Brought to date in open jar Unified State Examination problems in mathematics (mathege.ru), the solution of which is based on only one formula, which is classic definition probabilities.
The easiest way to understand the formula is with examples.
Example 1. There are 9 red balls and 3 blue balls in the basket. The balls differ only in color. We take out one of them at random (without looking). What is the probability that the ball chosen in this way will be blue?
Comment. In problems in probability theory, something happens (in in this case our action of pulling out the ball), which can have a different result - outcome. It should be noted that the result can be looked at in different ways. “We pulled out some kind of ball” is also a result. “We pulled out the blue ball” - the result. “We pulled out exactly this ball from all possible balls” - this least generalized view of the result is called an elementary outcome. It is the elementary outcomes that are meant in the formula for calculating the probability.
Solution. Now let's calculate the probability of choosing the blue ball.
Event A: “the selected ball turned out to be blue”
Total number of all possible outcomes: 9+3=12 (the number of all balls that we could draw)
Number of outcomes favorable for event A: 3 (the number of such outcomes in which event A occurred - that is, the number of blue balls)
P(A)=3/12=1/4=0.25
Answer: 0.25
For the same problem, let's calculate the probability of choosing a red ball.
The total number of possible outcomes will remain the same, 12. Number of favorable outcomes: 9. Probability sought: 9/12=3/4=0.75
The probability of any event always lies between 0 and 1.
Sometimes in everyday speech(but not in probability theory!) the probability of events is estimated as a percentage. The transition between math and conversational scores is accomplished by multiplying (or dividing) by 100%.
So,
Moreover, the probability is zero for events that cannot happen - incredible. For example, in our example this would be the probability of drawing a green ball from the basket. (The number of favorable outcomes is 0, P(A)=0/12=0, if calculated using the formula)
Probability 1 has events that are absolutely certain to happen, without options. For example, the probability that “the selected ball will be either red or blue” is for our task. (Number of favorable outcomes: 12, P(A)=12/12=1)
We have reviewed classic example, illustrating the definition of probability. All similar Unified State Examination tasks According to probability theory, they are solved by using this formula.
In place of the red and blue balls there may be apples and pears, boys and girls, learned and unlearned tickets, tickets containing and not containing a question on a certain topic (prototypes,), defective and high-quality bags or garden pumps (prototypes,) - the principle remains the same.
They differ slightly in the formulation of the theory problem probability of the Unified State Exam, where you need to calculate the probability of an event occurring on a specific day. ( , ) As in previous problems, you need to determine what is the elementary outcome, and then apply the same formula.
Example 2. The conference lasts three days. On the first and second days there are 15 speakers each, on the third day - 20. What is the probability that Professor M.’s report will fall on the third day if the order of reports is determined by drawing lots?
What is the elementary outcome here? – Assigning the professor’s report one of all possible serial numbers for the speech. 15+15+20=50 people participate in the draw. Thus, Professor M.'s report may receive one of 50 issues. That means elementary outcomes only 50.
What are the favorable outcomes? - Those in which it turns out that the professor will speak on the third day. That is, the last 20 numbers.
According to the formula, probability P(A)= 20/50=2/5=4/10=0.4
Answer: 0.4
The drawing of lots here represents the establishment of a random correspondence between people and ordered places. In example 2, the establishment of correspondence was considered from the point of view of which of the places could be taken specific person. You can approach the same situation from the other side: which of the people with what probability could get to a specific place (prototypes , , , ):
Example 3. The draw includes 5 Germans, 8 French and 3 Estonians. What is the probability that the first (/second/seventh/last – it doesn’t matter) will be a Frenchman.
Number of elementary outcomes – number of all possible people, which could, by drawing lots, get to this place. 5+8+3=16 people.
Favorable outcomes - French. 8 people.
Required probability: 8/16=1/2=0.5
Answer: 0.5
The prototype is slightly different. There are still problems about coins () and dice (), which are somewhat more creative. The solution to these problems can be found on the prototype pages.
Here are a few examples of tossing a coin or dice.
Example 4. When we toss a coin, what is the probability of landing on heads?
There are 2 outcomes – heads or tails. (it is believed that the coin never lands on its edge) A favorable outcome is tails, 1.
Probability 1/2=0.5
Answer: 0.5.
Example 5. What if we toss a coin twice? What is the probability of getting heads both times?
The main thing is to determine what elementary outcomes we will consider when tossing two coins. After tossing two coins, one of the following results can occur:
1) PP – both times it came up heads
2) PO – first time heads, second time heads
3) OP – heads the first time, tails the second time
4) OO – heads came up both times
There are no other options. This means that there are 4 elementary outcomes. Only the first one, 1, is favorable.
Probability: 1/4=0.25
Answer: 0.25
What is the probability that two coin tosses will result in tails?
The number of elementary outcomes is the same, 4. Favorable outcomes are the second and third, 2.
Probability of getting one tail: 2/4=0.5
In such problems, another formula may be useful.
If during one toss of a coin possible options we have 2 results, then for two throws the results will be 2 2 = 2 2 = 4 (as in example 5), for three throws 2 2 2 = 2 3 = 8, for four: 2 2 2 2 =2 4 =16, ... for N throws the possible results will be 2·2·...·2=2 N .
So, you can find the probability of getting 5 heads out of 5 coin tosses.
Total number of elementary outcomes: 2 5 =32.
Favorable outcomes: 1. (RRRRRR – heads all 5 times)
Probability: 1/32=0.03125
The same is true for dice. With one throw, there are 6 possible results. So, for two throws: 6 6 = 36, for three 6 6 6 = 216, etc.
Example 6. We throw the dice. What is the probability that an even number will be rolled?
Total outcomes: 6, according to the number of sides.
Favorable: 3 outcomes. (2, 4, 6)
Probability: 3/6=0.5
Example 7. We throw two dice. What is the probability that the total will be 10? (round to the nearest hundredth)
For one die there are 6 possible outcomes. This means that for two, according to the above rule, 6·6=36.
What outcomes will be favorable for the total to roll 10?
10 must be decomposed into the sum of two numbers from 1 to 6. This can be done in two ways: 10=6+4 and 10=5+5. This means that the following options are possible for the cubes:
(6 on the first and 4 on the second)
(4 on the first and 6 on the second)
(5 on the first and 5 on the second)
Total, 3 options. Required probability: 3/36=1/12=0.08
Answer: 0.08
Other types of B6 problems will be discussed in a future How to Solve article.
The need to act on probabilities occurs when the probabilities of some events are known, and it is necessary to calculate the probabilities of other events that are associated with these events.
Addition of probabilities is used when you need to calculate the probability of a combination or logical sum of random events.
Sum of events A And B denote A + B or A ∪ B. The sum of two events is an event that occurs if and only if at least one of the events occurs. This means that A + B- an event that occurs if and only if the event occurred during observation A or event B, or simultaneously A And B.
If events A And B are mutually inconsistent and their probabilities are given, then the probability that one of these events will occur as a result of one trial is calculated using the addition of probabilities.
Probability addition theorem. The probability that one of two things will happen is mutually exclusive joint events, is equal to the sum of the probabilities of these events:
For example, while hunting, two shots are fired. Event A– hitting a duck with the first shot, event IN– hit from the second shot, event ( A+ IN) – a hit from the first or second shot or from two shots. So, if two events A And IN– incompatible events, then A+ IN– the occurrence of at least one of these events or two events.
Example 1. There are 30 balls in a box same sizes: 10 red, 5 blue and 15 white. Calculate the probability that a colored (not white) ball will be picked up without looking.
Solution. Let us assume that the event A- “the red ball is taken”, and the event IN- “The blue ball was taken.” Then the event is “a colored (not white) ball is taken.” Let's find the probability of the event A:
and events IN:
Events A And IN– mutually incompatible, since if one ball is taken, then the balls cannot be taken different colors. Therefore, we use the addition of probabilities:
The theorem for adding probabilities for several incompatible events. If events constitute a complete set of events, then the sum of their probabilities is equal to 1:
The sum of the probabilities of opposite events is also equal to 1:
Opposite events form a complete set of events, and the probability of a complete set of events is 1.
Probabilities of opposite events are usually indicated in small letters p And q. In particular,
what follows following formulas probabilities of opposite events:
Example 2. The target in the shooting range is divided into 3 zones. The probability that a certain shooter will shoot at the target in the first zone is 0.15, in the second zone – 0.23, in the third zone – 0.17. Find the probability that the shooter will hit the target and the probability that the shooter will miss the target.
Solution: Find the probability that the shooter will hit the target:
Let's find the probability that the shooter will miss the target:
For more complex problems, in which you need to use both addition and multiplication of probabilities, see the page "Various problems involving addition and multiplication of probabilities".
Addition of probabilities of mutually simultaneous events
Two random events are called joint if the occurrence of one event does not exclude the occurrence of a second event in the same observation. For example, when throwing a die the event A The number 4 is considered to be rolled out, and the event IN– loss even number. Since 4 is an even number, the two events are compatible. In practice, there are problems of calculating the probabilities of the occurrence of one of the mutually simultaneous events.
Probability addition theorem for joint events. The probability that one of the joint events will occur is equal to the sum of the probabilities of these events, from which the probability is subtracted general offensive both events, that is, the product of probabilities. The formula for the probabilities of joint events has the following form:
Since events A And IN compatible, event A+ IN occurs if one of three possible events occurs: or AB. According to the theorem of addition of incompatible events, we calculate as follows:
Event A will occur if one of two incompatible events occurs: or AB. However, the probability of the occurrence of one event from several incompatible events is equal to the sum of the probabilities of all these events:
Likewise:
Substituting expressions (6) and (7) into expression (5), we obtain the probability formula for joint events:
When using formula (8), it should be taken into account that events A And IN may be:
- mutually independent;
- mutually dependent.
Probability formula for mutually independent events:
Probability formula for mutually dependent events:
If events A And IN are inconsistent, then their coincidence is an impossible case and, thus, P(AB) = 0. The fourth probability formula for incompatible events is:
Example 3. In auto racing, when you drive the first car, you have a better chance of winning, and when you drive the second car. Find:
- the probability that both cars will win;
- the probability that at least one car will win;
1) The probability that the first car will win does not depend on the result of the second car, so the events A(the first car wins) and IN(the second car will win) – independent events. Let's find the probability that both cars win:
2) Find the probability that one of the two cars will win:
For more complex problems, in which you need to use both addition and multiplication of probabilities, see the page "Various problems involving addition and multiplication of probabilities".
Solve the addition of probabilities problem yourself, and then look at the solution
Example 4. Two coins are tossed. Event A- loss of the coat of arms on the first coin. Event B- loss of the coat of arms on the second coin. Find the probability of an event C = A + B .
Multiplying Probabilities
Probability multiplication is used when the probability of a logical product of events must be calculated.
In this case, random events must be independent. Two events are said to be mutually independent if the occurrence of one event does not affect the probability of the occurrence of the second event.
Probability multiplication theorem for independent events. Probability of simultaneous occurrence of two independent events A And IN is equal to the product of the probabilities of these events and is calculated by the formula:
Example 5. The coin is tossed three times in a row. Find the probability that the coat of arms will appear all three times.
Solution. The probability that the coat of arms will appear on the first toss of a coin, the second time, and the third time. Let's find the probability that the coat of arms will appear all three times:
Solve probability multiplication problems on your own and then look at the solution
Example 6. There is a box of nine new tennis balls. To play, three balls are taken, and after the game they are put back. When choosing balls, played balls are not distinguished from unplayed balls. What is the probability that after three games will there be any unplayed balls left in the box?
Example 7. 32 letters of the Russian alphabet are written on cut-out alphabet cards. Five cards are drawn at random one after another and placed on the table in order of appearance. Find the probability that the letters will form the word "end".
Example 8. From a full deck of cards (52 sheets), four cards are taken out at once. Find the probability that all four of these cards will be of different suits.
Example 9. The same task as in example 8, but each card after being removed is returned to the deck.
More complex problems, in which you need to use both addition and multiplication of probabilities, as well as calculate the product of several events, can be found on the page "Various problems involving addition and multiplication of probabilities".
The probability that at least one of the mutually independent events will occur can be calculated by subtracting from 1 the product of the probabilities of opposite events, that is, using the formula:
Example 10. Cargo is delivered by three modes of transport: river, rail and road transport. Probability that the cargo will be delivered river transport, is 0.82, by rail 0.87, by motor transport 0.90. Find the probability that the cargo will be delivered by at least one of three types transport.
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.
Excellent definition
PROBABILITY
concept characterizing quantities. the measure of the possibility of the occurrence of a certain event at a certain conditions. In scientific in cognition there are three interpretations of V. Classic concept V., which arose from mathematics. analysis 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 total number all equally possible. 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 its appearance, 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 mass events, or collectives, 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. 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 probabilistic logics (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 psychological factors. 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 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 inclusion in the number basic concepts the existence of the concepts of randomness, independence and hierarchy (ideas of levels in the structure and determination of systems).
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, on the development of cognition 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 entered into the structures of such basic sciences about nature, such 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 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, not 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 characteristics individual elements And holistic characteristics systems: the latter seem to be 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 assess 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 strict methods mechanics, then claims arose that appeal to probabilistic methods and statistical laws are the result of the incompleteness of our knowledge. In the history of the development of classical statistical physics Numerous attempts have been made to substantiate it on the basis classical mechanics, however, 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. Basic models, represented by statistical theories, have a different, more general character: These include 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 fully determined 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.
Excellent definition
Incomplete definition ↓
probability- a number between 0 and 1 that reflects the chances that a random event will occur, where 0 is complete absence the probability of an event occurring, and 1 means that the event in question will definitely occur.
The probability of event E is a number from to 1.
The sum of the probabilities of mutually exclusive events is equal to 1.
empirical probability- probability, which is calculated as the relative frequency of an event in the past, extracted from the analysis of historical data.
The probability of very rare events cannot be calculated empirically.
subjective probability- probability based on personal subjective assessment events without regard to historical data. Investors who make decisions to buy and sell shares often act based on considerations of subjective probability.
prior probability -
The chance is 1 in... (odds) that an event will occur through the concept of probability. The chance of an event occurring is expressed through probability as follows: P/(1-P).
For example, if the probability of an event is 0.5, then the chance of the event is 1 out of 2 because 0.5/(1-0.5).
The chance that an event will not occur is calculated using the formula (1-P)/P
Inconsistent probability- for example, the price of shares of company A takes into account 85% possible event E, and in the share price of company B by only 50%. This is called inconsistent probability. According to the Dutch Betting Theorem, inconsistent probability creates profit opportunities.
Unconditional probability is the answer to the question “What is the probability that the event will occur?”
Conditional probability - this is the answer to the question: “What is the probability of event A if event B occurs.” Conditional probability is denoted as P(A|B).
Joint probability- the probability that events A and B will occur simultaneously. Denoted as P(AB).
P(A|B) = P(AB)/P(B) (1)
P(AB) = P(A|B)*P(B)
Rule for summing up probabilities:
The probability that either event A or event B will happen is
P (A or B) = P(A) + P(B) - P(AB) (2)
If events A and B are mutually exclusive, then
P (A or B) = P(A) + P(B)
Independent events - events A and B are independent if
P(A|B) = P(A), P(B|A) = P(B)
That is, it is a sequence of results where the probability value is constant from one event to the next.
A coin toss is an example of such an event - the result of each subsequent toss does not depend on the result of the previous one.
Dependent Events - these are events where the probability of the occurrence of one depends on the probability of the occurrence of another.
The rule for multiplying the probabilities of independent events:
If events A and B are independent, then
P(AB) = P(A) * P(B) (3)
Total probability rule:
P(A) = P(AS) + P(AS") = P(A|S")P(S) + P (A|S")P(S") (4)
S and S" are mutually exclusive events
expected value random variable is the average of possible outcomes random variable. For event X, the expectation is denoted as E(X).
Let’s say we have 5 values of mutually exclusive events with a certain probability (for example, a company’s income was such and such an amount with such a probability). The expected value is the sum of all outcomes multiplied by their probability:
Dispersion of a random variable is the expectation of square deviations of a random variable from its expectation:
s 2 = E( 2 ) (6)
Conditional expected value is the expected value of a random variable X, provided that the event S has already occurred.
Probability event is called the ratio of the number of elementary outcomes favorable this event, to the number of all equally possible outcomes of the experience in which this event may appear. The probability of event A is denoted by P(A) (here P is the first letter French word probabilite - probability). According to the definition
(1.2.1)
where is the number of elementary outcomes favorable to event A; - the number of all equally possible elementary outcomes of the experiment, forming full group events.
This definition of probability is called classical. It arose on initial stage development of probability theory.
The probability of an event has the following properties:
1. Probability reliable event equal to one. Let us denote a reliable event by the letter . For a certain event, therefore
(1.2.2)
2. The probability of an impossible event is zero. Let us denote an impossible event by the letter . For an impossible event, therefore
(1.2.3)
3. Probability random event is expressed positive number, less than one. Since for a random event the inequalities , or , are satisfied, 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 equal 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 will be blue?
Solution. We denote the event “the drawn ball turned out to be blue” by the letter A. This test has 10 equally possible elementary outcomes, of which 6 favor 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 shuffling the cards, one card is removed from the urn. What is the probability that the number on the card taken is a multiple of 5?
Solution. Let us 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 event A is favored by 6 outcomes (the numbers 5, 10, 15, 20, 25, 30). Hence, 
Example 3. Two dice are tossed and the total points are calculated. upper faces. Find the probability of event B such that the top faces of the dice have a total of 9 points.
Solution. In this test there are only 6 2 = 36 equally possible elementary outcomes. Event B is favored by 4 outcomes: (3;6), (4;5), (5;4), (6;3), therefore 
Example 4. Selected at random natural number, not exceeding 10. What is the probability that this number is prime?
Solution. Let us denote by the letter C the event “the chosen number is prime”. In this case n = 10, m = 4 ( prime numbers 2, 3, 5, 7). Therefore, the required probability 
Example 5. Two symmetrical coins are tossed. What is the probability that there are numbers on the top sides of both coins?
Solution. Let us denote by the letter D the event “there is a number on the top side of each coin.” In this test there are 4 equally possible elementary outcomes: (G, G), (G, C), (C, G), (C, C). (The notation (G, C) means that the first coin has a coat of arms, the second one has 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 a two-digit number chosen at random has the same digits?
Solution. Double digit numbers are numbers from 10 to 99; There are 90 such numbers in total. 9 numbers have identical digits (these are 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 identical 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?
Solution. The word differential has 12 letters, of which 5 are vowels and 7 are consonants. Letters h there is no in this word. Let us denote the events: A - “vowel letter”, B - “consonant letter”, C - “letter h". The number of favorable elementary outcomes: - for event A, - for event B, - for event C. Since n = 12, then
, And .
Example 8. Two dice are tossed and the number of points on the top of each dice is noted. Find the probability that both dice roll same number points.
Solution. Let's 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). The total number of equally possible elementary outcomes that form a complete group of events, in this case n=6 2 =36. This means that the required probability 
Example 9. The book has 300 pages. What is the probability that a randomly opened page will have a serial number divisible by 5?
Solution. From the conditions of the problem it follows that all equally possible elementary outcomes that form a complete group of events will be n = 300. 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 tossed and the sum of points on the top faces is calculated. What is more likely - getting a total of 7 or 8?
Solution. Let us denote the events: A - “7 points are rolled”, B – “8 points are rolled”. Event A is favored by 6 elementary outcomes: (1; 6), (2; 5), (3; 4), (4; 3), (5; 2), (6; 1), and event B is favored by 5 outcomes: (2; 6), (3; 5), (4; 4), (5; 3), (6; 2). All equally possible elementary outcomes are n = 6 2 = 36. 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, identical in size and weight. What is the probability that a ball drawn at random from this urn will be blue?
3. A number not exceeding 30 is chosen at random. What is the probability that this number is a divisor of 30?
4. In the urn A blue and b red balls, identical in size and weight. One ball is taken from this urn and set aside. This ball turned out to be red. After this, another ball is drawn from the urn. Find the probability that the second ball is also red.
5. A national number not exceeding 50 is chosen at random. What is the probability that this number is prime?
6. Three dice are tossed and the sum of points on the top faces is calculated. What is more likely - to get a total of 9 or 10 points?
7. Three dice are tossed and the sum of the points rolled 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 = 25/216 - probability of getting 9 points in total; p 2 = 27/216 - probability of getting 10 points in total; p 2 > p 1 7 . P(A) = 27/216, P(B) = 25/216, P(A) > P(B).
Questions
1. What is the probability of an event called?
2. What is the probability of a reliable 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?
