corresponding probabilities. The language of probability theory is the language of probability distributions. Accordingly, the theory of probability is defined as the abstract science of operating with distributions.
Probability gave rise in science to ideas about statistical regularities and statistical systems. The latter are 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 in order to form systems that have integral characteristics, it is necessary that sufficiently stable bonds exist between their elements, which cement the systems. The stability of statistical systems is given by the presence of external conditions, the 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 that characterizes 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 significance of probabilistic methods in cognition lies in the fact that they allow us to explore 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, for a very long time, such an understanding of probability dominated in science, 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 assess the degree of confirmation (reliability, truth) of an inductive conclusion (hypothetical conclusion) in a quantitative form? In the course of the formation of the theory of probability, 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 information at the disposal of a given person, his experience, views on the world and the psychological mindset. In all such cases, the magnitude of the probability is not amenable to strict measurements and practically lies outside the competence of probability theory as a consistent mathematical discipline.
An objective, frequency interpretation of probability was established in science with considerable difficulty. 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 formation of probabilistic methods in physics occurred under the decisive influence of the ideas of mechanics: statistical systems were treated simply as mechanical ones. Since the corresponding problems were not solved by strict methods of mechanics, statements arose that the appeal to probabilistic methods and statistical regularities is the result of the incompleteness of our knowledge. In the history of the development of classical statistical physics, numerous attempts have been made to justify it on the basis of classical mechanics, but they all failed. The basis of probability is that it expresses the features of the structure of a certain class of systems, other than systems of mechanics: 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 cognition leads to the denial of the concept of rigid determinism, to the denial of the basic model of being and cognition 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 connected with the disclosure of the internal dynamics of objects and systems, which cannot be completely 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 of modern science to analytical methods for studying complexly organized systems and the physical and mathematical foundations of self-organization phenomena.
Great Definition
Incomplete definition ↓
Probability event is the ratio of the number of elementary outcomes that favor a given event to the number of all equally possible outcomes of experience in which this event may occur. The probability of an event A is denoted by P(A) (here P is the first letter of the French word probabilite - probability). According to the definition
(1.2.1)
where is the number of elementary outcomes favoring event A; - the number of all equally possible elementary outcomes of experience, forming a complete group of events.
This definition of probability is called classical. It arose at the initial stage of the development of probability theory.
The probability of an event has the following properties:
1. The probability of a certain event is equal to one. Let's designate a certain event by the letter . For a certain event, therefore
(1.2.2)
2. The probability of an impossible event is zero. We denote the impossible event by the letter . For an impossible event, therefore
(1.2.3)
3. The probability of a random event is expressed as a positive number less than one. Since the inequalities , or are satisfied for a random event, then
(1.2.4)
4. The probability of any event satisfies the inequalities
(1.2.5)
This follows from relations (1.2.2) -(1.2.4).
Example 1 An urn contains 10 balls of the same size and weight, of which 4 are red and 6 are blue. One ball is drawn from the urn. What is the probability that the drawn ball is blue?
Decision. The event "the drawn ball turned out to be blue" will be denoted by the letter A. This test has 10 equally possible elementary outcomes, of which 6 favor the event A. In accordance with formula (1.2.1), we obtain 
Example 2 All natural numbers from 1 to 30 are written on identical cards and placed in an urn. After thoroughly mixing the cards, one card is removed from the urn. What is the probability that the number on the card drawn is a multiple of 5?
Decision. Denote by A the event "the number on the taken card is a multiple of 5". In this test, there are 30 equally possible elementary outcomes, of which 6 outcomes favor event A (numbers 5, 10, 15, 20, 25, 30). Hence, 
Example 3 Two dice are thrown, the sum of points on the upper faces is calculated. Find the probability of the event B, consisting in the fact that the top faces of the cubes will have a total of 9 points.
Decision. There are 6 2 = 36 equally possible elementary outcomes in this trial. Event B is favored by 4 outcomes: (3;6), (4;5), (5;4), (6;3), so 
Example 4. A natural number not exceeding 10 is chosen at random. What is the probability that this number is prime?
Decision. Denote by the letter C the event "the chosen number is prime". In this case, n = 10, m = 4 (primes 2, 3, 5, 7). Therefore, the desired probability 
Example 5 Two symmetrical coins are tossed. What is the probability that both coins have digits on the top sides?
Decision. Let's denote by the letter D the event "there was a number on the top side of each coin". There are 4 equally possible elementary outcomes in this test: (G, G), (G, C), (C, G), (C, C). (The notation (G, C) means that on the first coin there is a coat of arms, on the second - a number). Event D is favored by one elementary outcome (C, C). Since m = 1, n = 4, then 
Example 6 What is the probability that the digits in a randomly chosen two-digit number are the same?
Decision. Two-digit numbers are numbers from 10 to 99; there are 90 such numbers in total. 9 numbers have the same digits (these are the numbers 11, 22, 33, 44, 55, 66, 77, 88, 99). Since in this case m = 9, n = 90, then 
,
where A is the "number with the same digits" event.
Example 7 From the letters of the word differential one letter is chosen at random. What is the probability that this letter will be: a) a vowel b) a consonant c) a letter h?
Decision. There are 12 letters in the word differential, of which 5 are vowels and 7 are consonants. Letters h this word does not. Let's denote the events: A - "vowel", B - "consonant", C - "letter h". The number of favorable elementary outcomes: - for event A, - for event B, - for event C. Since n \u003d 12, then
, and .
Example 8 Two dice are tossed, the number of points on the top face of each dice is noted. Find the probability that both dice have the same number of points.
Decision. Let us denote this event by the letter A. Event A is favored by 6 elementary outcomes: (1;]), (2;2), (3;3), (4;4), (5;5), (6;6). In total there are equally possible elementary outcomes that form a complete group of events, in this case n=6 2 =36. So the desired probability 
Example 9 The book has 300 pages. What is the probability that a randomly opened page will have a sequence number that is a multiple of 5?
Decision. It follows from the conditions of the problem that there will be n = 300 of all equally possible elementary outcomes that form a complete group of events. Of these, m = 60 favor the occurrence of the specified event. Indeed, a number that is a multiple of 5 has the form 5k, where k is a natural number, and , whence 
. Hence, 
, where A - the "page" event has a sequence number that is a multiple of 5".
Example 10. Two dice are thrown, the sum of points on the upper faces is calculated. What is more likely to get a total of 7 or 8?
Decision. Let's designate the events: A - "7 points fell out", B - "8 points fell out". Event A is favored by 6 elementary outcomes: (1; 6), (2; 5), (3; 4), (4; 3), (5; 2), (6; 1), and event B - by 5 outcomes: (2; 6), (3; 5), (4; 4), (5; 3), (6; 2). There are n = 6 2 = 36 of all equally possible elementary outcomes. Hence, and .
So, P(A)>P(B), that is, getting a total of 7 points is a more likely event than getting a total of 8 points.
Tasks
1. A natural number not exceeding 30 is chosen at random. What is the probability that this number is a multiple of 3?
2. In the urn a red and b blue balls of the same size and weight. What is the probability that a randomly drawn ball from this urn is blue?
3. A number not exceeding 30 is chosen at random. What is the probability that this number is a divisor of zo?
4. In the urn a blue and b red balls of the same size and weight. One ball is drawn from this urn and set aside. This ball is red. Then another ball is drawn from the urn. Find the probability that the second ball is also red.
5. A natural number not exceeding 50 is chosen at random. What is the probability that this number is prime?
6. Three dice are thrown, the sum of points on the upper faces is calculated. What is more likely - to get a total of 9 or 10 points?
7. Three dice are tossed, the sum of the dropped points is calculated. What is more likely to get a total of 11 (event A) or 12 points (event B)?
Answers
1. 1/3. 2 . b/(a+b). 3 . 0,2. 4 . (b-1)/(a+b-1). 5 .0,3.6 . p 1 \u003d 25/216 - the probability of getting 9 points in total; p 2 \u003d 27/216 - the probability of getting 10 points in total; p2 > p1 7 . P(A) = 27/216, P(B) = 25/216, P(A) > P(B).
Questions
1. What is called the probability of an event?
2. What is the probability of a certain event?
3. What is the probability of an impossible event?
4. What are the limits of the probability of a random event?
5. What are the limits of the probability of any event?
6. What definition of probability is called classical?
A professional better should be well versed in odds, quickly and correctly evaluate the probability of an event by a coefficient and, if necessary, be able convert odds from one format to another. In this manual, we will talk about what types of coefficients are, as well as using examples, we will analyze how you can calculate the probability from a known coefficient and vice versa.
What are the types of coefficients?
There are three main types of odds offered by bookmakers: decimal odds, fractional odds(English) and american odds. The most common odds in Europe are decimal. American odds are popular in North America. Fractional odds are the most traditional type, they immediately reflect information about how much you need to bet in order to get a certain amount.
Decimal Odds
Decimals or else they are called European odds- this is the usual number format, represented by a decimal fraction with an accuracy of hundredths, and sometimes even thousandths. An example of a decimal odd is 1.91. Calculating your profit with decimal odds is very easy, just multiply your bet amount by that odd. For example, in the match "Manchester United" - "Arsenal", the victory of "MU" is set with a coefficient of 2.05, a draw is estimated at a coefficient of 3.9, and the victory of "Arsenal" is equal to - 2.95. Let's say we're confident United will win and bet $1,000 on them. Then our possible income is calculated as follows:
2.05 * $1000 = $2050;
Isn't it really that difficult? In the same way, the possible income is calculated when betting on a draw and the victory of Arsenal.
Draw: 3.9 * $1000 = $3900;
Arsenal win: 2.95 * $1000 = $2950;
How to calculate the probability of an event by decimal odds?
Imagine now that we need to determine the probability of an event by the decimal odds set by the bookmaker. This is also very easy to do. To do this, we divide the unit by this coefficient.
Let's take the data we already have and calculate the probability of each event:
Manchester United win: 1 / 2.05 = 0,487 = 48,7%;
Draw: 1 / 3.9 = 0,256 = 25,6%;
Arsenal win: 1 / 2.95 = 0,338 = 33,8%;
Fractional Odds (English)
As the name implies fractional coefficient represented by an ordinary fraction. An example of an English odd is 5/2. The numerator of the fraction contains a number that is the potential amount of net winnings, and the denominator contains a number indicating the amount that you need to bet in order to receive this winnings. Simply put, we have to wager $2 dollars to win $5. Odds of 3/2 means that in order to get $3 of net winnings, we will have to bet $2.
How to calculate the probability of an event by fractional odds?
The probability of an event by fractional coefficients is also not difficult to calculate, you just need to divide the denominator by the sum of the numerator and denominator.
For the fraction 5/2, we calculate the probability: 2 / (5+2) = 2 / 7 = 0,28 = 28%;
For the fraction 3/2, we calculate the probability:
American odds
American odds unpopular in Europe, but very unpopular in North America. Perhaps this type of coefficients is the most difficult, but this is only at first glance. In fact, there is nothing complicated in this type of coefficients. Now let's take a look at everything in order.
The main feature of American odds is that they can be either positive, and negative. An example of American odds is (+150), (-120). The American odds (+150) means that in order to earn $150 we need to bet $100. In other words, a positive American multiplier reflects potential net earnings at a $100 bet. The negative American coefficient reflects the amount of the bet that must be made in order to receive a net winning of $100. For example, the coefficient (- 120) tells us that by betting $120 we will win $100.
How to calculate the probability of an event using American odds?
The probability of an event according to the American odds is calculated according to the following formulas:
(-(M)) / ((-(M)) + 100), where M is a negative American coefficient;
100/(P+100), where P is a positive American coefficient;
For example, we have a coefficient (-120), then the probability is calculated as follows:
(-(M)) / ((-(M)) + 100); we substitute the value (-120) instead of "M";
(-(-120)) / ((-(-120)) + 100 = 120 / (120 + 100) = 120 / 220 = 0,545 = 54,5%;
Thus, the probability of an event with an American coefficient (-120) is 54.5%.
For example, we have a coefficient (+150), then the probability is calculated as follows:
100/(P+100); we substitute the value (+150) instead of "P";
100 / (150 + 100) = 100 / 250 = 0,4 = 40%;
Thus, the probability of an event with an American coefficient (+150) is 40%.
How, knowing the percentage of probability, translate it into a decimal coefficient?
In order to calculate the decimal coefficient for a known percentage of probability, you need to divide 100 by the probability of an event in percent. For example, if the probability of an event is 55%, then the decimal coefficient of this probability will be equal to 1.81.
100 / 55% = 1,81
How, knowing the percentage of probability, translate it into a fractional coefficient?
In order to calculate a fractional coefficient from a known percentage of probability, you need to subtract one from dividing 100 by the probability of an event in percent. For example, we have a probability percentage of 40%, then the fractional coefficient of this probability will be equal to 3/2.
(100 / 40%) - 1 = 2,5 - 1 = 1,5;
The fractional coefficient is 1.5/1 or 3/2.
How, knowing the percentage of probability, translate it into an American coefficient?
If the probability of an event is more than 50%, then the calculation is made according to the formula:
- ((V) / (100 - V)) * 100, where V is the probability;
For example, we have an 80% probability of an event, then the American coefficient of this probability will be equal to (-400).
- (80 / (100 - 80)) * 100 = - (80 / 20) * 100 = - 4 * 100 = (-400);
If the probability of an event is less than 50%, then the calculation is made according to the formula:
((100 - V) / V) * 100, where V is the probability;
For example, if we have a probability percentage of an event of 20%, then the American coefficient of this probability will be equal to (+400).
((100 - 20) / 20) * 100 = (80 / 20) * 100 = 4 * 100 = 400;
How to convert the coefficient to another format?
There are times when it is necessary to convert coefficients from one format to another. For example, we have a fractional coefficient 3/2 and we need to convert it to decimal. To convert a fractional odds to decimal odds, we first determine the probability of an event with a fractional odds, and then convert this probability to a decimal odds.
The probability of an event with a fractional coefficient of 3/2 is 40%.
2 / (3+2) = 2 / 5 = 0,4 = 40%;
Now we translate the probability of an event into a decimal coefficient, for this we divide 100 by the probability of an event as a percentage:
100 / 40% = 2.5;
Thus, a fractional odd of 3/2 is equal to a decimal odd of 2.5. In a similar way, for example, American odds are converted to fractional, decimal to American, etc. The hardest part of all this is just the calculations.
I understand that everyone wants to know in advance how a sporting event will end, who will win and who will lose. With this information, you can bet on sports events without fear. But is it possible at all, and if so, how to calculate the probability of an event?
Probability is a relative value, therefore it cannot speak with accuracy about any event. This value allows you to analyze and evaluate the need to place a bet on a particular competition. The definition of probabilities is a whole science that requires careful study and understanding.
Probability coefficient in probability theory
In sports betting, there are several options for the outcome of the competition:
- victory of the first team;
- victory of the second team;
- draw;
- total
Each outcome of the competition has its own probability and frequency with which this event will occur, provided that the initial characteristics are preserved. As mentioned 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 can be no exact 100% prediction of the results of the competition, since many factors influence the outcome of the match. Naturally, the bookmakers do not know the outcome of the match in advance and only assume the result, making a decision on their analysis system and offer certain odds for bets.
How to calculate the probability of an event?
Let's say that the odds of the bookmaker is 2.1/2 - we get 50%. It turns out that the coefficient 2 is equal to the probability of 50%. By the same principle, you can get a break-even probability ratio - 1 / probability.
Many players think that after a few repeated losses, a win will definitely happen - this is an erroneous opinion. The probability of winning a bet does not depend on the number of losses. Even if you throw several heads in a row in a coin game, the probability of throwing tails remains the same - 50%.
