In the economy, as well as in other areas of human activity or in nature, we constantly have to deal with events that cannot be accurately predicted. Thus, the volume of sales of goods depends on demand, which can vary significantly, and on a number of other factors that are almost impossible to take into account. Therefore, when organizing production and sales, one has to predict the outcome of such activities on the basis of either one's own previous experience, or similar experience of other people, or intuition, which is also largely based on experimental data.
In order to somehow evaluate the event under consideration, it is necessary to take into account or specially organize the conditions in which this event is recorded.
The implementation of certain conditions or actions to identify the event in question is called experience or experiment.
The event is called random if, as a result of the experiment, it may or may not occur.
The event is called authentic, if it necessarily appears as a result of this experience, and impossible if it cannot appear in this experience.
For example, snowfall in Moscow on November 30th is a random event. The daily sunrise can be considered a certain event. Snowfall at the equator can be seen as an impossible event.
One of the main problems in probability theory is the problem of determining a quantitative measure of the possibility of an event occurring.
Events are called incompatible if they cannot be observed together in the same experience. Thus, the presence of two and three cars in one store for sale at the same time are two incompatible events.
sum events is an event consisting in the occurrence of at least one of these events
An example of a sum of events is the presence of at least one of two products in a store.
work events is called an event consisting in the simultaneous occurrence of all these events
An event consisting in the appearance of two goods at the same time in the store is a product of events: - the appearance of one product, - the appearance of another product.
Events form a complete group of events if at least one of them necessarily occurs in the experience.
Example. The port has two berths for ships. Three events can be considered: - the absence of vessels at the berths, - the presence of one vessel at one of the berths, - the presence of two vessels at two berths. These three events form a complete group of events.
Opposite two unique possible events that form a complete group are called.
If one of the events that are opposite is denoted by , then the opposite event is usually denoted by .
Each of the equally possible test results (experiments) is called an elementary outcome. They are usually denoted by letters . For example, a dice is thrown. There can be six elementary outcomes according to the number of points on the sides.
From elementary outcomes, you can compose a more complex event. So, the event of an even number of points is determined by three outcomes: 2, 4, 6.
A quantitative measure of the possibility of occurrence of the event under consideration is the probability.
Two definitions of the probability of an event are most widely used: classic And statistical.
The classical definition of probability is related to the notion of a favorable outcome.
Exodus is called favorable this event, if its occurrence entails the occurrence of this event.
In the given example, the event under consideration is an even number of points on the dropped edge, has three favorable outcomes. In this case, the general
the number of possible outcomes. So, here you can use the classical definition of the probability of an event.
Classic definition equals the ratio of the number of favorable outcomes to the total number of possible outcomes
where is the probability of the event , is the number of favorable outcomes for the event, is the total number of possible outcomes.
In the considered example
The statistical definition of probability is associated with the concept of the relative frequency of occurrence of an event in experiments.
The relative frequency of occurrence of an event is calculated by the formula
where is the number of occurrence of an event in a series of experiments (tests).
Statistical definition. The probability of an event is the number relative to which the relative frequency is stabilized (established) with an unlimited increase in the number of experiments.
In practical problems, the relative frequency for a sufficiently large number of trials is taken as the probability of an event.
From these definitions of the probability of an event, it can be seen that the inequality always holds
To determine the probability of an event based on formula (1.1), combinatorics formulas are often used to find the number of favorable outcomes and the total number of possible outcomes.
In the third lesson, we will consider various problems related to the direct application of the classical definition of probability. To effectively study the materials of this article, I recommend that you familiarize yourself with the basic concepts probability theory And basics of combinatorics. The problem of the classical determination of probability with a probability tending to one will be present in your independent / control work on terver, so we are getting ready for serious work. What's so serious, you ask? ... just one primitive formula. I warn against frivolity - thematic tasks are quite diverse, and many of them can easily confuse. In this regard, in addition to working out the main lesson, try to study additional tasks on the topic that are in the piggy bank ready-made solutions in higher mathematics. Decision methods are decision methods, but “friends” still “need to be known by sight”, because even a rich imagination is limited and there are also enough typical tasks. Well, I will try to make out the maximum number of them in good quality.
Let's remember the classics of the genre:
The probability of an event occurring in some trial is equal to the ratio , where:
is the total number of all equally possible, elementary outcomes of this test, which form full group of events;
- number elementary outcomes favoring the event.
And immediately an immediate pit stop. Do you understand the underlined terms? It means clear, not intuitive understanding. If not, then it is still better to return to the 1st article on probability theory and only then move on.
Please do not skip the first examples - in them I will repeat one fundamentally important point, and also tell you how to properly format a solution and in what ways it can be done:
Task 1
An urn contains 15 white, 5 red and 10 black balls. 1 ball is drawn at random, find the probability that it will be: a) white, b) red, c) black.
Solution: the most important prerequisite for using the classical definition of probability is the ability to calculate the total number of outcomes.
There are 15 + 5 + 10 = 30 balls in the urn, and obviously the following facts are true:
– extraction of any ball is equally possible (equal opportunity outcomes), while the outcomes elementary and form full group of events (i.e. as a result of the test, one of the 30 balls will definitely be removed).
Thus, the total number of outcomes:
Consider the following event: – a white ball will be drawn from the urn. This event is favored elementary outcomes, so by the classical definition: 
is the probability that a white ball will be drawn from the urn.
Oddly enough, even in such a simple problem, one can make a serious inaccuracy, which I already focused on in the first article on probability theory. Where is the pitfall here? It is incorrect to argue here that "since half of the balls are white, then the probability of drawing a white ball» . The classic definition of probability is ELEMENTARY outcomes, and the fraction must be written!
With other points similarly, consider the following events:
- a red ball will be drawn from the urn;
- A black ball will be drawn from the urn.
The event is favored by 5 elementary outcomes, and the event is favored by 10 elementary outcomes. So the corresponding probabilities are: 
A typical verification of many terver problems is done using theorems on the sum of probabilities of events forming a complete group. In our case, the events form a complete group, which means that the sum of the corresponding probabilities must necessarily be equal to one: .
Let's check if this is so: , which I wanted to make sure of.
Answer: 
In principle, the answer can be written in more detail, but personally I’m used to putting only numbers there - for the reason that when you start “stamping” tasks in hundreds and thousands, you strive to minimize the solution entry. By the way, about brevity: in practice, a “high-speed” design option is common. solutions:
Total: 15 + 5 + 10 = 30 balls in the urn. According to the classical definition:
is the probability that a white ball will be drawn from the urn;
is the probability that a red ball will be drawn from the urn;
is the probability that a black ball will be drawn from the urn.
Answer: 
However, if there are several points in the condition, then the solution is often more convenient to draw up in the first way, which takes a little more time, but then it “puts everything on the shelves” and makes it easier to navigate the task.
Warm up:
Task 2
The store received 30 refrigerators, five of which have a factory defect. One refrigerator is randomly selected. What is the probability that it will be defect free?
Choose the design option that suits you and check the template at the bottom of the page.
In the simplest examples, the number of common and the number of favorable outcomes lie on the surface, but in most cases you have to dig up the potatoes yourself. The canonical series of problems about the forgetful subscriber:
Task 3
When dialing a phone number, the subscriber forgot the last two digits, but remembers that one of them is zero, and the other is odd. Find the probability that he will dial the correct number.
Note : zero is an even number (divisible by 2 without a remainder)
Solution: first find the total number of outcomes. By condition, the subscriber remembers that one of the digits is zero, and the other digit is odd. Here it is more rational not to be wiser with combinatorics and use direct enumeration of outcomes
. That is, when making a decision, we simply write down all the combinations:
01, 03, 05, 07, 09
10, 30, 50, 70, 90
And we count them - in total: 10 outcomes.
There is only one favorable outcome: the right number.
According to the classical definition:
is the probability that the subscriber will dial the correct number
Answer: 0,1
Decimal fractions in probability theory look quite appropriate, but you can also follow the traditional Vyshmatov style, operating only with ordinary fractions.
Advanced task for independent solution:
Task 4
The subscriber forgot the pin code for his SIM card, but remembers that it contains three "fives", and one of the numbers is either "seven" or "eight". What is the probability of successful authorization on the first attempt?
Here you can still develop the idea of the probability that a punishment in the form of a fart code is waiting for the subscriber, but, unfortunately, the reasoning will already go beyond the scope of this lesson.
Solution and answer below.
Sometimes listing combinations turns out to be a very painstaking task. In particular, this is the case in the next, no less popular group of problems, where 2 dice are thrown (less often - more):
Task 5
Find the probability that when two dice are thrown, the total will be:
a) five points
b) no more than four points;
c) from 3 to 9 points inclusive.
Solution: find the total number of outcomes:
Ways can drop the face of the 1st die And the face of the 2nd die can fall out in ways; on combination multiplication rule, Total: 
possible combinations. In other words, each the face of the 1st cube can be orderly couple with each face of the 2nd cube. We agree to write such a pair in the form , where is the number that fell on the 1st die, is the number that fell on the 2nd die. For example:
- 3 points on the first die, 5 points on the second, total points: 3 + 5 = 8;
- on the first die 6 points fell out, on the second - 1 point, the sum of points: 6 + 1 = 7;
- both dice rolled 2 points, sum: 2 + 2 = 4.
Obviously, the smallest amount is given by a pair, and the largest by two "sixes".
a) Consider the event: - when throwing two dice, 5 points will fall out. Let's write down and count the number of outcomes that favor this event:
Total: 4 favorable outcomes. According to the classical definition:
is the desired probability.
b) Consider the event: - no more than 4 points will fall out. That is, either 2, or 3, or 4 points. Again, we list and count the favorable combinations, on the left I will write down the total number of points, and after the colon - suitable pairs: 
Total: 6 favorable combinations. In this way:
- the probability that no more than 4 points will fall out.
c) Let's consider the event: - from 3 to 9 points will fall out inclusive. Here you can go a straight road, but ... something does not feel like it. Yes, some pairs are already listed in the previous paragraphs, but there is still a lot of work to be done.
What's the best way to do it? In such cases, a detour turns out to be rational. Consider opposite event: - 2 or 10 or 11 or 12 points will fall out.
What's the point? The opposite event is favored by a much smaller number of pairs: 
Total: 7 favorable outcomes.
According to the classical definition:
- the probability that less than three or more than 9 points will fall out.
In addition to direct enumeration and calculation of outcomes, various combinatorial formulas. And again the epic task about the elevator:
Task 7
3 people entered the elevator of a 20-storey building on the first floor. And let's go. Find the probability that:
a) they will go out on different floors
b) two will exit on the same floor;
c) everyone will exit on the same floor.
Our fascinating lesson has come to an end, and finally, once again, I strongly recommend, if not to solve, then at least to understand additional tasks on the classical definition of probability. As I noted, "stuffing the hand" also matters!
Further down the course - Geometric definition of probability And Theorems of addition and multiplication of probabilities and ... luck in the main!
Solutions and answers:
Task 2: Solution: 30 - 5 = 25 refrigerators have no defect.
is the probability that a randomly selected refrigerator does not have a defect.
Answer
:
Task 4: Solution: find the total number of outcomes:
ways you can choose the place where the dubious figure is located and on each of these 4 places, 2 digits can be located (seven or eight). According to the rule of multiplication of combinations, the total number of outcomes: 
.
Alternatively, in the solution, you can simply list all the outcomes (fortunately there are not many of them):
7555, 8555, 5755, 5855, 5575, 5585, 5557, 5558
There is only one favorable outcome (correct pin code).
Thus, by the classical definition:
- the probability that the subscriber is authorized on the 1st attempt
Answer
:
Task 6: Solution: find the total number of outcomes:
ways can drop numbers on 2 dice.
a) Consider the event: - when throwing two dice, the product of points will be equal to seven. For this event, there are no favorable outcomes, according to the classical definition of probability:
, i.e. this event is impossible.
b) Let's consider the event: - when throwing two dice, the product of points will be at least 20. This event is favored by the following outcomes:
Total: 8
According to the classical definition:
is the desired probability.
c) Consider opposite events:
– the product of points will be even;
– the product of points will be odd.
Let's list all the outcomes that favor the event:
Total: 9 favorable outcomes.
According to the classical definition of probability: 
Opposite events form a complete group, so:
is the desired probability.
Answer
: 
Task 8: Solution: calculate the total number of outcomes: 
10 coins can fall in ways.
Another way: 1st coin can fall in ways And 2nd coin can fall in ways And … And ways the 10th coin can fall. According to the rule of multiplying combinations, 10 coins can fall 
ways.
a) Consider the event: - all coins will fall heads. This event is favored by a single outcome, according to the classical definition of probability: .
b) Consider the event: - 9 coins will come up heads, and one will come up tails.
There are coins that can land tails. According to the classical definition of probability: 
.
c) Let's consider the following event: - heads will fall on half of the coins.
Exists 
unique combinations of five coins that can land heads. According to the classical definition of probability: 
Answer
:
The probability of an event is understood as some numerical characteristic of the possibility of the occurrence of this event. There are several approaches to determining probability.
Probability of an event BUT is the ratio of the number of outcomes favorable to this event to the total number of all equally possible incompatible elementary outcomes that form a complete group. So the probability of an event BUT is determined by the formula
where m is the number of elementary outcomes favoring BUT, n- the number of all possible elementary outcomes of the test.
Example 3.1. In the experiment with throwing a dice, the number of all outcomes n is 6 and they are all equally possible. Let the event BUT means the appearance of an even number. Then for this event, favorable outcomes will be the appearance of numbers 2, 4, 6. Their number is 3. Therefore, the probability of the event BUT is equal to
Example 3.2. What is the probability that the digits in a randomly chosen two-digit number are the same?
Two-digit numbers are numbers from 10 to 99, there are 90 such numbers in total. 9 numbers have the same numbers (these are the numbers 11, 22, ..., 99). Since in this case m=9, n=90, then
where BUT- event, "a number with the same digits."
Example 3.3. There are 7 standard parts in a lot of 10 parts. Find the probability that there are 4 standard parts among six randomly selected parts.
The total number of possible elementary outcomes of the test is equal to the number of ways in which 6 parts can be extracted from 10, i.e., the number of combinations of 10 elements of 6 elements. Determine the number of outcomes that favor the event of interest to us BUT(among the six parts taken, 4 are standard). Four standard parts can be taken from seven standard parts in ways; at the same time, the remaining 6-4=2 parts must be non-standard, but you can take two non-standard parts from 10-7=3 non-standard parts in different ways. Therefore, the number of favorable outcomes is .
Then the desired probability is equal to
The following properties follow from the definition of probability:
1. The probability of a certain event is equal to one.
Indeed, if the event is reliable, then each elementary outcome of the test favors the event. In this case m=n, hence
2. The probability of an impossible event is zero.
Indeed, if the event is impossible, then none of the elementary outcomes of the trial favors the event. In this case it means
3. The probability of a random event is a positive number between zero and one.
Indeed, only a part of the total number of elementary outcomes of the test favors a random event. In this case< m< n, means 0 < m/n < 1, i.e. 0< P(A) < 1. Итак, вероятность любого события удовлетворяет двойному неравенству
The construction of a logically complete probability theory is based on the axiomatic definition of a random event and its probability. In the system of axioms proposed by A. N. Kolmogorov, undefined concepts are an elementary event and probability. Here are the axioms that define the probability:
1. Every event BUT assigned a non-negative real number P(A). This number is called the probability of the event. BUT.
2. The probability of a certain event is equal to one.
3. The probability of occurrence of at least one of the pairwise incompatible events is equal to the sum of the probabilities of these events.
Based on these axioms, the properties of probabilities and the relationships between them are derived as theorems.
Questions for self-examination
1. What is the name of the numerical characteristic of the possibility of an event?
2. What is called the probability of an event?
3. What is the probability of a certain event?
4. What is the probability of an impossible event?
5. What are the limits of the probability of a random event?
6. What are the limits of the probability of any event?
7. What definition of probability is called classical?
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?
Solution. 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?
Solution. 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). Consequently, 
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.
Solution. 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?
Solution. 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?
Solution. 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?
Solution. 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?
Solution. 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.
Solution. 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?
Solution. 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 
. Consequently, 
, where A - the "page" event has a sequence number that is a multiple of 5".
Example 10. Two dice are tossed, the sum of the points on the upper faces is calculated. What is more likely to get a total of 7 or 8?
Solution. 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 but 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 thrown, 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?
