This is the ratio of the number of those observations in which the event in question occurred to the total number of observations. This interpretation is acceptable in the case of a sufficiently large number of observations or experiments. For example, if about half of the people you meet on the street are women, then you can say that the probability that the person you meet on the street will be a woman is 1/2. In other words, an estimate of the probability of an event can be the frequency of its occurrence in a long series of independent repetitions of a random experiment.

Probability in mathematics

In the modern mathematical approach, classical (that is, not quantum) probability is given by Kolmogorov’s axiomatics. Probability is a measure P, which is defined on the set X, called probability space. This measure must have the following properties:

From these conditions it follows that the probability measure P also has the property additivity: if sets A 1 and A 2 do not intersect, then . To prove you need to put everything A 3 , A 4 , ... equal to the empty set and apply the property of countable additivity.

The probability measure may not be defined for all subsets of the set X. It is enough to define it on a sigma algebra, consisting of some subsets of the set X. In this case, random events are defined as measurable subsets of space X, that is, as elements of sigma algebra.

Probability sense

When we find that the reasons for some possible fact actually occurring outweigh the contrary reasons, we consider that fact probable, otherwise - incredible. This preponderance of positive bases over negative ones, and vice versa, can represent an indefinite set of degrees, as a result of which probability(And improbability) It happens more or less .

Complex individual facts do not allow for an exact calculation of the degrees of their probability, but even here it is important to establish some large subdivisions. So, for example, in the legal field, when a personal fact subject to trial is established on the basis of testimony, it always remains, strictly speaking, only probable, and it is necessary to know how significant this probability is; in Roman law, a quadruple division was adopted here: probatio plena(where the probability practically turns into reliability), Further - probatio minus plena, then - probatio semiplena major and finally probatio semiplena minor .

In addition to the question of the probability of the case, the question may arise, both in the field of law and in the moral field (with a certain ethical point of view), of how likely it is that a given particular fact constitutes a violation of the general law. This question, which served as the main motive in the religious jurisprudence of the Talmud, also gave rise to Roman Catholic moral theology (especially with late XVI centuries) very complex systematic constructions and a huge literature, dogmatic and polemical (see Probabilism).

The concept of probability allows for a certain numerical expression when applied only to such facts that are part of certain homogeneous series. So (in the simplest example), when someone throws a coin a hundred times in a row, we find here one general or large series (the sum of all falls of the coin), consisting of two private or smaller, in this case numerically equal, series (falls " heads" and falls "tails"); The probability that in this time the coin will land on heads, that is, that this new member of the general series will belong to this of the two smaller series, is equal to the fraction expressing the numerical relation between this small series and the larger one, namely 1/2, that is, the same probability belongs to one or the other of the two particular series. In less simple examples, the conclusion cannot be deduced directly from the data of the problem itself, but requires prior induction. So, for example, the question is: what is the probability for a given newborn to live to be 80 years old? Here there should be a general, or large, series of known number people born in similar conditions and dying at different ages (this number should be large enough to eliminate random deviations, and small enough to maintain the homogeneity of the series, because for a person born, for example, in St. Petersburg in a wealthy cultural family , the entire million-strong population of the city, a significant part of which consists of people from various groups who may die prematurely - soldiers, journalists, workers in dangerous professions - represents a group too heterogeneous for a real determination of probability); let this general row consist of ten thousand human lives; it includes smaller series representing the number of people surviving to a particular age; one of these smaller series represents the number of people living to age 80. But it is impossible to determine the number of this smaller series (like all others) a priori; this is done purely inductively, through statistics. Let's say statistical research found that out of 10,000 middle-class St. Petersburg residents, only 45 live to be 80; Thus, this smaller series is related to the larger one as 45 is to 10,000, and the probability for a given person to belong to this smaller series, that is, to live to be 80 years old, is expressed as a fraction of 0.0045. The study of probability from a mathematical point of view constitutes a special discipline - probability theory.

see also

Notes

Literature

• Alfred Renyi. Letters on probability / trans. from Hungarian D. Saas and A. Crumley, eds. B.V. Gnedenko. M.: Mir. 1970
• Gnedenko B.V. Probability theory course. M., 2007. 42 p.
• Kuptsov V.I. Determinism and probability. M., 1976. 256 p.

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See what “Probability” is in other dictionaries: General scientific and philosophical. a category denoting the quantitative degree of possibility of the occurrence of mass random events under fixed observation conditions, characterizing the stability of their relative frequencies. In logic, semantic degree... ...

Philosophical Encyclopedia PROBABILITY, a number in the range from zero to one inclusive, representing the possibility of a given event occurring. The probability of an event is defined as the ratio of the number of chances that an event can occur to the total number of possible... ...

In all likelihood.. Dictionary of Russian synonyms and similar expressions. under. ed. N. Abramova, M.: Russian Dictionaries, 1999. probability possibility, likelihood, chance, objective possibility, maza, admissibility, risk. Ant. impossibility... ... Synonym dictionary

probability- A measure that an event is likely to occur. Note The mathematical definition of probability is: “a real number between 0 and 1 that is associated with a random event.” The number may reflect the relative frequency in a series of observations... ... Technical Translator's Guide

Probability- “a mathematical, numerical characteristic of the degree of possibility of the occurrence of any event in certain specific conditions that can be repeated an unlimited number of times.” Based on this classic... ... Economic and mathematical dictionary

- (probability) The possibility of the occurrence of an event or a certain result. It can be presented in the form of a scale with divisions from 0 to 1. If the probability of an event is zero, its occurrence is impossible. With a probability equal to 1, the onset of... Dictionary of business terms

Classical and statistical definition of probability

For practical activities it is necessary to be able to compare events according to the degree of possibility of their occurrence. Let's consider a classic case. There are 10 balls in the urn, 8 of them white, 2 black. Obviously, the event “a white ball will be drawn from the urn” and the event “a black ball will be drawn from the urn” have different degrees of possibility of their occurrence. Therefore, to compare events, a certain quantitative measure is needed.

A quantitative measure of the possibility of an event occurring is probability . The most widely used definitions of the probability of an event are classical and statistical.

Classic definition probability is associated with the concept of a favorable outcome. Let's look at this in more detail.

Let the outcomes of some test form a complete group of events and are equally possible, i.e. uniquely possible, incompatible and equally possible. Such outcomes are called elementary outcomes, or cases. It is said that the test boils down to case diagram or " urn scheme", because Any probability problem for such a test can be replaced by an equivalent problem with urns and balls of different colors.

The outcome is called favorable event A, if the occurrence of this case entails the occurrence of the event A.

According to the classical definition probability of an event A is equal to the ratio of the number of outcomes favorable to this event to the total number of outcomes, i.e.

Where P(A)– probability of event A; m– number of cases favorable to the event A; n – total number cases.

Example 1.1. When throwing a dice, there are six possible outcomes: 1, 2, 3, 4, 5, 6 points. What is the probability of getting an even number of points?

Solution. All n= 6 outcomes form a complete group of events and are equally possible, i.e. uniquely possible, incompatible and equally possible. Event A - “the appearance of an even number of points” - is favored by 3 outcomes (cases) - the loss of 2, 4 or 6 points. Using the classical formula for the probability of an event, we obtain

P(A) = = . ◄

Based classical definition probability of an event, we note its properties:

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

0 ≤ R(A) ≤ 1.

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

3. The probability of an impossible event is zero.

As stated earlier, the classical definition of probability is applicable only to those events that can occur as a result of tests that have symmetry possible outcomes, i.e. reducible to a pattern of cases. However, there is big class events whose probabilities cannot be calculated using the classical definition.

For example, if we assume that the coin is flattened, then it is obvious that the events “appearance of a coat of arms” and “appearance of heads” cannot be considered equally possible. Therefore, the formula for determining the probability according to the classical scheme is not applicable in this case.

However, there is another approach to estimating the probability of events, based on how often a given event will occur in the trials performed. In this case, the statistical definition of probability is used.

Statistical probabilityevent A is the relative frequency (frequency) of occurrence of this event in n trials performed, i.e.

,

(1.2)

Where P*(A)– statistical probability of an event A; w(A)– relative frequency of the event A; m– number of trials in which the event occurred A; n– total number of tests.

Unlike mathematical probability P(A), considered in the classical definition, statistical probability P*(A) is a characteristic experienced, experimental. In other words, the statistical probability of an event A is the number around which the relative frequency is stabilized (set) w(A) with an unlimited increase in the number of tests carried out under the same set of conditions.

For example, when they say about a shooter that he hits the target with a probability of 0.95, this means that out of hundreds of shots fired by him under certain conditions (the same target at the same distance, the same rifle, etc. .), on average there are about 95 successful ones. Naturally, not every hundred will have 95 successful shots, sometimes there will be fewer, sometimes more, but on average, when shooting is repeated many times under the same conditions, this percentage of hits will remain unchanged. The figure of 0.95, which serves as an indicator of the shooter's skill, is usually very stable, i.e. the percentage of hits in most shootings will be almost the same for a given shooter, only in rare cases deviating any significantly from its average value.

Another disadvantage of the classical definition of probability ( 1.1 ) limiting its use is that it assumes a finite number of possible test outcomes. In some cases, this disadvantage can be overcome by using a geometric definition of probability, i.e. finding the probability of a point falling into a certain area (segment, part of a plane, etc.).

Let the flat figure g forms part of a flat figure G(Fig. 1.1). Fit G a dot is thrown at random. This means that all points in the region G“equal rights” with respect to whether a thrown random point hits it. Assuming that the probability of an event A– the thrown point hits the figure g– is proportional to the area of ​​this figure and does not depend on its location relative to G, neither from the form g, we'll find

A professional bettor must have a good understanding of the odds, quickly and correctly estimate the probability of an event by coefficient and, if necessary, be able to convert odds from one format to another. In this manual we will talk about what types of coefficients there are, and also use examples to show how you can calculate the probability using a known coefficient and vice versa.

What types of odds are there?

There are three main types of odds that bookmakers offer players: decimal odds, fractional odds(English) and American odds. The most common odds in Europe are decimal. IN North America American odds are popular. Fractional odds are the most traditional look, they immediately reflect information about how much you need to bet to get a certain amount.

Decimal odds

Decimal or they are also called European odds is the familiar number format represented by decimal accurate to hundredths, and sometimes even thousandths. An example of a decimal odd is 1.91. Calculating profit in the case of decimal odds is very simple; you just need to multiply the amount of your bet by this odds. For example, in the match “Manchester United” - “Arsenal”, the victory of “Manchester United” is set with a coefficient of 2.05, a draw is estimated with 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 we bet $1,000 on them. Then our possible income is calculated as follows:

2.05 * $1000 = $2050;

It's really not that complicated, is it?! The possible income is calculated in the same way when betting on a draw or victory for Arsenal.

Draw: 3.9 * $1000 = $3900;
Arsenal win: 2.95 * $1000 = $2950;

How to calculate the probability of an event using decimal odds?

Now imagine that we need to determine the probability of an event based on the decimal odds set by the bookmaker. This is also done very simply. To do this, we divide one by this coefficient.

Let's take the existing data 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 suggests fractional coefficient represented by an ordinary fraction. An example of English odds is 5/2. The numerator of the fraction contains a number that is the potential amount of the net winnings, and the denominator contains a number indicating the amount that must be bet in order to receive this winning. Simply put, we have to bet $2 dollars to win $5. Odds of 3/2 means that in order to get $3 in net winnings, we will have to bet $2.

How to calculate the probability of an event using fractional odds?

It is also not difficult to calculate the probability of an event using fractional odds; 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 much so in North America. Perhaps this type of coefficients is the most complex, but this is only at first glance. In fact, there is nothing complicated in this type of coefficients. Now let's figure it all out in order.

The main feature of American odds is that they can be either positive, so negative. Example of American odds - (+150), (-120). The American odds (+150) means that in order to earn $150 we need to bet $100. In other words, a positive American coefficient reflects the potential net earnings at a bet of $100. A negative American odds reflect the amount of bet that needs to be made in order to get a net win 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 using the American coefficient is calculated using 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); substitute the value (-120) for “M”;
(-(-120)) / ((-(-120)) + 100 = 120 / (120 + 100) = 120 / 220 = 0,545 = 54,5%;

Thus, the probability of an event with American odds (-120) is 54.5%.

For example, we have a coefficient (+150), then the probability is calculated as follows:

100/(P+100); substitute the value (+150) for “P”;
100 / (150 + 100) = 100 / 250 = 0,4 = 40%;

Thus, the probability of an event with American odds (+150) is 40%.

How, knowing the percentage of probability, convert it into a decimal coefficient?

In order to calculate the decimal coefficient by known percentage the probability must be divided by 100 by the probability of the event as a percentage. For example, 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, convert it into a fractional coefficient?

In order to calculate the fractional coefficient based on a known percentage of probability, you need to subtract one from dividing 100 by the probability of an event as a percentage. For example, if 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, convert it into an American coefficient?

If the probability of an event is more than 50%, then the calculation is made using the formula:

- ((V) / (100 - V)) * 100, where V is probability;

For example, if the probability of an event is 80%, 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 using the formula:

((100 - V) / V) * 100, where V is probability;

For example, if we have a percentage probability 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 odds from one format to another. For example, we have a fractional odds of 3/2 and we need to convert it to decimal. To convert a fractional odds to a decimal odds, we first determine the probability of an event with a fractional odds, and then convert this probability into a decimal odds.

The probability of an event with a fractional odds of 3/2 is 40%.

2 / (3+2) = 2 / 5 = 0,4 = 40%;

Now let’s convert the probability of an event into a decimal coefficient; to do this, divide 100 by the probability of the event as a percentage:

100 / 40% = 2.5;

Thus, the fractional odds of 3/2 are equal to the decimal odds of 2.5. In a similar way, for example, American odds are converted to fractional, decimal to American, etc. The most difficult thing in all this is just the calculations.

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

For a long time probability theory did not have a clear definition. It was formulated only in 1929. The emergence of probability theory as a science dates back to the Middle Ages and the first attempts at mathematical analysis of gambling (flake, dice, roulette). French mathematicians of the 17th century Blaise Pascal and Pierre Fermat, while studying the prediction of winnings in gambling, discovered the first probabilistic patterns that arise when throwing dice.

Probability theory arose as a science from the belief that mass random events are based on certain patterns. Probability theory studies these patterns.

Probability theory deals with the study of events whose occurrence is not known with certainty. It allows you to judge the degree of probability of the occurrence of some events compared to others.

For example: it is impossible to determine unambiguously the result of “heads” or “tails” as a result of tossing a coin, but with repeated tossing, approximately the same number of “heads” and “tails” appear, which means that the probability that “heads” or “tails” will fall ", is equal to 50%.

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

The test result is event. The event happens:

1. Reliable (always occurs as a result of testing).
2. Impossible (never happens).
3. Random (may or may not occur as a result of the test).

For example, when tossing a coin possible event- the coin will be on its edge, a random event - the appearance of “heads” or “tails”. The specific test result is called elementary event. As a result of the test, only elementary events occur. The set of all possible, different, specific test outcomes is called space of elementary events.

Basic concepts of the theory

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

Random value- this is a quantity that, as a result of testing, can take one or another value, and it is not known in advance which one. For example: number per fire station per day, number of hits with 10 shots, etc.

Random variables can be divided into two categories.

1. Discrete random variable is a quantity that, as a result of testing, can take on certain values ​​with a certain probability, forming a countable set (a set whose elements can be numbered). This set can be either finite or infinite. For example, the number of shots before the first hit on the target is a discrete random variable, because this quantity can take on an infinite, albeit countable, number of values.
2. Continuous random variable is a quantity that can take any value from some finite or infinite interval. Obviously, the number of possible values ​​of a continuous random variable is infinite.

Probability space- concept introduced by A.N. Kolmogorov in the 30s of the 20th century to formalize the concept of probability, which gave rise to the rapid development of probability theory as a strict mathematical discipline.

A probability space is a triple (sometimes enclosed in angle brackets: , where

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

De Moivre-Laplace theorem- one of the limit theorems of probability theory, established by Laplace in 1812. It states that the number of successes when repeating the same random experiment over and over again with two possible outcomes is approximately normally distributed. It allows you to find an approximate probability value.

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

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

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

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

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

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

where the symbol denotes the mathematical expectation.

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

Simplest form of law large numbers- This is Bernoulli's theorem, which states that if the probability of an event is the same in all trials, then as the number of trials increases, the frequency of the event tends to the probability of the event and ceases to be random.

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

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

Methods for estimating probability based on finite sample analysis are based on this property. A clear example is the forecast of election results based on a survey of a sample of voters.

Central limit theorems- a class of theorems in probability theory stating that the sum of a sufficiently large number of weakly dependent random variables that have approximately the same scales (none of the terms dominates or makes a determining contribution to the sum) has a distribution close to normal.

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

Initially, being just a collection of information and empirical observations about the game of dice, the theory of probability became a thorough science. The first to give it a mathematical framework were Fermat and Pascal.

From thinking about the eternal to the theory of probability

The two individuals to whom probability theory owes many of its fundamental formulas, Blaise Pascal and Thomas Bayes, are known as deeply religious people, the latter being a Presbyterian minister. Apparently, the desire of these two scientists to prove the fallacy of the opinion about a certain Fortune giving good luck to her favorites gave impetus to research in this area. After all, in fact, any gambling with its wins and losses, it is just a symphony of mathematical principles.

Thanks to the passion of the Chevalier de Mere, who was equally a gambler and a man not indifferent to science, Pascal was forced to find a way to calculate probability. De Mere was interested in the following question: “How many times do you need to throw two dice in pairs so that the probability of getting 12 points exceeds 50%?” The second question, which was of great interest to the gentleman: “How to divide the bet between the participants in the unfinished game?” Of course, Pascal successfully answered both questions of de Mere, who became the unwitting initiator of the development of probability theory. It is interesting that the person of de Mere remained known in this area, and not in literature.

Previously, no mathematician had ever attempted to calculate the probabilities of events, since it was believed that this was only a guessing solution. Blaise Pascal gave the first definition of the probability of an event and showed that it is a specific figure that can be justified mathematically. Probability theory has become the basis for statistics and is widely used in modern science.

What is randomness

If we consider a test that can be repeated an infinite number of times, then we can define a random event. This is one of probable outcomes experience.

Experience is the implementation of specific actions under constant conditions.

To be able to work with the results of the experiment, events are usually designated by the letters A, B, C, D, E...

Probability of a random event

In order to begin the mathematical part of probability, it is necessary to define all its components.

The probability of an event is a numerical measure of the possibility of some event (A or B) occurring as a result of an experience. The probability is denoted as P(A) or P(B).

In probability theory they distinguish:

• reliable the event is guaranteed to occur as a result of the experience P(Ω) = 1;
• impossible the event can never happen P(Ø) = 0;
• random an event lies between reliable and impossible, that is, the probability of its occurrence is possible, but not guaranteed (the probability of a random event is always within the range 0≤Р(А)≤ 1).

Relationships between events

Both one and the sum of events A+B are considered, when the event is counted when at least one of the components, A or B, or both, A and B, is fulfilled.

In relation to each other, events can be:

• Equally possible.
• Compatible.
• Incompatible.
• Opposite (mutually exclusive).
• Dependent.

If two events can happen with equal probability, then they equally possible.

If the occurrence of event A does not reduce to zero the probability of the occurrence of event B, then they compatible.

If events A and B never occur simultaneously in the same experience, then they are called incompatible. Coin toss - good example: the appearance of heads is automatically the non-appearance of heads.

The probability for the sum of such incompatible events consists of the sum of the probabilities of each of the events:

P(A+B)=P(A)+P(B)

If the occurrence of one event makes the occurrence of another impossible, then they are called opposite. Then one of them is designated as A, and the other - Ā (read as “not A”). The occurrence of event A means that Ā did not occur. These two events form a complete group with a sum of probabilities equal to 1.

Dependent events have mutual influence, decreasing or increasing the probability of each other.

Relationships between events. Examples

Using examples it is much easier to understand the principles of probability theory and combinations of events.

The experiment that will be carried out consists of taking balls out of a box, and the result of each experiment is an elementary outcome.

An event is one of the possible outcomes of an experiment - a red ball, a blue ball, a ball with number six, etc.

Test No. 1. There are 6 balls involved, three of which are blue with odd numbers on them, and the other three are red with even numbers.

Test No. 2. 6 balls involved of blue color with numbers from one to six.

Based on this example, we can name combinations:

• Reliable event. In Spanish No. 2 the event “get the blue ball” is reliable, since the probability of its occurrence is equal to 1, since all the balls are blue and there can be no miss. Whereas the event “get the ball with the number 1” is random.
• Impossible event. In Spanish No. 1 with blue and red balls, the event “getting a purple ball” is impossible, since the probability of its occurrence is 0.
• Equally possible events. In Spanish No. 1, the events “get the ball with the number 2” and “get the ball with the number 3” are equally possible, and the events “get the ball with an even number” and “get the ball with the number 2” have different probabilities.
• Compatible Events. Getting a six twice in a row while throwing a die is a compatible event.
• Incompatible events. In the same Spanish No. 1, the events “get a red ball” and “get a ball with an odd number” cannot be combined in the same experience.
• Opposite events. Most shining example This is coin tossing, where drawing heads is equivalent to not drawing tails, and the sum of their probabilities is always 1 (full group).
• Dependent Events. So, in Spanish No. 1, you can set the goal of drawing the red ball twice in a row. Retrieving it or not retrieving it the first time affects the probability of retrieving it the second time.

It can be seen that the first event significantly affects the probability of the second (40% and 60%).

Event probability formula

The transition from fortune-telling to precise data occurs through the translation of the topic into a mathematical plane. That is, judgments about a random event such as “high probability” or “minimal probability” can be translated into specific numerical data. It is already permissible to evaluate, compare and enter such material into more complex calculations.

From a calculation point of view, determining the probability of an event is the ratio of the number of elementary positive outcomes to the number of all possible outcomes of experience regarding a specific event. Probability is denoted by P(A), where P stands for the word “probabilite”, which is translated from French as “probability”.

So, the formula for the probability of an event is:

Where m is the number of favorable outcomes for event A, n is the sum of all outcomes possible for this experience. In this case, the probability of an event always lies between 0 and 1:

0 ≤ P(A)≤ 1.

Calculation of the probability of an event. Example

Let's take Spanish. No. 1 with balls, which was described earlier: 3 blue balls with the numbers 1/3/5 and 3 red balls with the numbers 2/4/6.

Based on this test, several different problems can be considered:

• A - red ball falling out. There are 3 red balls, and there are 6 options in total. This is simplest example, in which the probability of the event is equal to P(A)=3/6=0.5.
• B - rolling an even number. There are 3 even numbers (2,4,6), and the total number of possible numerical options is 6. The probability of this event is P(B)=3/6=0.5.
• C - the occurrence of a number greater than 2. There are 4 such options (3,4,5,6) out of a total number of possible outcomes of 6. The probability of event C is equal to P(C)=4/6=0.67.

As can be seen from the calculations, event C has a higher probability, since the number of probable positive outcomes is higher than in A and B.

Incompatible events

Such events cannot appear simultaneously in the same experience. As in Spanish No. 1 it is impossible to get a blue and a red ball at the same time. That is, you can get either a blue or a red ball. In the same way, an even and an odd number cannot appear in a dice at the same time.

The probability of two events is considered as the probability of their sum or product. The sum of such events A+B is considered to be an event that consists of the occurrence of event A or B, and the product of them AB is the occurrence of both. For example, the appearance of two sixes at once on the faces of two dice in one throw.

The sum of several events is an event that presupposes the occurrence of at least one of them. The production of several events is the joint occurrence of them all.

In probability theory, as a rule, the use of the conjunction “and” denotes a sum, and the conjunction “or” - multiplication. Formulas with examples will help you understand the logic of addition and multiplication in probability theory.

Probability of the sum of incompatible events

If the probability is considered joint events, then the probability of the sum of events is equal to the addition of their probabilities:

P(A+B)=P(A)+P(B)

For example: let's calculate the probability that in Spanish. No. 1 with blue and red balls, a number between 1 and 4 will appear. We will calculate not in one action, but by the sum of the probabilities of the elementary components. So, in such an experiment there are only 6 balls or 6 of all possible outcomes. The numbers that satisfy the condition are 2 and 3. The probability of getting a 2 is 1/6, the probability of getting a 3 is also 1/6. The probability of getting a number between 1 and 4 is:

The probability of the sum of incompatible events of a complete group is 1.

So, if in an experiment with a cube we add up the probabilities of all numbers appearing, the result will be one.

This is also true for opposite events, for example in the experiment with a coin, where one side is event A, and the other is opposite eventĀ, as is known,

P(A) + P(Ā) = 1

Probability of incompatible events occurring

Probability multiplication is used when considering the occurrence of two or more incompatible events in one observation. The probability that events A and B will appear in it simultaneously is equal to the product of their probabilities, or:

P(A*B)=P(A)*P(B)

For example, the probability that in Spanish No. 1, as a result of two attempts, a blue ball will appear twice, equal to

That is, the probability of an event occurring when, as a result of two attempts to extract balls, only blue balls are extracted is 25%. It is very easy to do practical experiments on this problem and see if this is actually the case.

Joint events

Events are considered joint when the occurrence of one of them can coincide with the occurrence of another. Despite the fact that they are joint, the probability of independent events is considered. For example, throwing two dice can give a result when the number 6 appears on both of them. Although the events coincided and appeared at the same time, they are independent of each other - only one six could fall out, the second die has no influence on it.

The probability of joint events is considered as the probability of their sum.

Probability of the sum of joint events. Example

The probability of the sum of events A and B, which are joint in relation to each other, is equal to the sum of the probabilities of the event minus the probability of their occurrence (that is, their joint occurrence):

R joint (A+B)=P(A)+P(B)- P(AB)

Let's assume that the probability of hitting the target with one shot is 0.4. Then event A is hitting the target in the first attempt, B - in the second. These events are joint, since it is possible that you can hit the target with both the first and second shots. But events are not dependent. What is the probability of the event of hitting the target with two shots (at least with one)? According to the formula:

0,4+0,4-0,4*0,4=0,64

The answer to the question is: “The probability of hitting the target with two shots is 64%.”

This formula for the probability of an event can also be applied to incompatible events, where the probability of the joint occurrence of an event P(AB) = 0. This means that the probability of the sum of incompatible events can be considered a special case of the proposed formula.

Geometry of probability for clarity

Interestingly, the probability of the sum of joint events can be represented as two areas A and B, which intersect with each other. As can be seen from the picture, the area of ​​their union is equal to the total area minus the area of ​​their intersection. This geometric explanation makes the seemingly illogical formula more understandable. Note that geometric solutions are not uncommon in probability theory.

Determining the probability of the sum of many (more than two) joint events is quite cumbersome. To calculate it, you need to use the formulas that are provided for these cases.

Dependent Events

Events are called dependent if the occurrence of one (A) of them affects the probability of the occurrence of another (B). Moreover, the influence of both the occurrence of event A and its non-occurrence is taken into account. Although events are called dependent by definition, only one of them is dependent (B). Ordinary probability was denoted as P(B) or the probability of independent events. In the case of dependent events, a new concept is introduced - conditional probability P A (B), which is the probability of a dependent event B subject to the occurrence of event A (hypothesis) on which it depends.

But event A is also random, so it also has a probability that needs and can be taken into account in the calculations performed. The following example will show how to work with dependent events and a hypothesis.

An example of calculating the probability of dependent events

A good example for calculating dependent events would be a standard deck of cards.

Using a deck of 36 cards as an example, let’s look at dependent events. We need to determine the probability that the second card drawn from the deck will be of diamonds if the first card drawn is:

1. Bubnovaya.
2. A different color.

Obviously, the probability of the second event B depends on the first A. So, if the first option is true, that there is 1 card (35) and 1 diamond (8) less in the deck, the probability of event B:

R A (B) =8/35=0.23

If the second option is true, then the deck has 35 cards, and the full number of diamonds (9) is still retained, then the probability of the following event B:

R A (B) =9/35=0.26.

It can be seen that if event A is conditioned on the fact that the first card is a diamond, then the probability of event B decreases, and vice versa.

Multiplying dependent events

Guided by the previous chapter, we accept the first event (A) as a fact, but in essence, it is of a random nature. The probability of this event, namely drawing a diamond from a deck of cards, is equal to:

P(A) = 9/36=1/4

Since the theory does not exist on its own, but is intended to serve for practical purposes, it is fair to note that what is most often needed is the probability of producing dependent events.

According to the theorem on the product of probabilities of dependent events, the probability of the occurrence of jointly dependent events A and B is equal to the probability of one event A, multiplied by the conditional probability of event B (dependent on A):

P(AB) = P(A) *P A(B)

Then, in the deck example, the probability of drawing two cards with the suit of diamonds is:

9/36*8/35=0.0571, or 5.7%

And the probability of extracting not diamonds first, and then diamonds, is equal to:

27/36*9/35=0.19, or 19%

It can be seen that the probability of event B occurring is greater provided that the first card of a suit other than diamonds is drawn. This result is quite logical and understandable.

Total probability of an event

When a problem with conditional probabilities becomes multifaceted, it cannot be calculated using conventional methods. When there are more than two hypotheses, namely A1,A2,…,A n, ..forms a complete group of events provided:

• P(A i)>0, i=1,2,…
• A i ∩ A j =Ø,i≠j.
• Σ k A k =Ω.

So, the formula for the total probability for event B at full group random events A1,A2,…,And n is equal to:

A look into the future

The probability of a random event is extremely necessary in many areas of science: econometrics, statistics, physics, etc. Since some processes cannot be described deterministically, since they themselves are probabilistic in nature, special working methods are required. The probability theory of an event can be used in any technological field as a way to determine the possibility of an error or malfunction.

We can say that by recognizing probability, we in some way take a theoretical step into the future, looking at it through the prism of formulas.