Brief theory

For a quantitative comparison of events according to the degree of possibility of their occurrence, a numerical measure is introduced, which is called the probability of an event. The probability of a random event a number is called, which is an expression of a measure of the objective possibility of the occurrence of an event.

The values ​​that determine how significant are the objective grounds for counting on the occurrence of an event are characterized by the probability of the event. It must be emphasized that probability is an objective quantity that exists independently of the cognizer and is conditioned by the totality of conditions that contribute to the occurrence of an event.

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

The classical definition of the probability of an event reduces this concept to a more elementary concept of equally probable events, which is no longer subject to definition and is assumed to be intuitively clear. For example, if a dice is a homogeneous cube, then the fallout of any of the faces of this cube will be equally probable events.

Let a certain event be divided into equally probable cases, the sum of which gives the event. That is, the cases from , into which it breaks up, are called favorable for the event, since the appearance of one of them ensures the offensive.

The probability of an event will be denoted by the symbol .

The probability of an event is equal to the ratio of the number of cases favorable to it, out of the total number of unique, equally possible and incompatible cases, to the number, i.e.

This is the classical definition of probability. Thus, to find the probability of an event, it is necessary, after considering the various outcomes of the test, to find a set of the only possible, equally possible and incompatible cases, calculate their total number n, the number of cases m that favor this event, and then perform the calculation according to the above formula.

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

The following properties of probability follow from the definition:

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

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

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

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

Property 5. The probability of the occurrence of the opposite event is defined in the same way as the probability of the occurrence of event A.

The number of occurrences that favor the occurrence of the opposite event. Hence, the probability of the opposite event occurring is equal to the difference between unity and the probability of the event A occurring:

An important advantage of the classical definition of the probability of an event is that with its help the probability of an event can be determined without resorting to experience, but on the basis of logical reasoning.

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

Problem solution example

Example 1

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

The solution of the problem

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

Find the probability of an event– drawn at least 1 red ball (1,2 or 3 red balls)

Required probability:

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

Number of outcomes favoring the event:

Required probability:

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

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

Number of outcomes favoring the event:

Required probability:

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

Example 2

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

Decision

Let the event be the sum of points not less than 5

Let's use the classical definition of probability:

Total number of possible trial outcomes

The number of trials that favor the event of interest to us

On the dropped face of the first dice, one point, two points ..., six points can appear. similarly, six outcomes are possible on the second die roll. Each of the outcomes of the first die can be combined with each of the outcomes of the second. Thus, the total number of possible elementary outcomes of the test is equal to the number of placements with repetitions (selection with placements of 2 elements from a set of volume 6):

Find the probability of the opposite event - the sum of points is less than 5

The following combinations of dropped points will favor the event:

№

1st bone

2nd bone

1

1

1

2

1

2

3

2

1

4

3

1

5

1

3

The geometric definition of probability is presented and the solution of the well-known meeting problem is given.

In order to quantitatively compare events with each other according to the degree of their possibility, it is obviously necessary to associate a certain number with each event, which is the greater, the more possible the event is. We call this number the probability of the event. Thus, event probability is a numerical measure of the degree of objective possibility of this event.

The classical definition of probability, which arose from the analysis of gambling and was initially applied intuitively, should be considered the first definition of probability.

The classical method of determining probability is based on the concept of equally probable and incompatible events, which are the outcomes of a given experience and form a complete group of incompatible events.

The simplest example of equally possible and incompatible events that form a complete group is the appearance of one or another ball from an urn containing several balls of the same size, weight and other tangible features, differing only in color, thoroughly mixed before being taken out.

Therefore, a trial, the outcomes of which form a complete group of incompatible and equally probable events, is said to be reduced to the scheme of urns, or the scheme of cases, or fit into the classical scheme.

Equally possible and incompatible events that make up a complete group will be called simply cases or chances. Moreover, in each experiment, along with cases, more complex events can occur.

Example: When throwing a dice, along with cases A i - i-points falling on the upper face, events such as B - an even number of points falling out, C - a multiple of three points falling out ...

In relation to each event that can occur during the implementation of the experiment, the cases are divided into favorable, at which this event occurs, and unfavorable, at which the event does not occur. In the previous example, event B is favored by cases A 2 , A 4 , A 6 ; event C - cases A 3 , A 6 .

classical probability the occurrence of some event is the ratio of the number of cases that favor the appearance of this event to the total number of cases of equally possible, incompatible, constituting a complete group in a given experience:

where P(A)- probability of occurrence of event A; m- number of cases favorable for event A; n is the total number of cases.

Examples:

1) (see example above) P(B)= , P(C) =.

2) An urn contains 9 red and 6 blue balls. Find the probability that one or two balls drawn at random will be red.

BUT- a red ball drawn at random:

m= 9, n= 9 + 6 = 15, P(A)=

B- two red balls drawn at random:

The following properties follow from the classical definition of probability (show yourself):

1) The probability of an impossible event is 0;

2) The probability of a certain event is 1;

3) The probability of any event lies between 0 and 1;

4) The probability of an event opposite to event A,

The classical definition of probability assumes that the number of outcomes of a trial is finite. In practice, however, very often there are trials, the number of possible cases of which is infinite. In addition, the weakness of the classical definition is that it is very often impossible to represent the result of a test as a set of elementary events. It is even more difficult to indicate the grounds for considering the elementary outcomes of the test as equally probable. Usually, the equality of the elementary outcomes of the test is concluded from considerations of symmetry. However, such tasks are very rare in practice. For these reasons, along with the classical definition of probability, other definitions of probability are also used.

Statistical Probability event A is the relative frequency of occurrence of this event in the tests performed:

where is the probability of occurrence of event A;

Relative frequency of occurrence of event A;

The number of trials in which event A appeared;

The total number of trials.

Unlike classical probability, statistical probability is a characteristic of an experimental one.

Example: To control the quality of products from a batch, 100 products were randomly selected, among which 3 products turned out to be defective. Determine the probability of marriage.

.

The statistical method of determining the probability is applicable only to those events that have the following properties:

The events under consideration should be the outcomes of only those trials that can be reproduced an unlimited number of times under the same set of conditions.

Events must have statistical stability (or stability of relative frequencies). This means that in different series of tests, the relative frequency of the event does not change significantly.

The number of trials that result in event A must be large enough.

It is easy to verify that the properties of probability, which follow from the classical definition, are also preserved in the statistical definition of probability.

3) P (Æ )=0.

We will say what is given probability space, if the space of elementary outcomes9 is given and the correspondence

w i ® P(w i ) =Pi .

The question arises: how to determine the probability P (w i ) of individual elementary outcomes from the specific conditions of the problem being solved?

The classic definition of probability.

The probabilities P (w i ) can be calculated using an a priori approach, which consists in analyzing the specific conditions of a given experiment (before the experiment itself).

A situation is possible when the space of elementary outcomes consists of a finite number N of elementary outcomes, and a random experiment is such that the probabilities of each of these N elementary outcomes appear to be equal. Examples of such random experiments are: tossing a symmetrical coin, throwing a regular dice, randomly removing a playing card from a shuffled deck. By virtue of the introduced axiom, the probability of each elementary

outcomes in this case are equal to N . It follows from this that if event A contains N A elementary outcomes, then in accordance with the definition (*)

P(A) = A

In this class of situations, the probability of an event is defined as the ratio of the number of favorable outcomes to the total number of all possible outcomes.

Example. From a set containing 10 identical-looking electric lamps, among which 4 are defective, 5 lamps are randomly selected. What is the probability that among the selected lamps there will be 2 defective ones?

First of all, we note that the choice of any five lamps has the same probability. In total, there are C 10 5 ways to make such a five, that is, a random experiment in this case has C 10 5 equiprobable outcomes.

How many of these outcomes satisfy the condition "there are two defective lamps in the five", that is, how many outcomes belong to the event of interest to us?

Each five we are interested in can be composed as follows: choose two defective lamps, which can be done in a number of ways equal to C 4 2 . Each pair of defective lamps can occur as many times as there are ways to complement it with three non-defective lamps, that is, 6 3 times. It turns out that the number of fives containing two

Statistical definition of probability.

Consider a random experiment in which a dice made of a non-homogeneous material is tossed. Its center of gravity is not in the geometric center. In this case, we cannot consider the outcomes (rolling one, two, etc.) equally probable. It is known from physics that the bone will fall more often on the face that is closer to the center of gravity. How to determine the probability of getting, for example, three points? The only thing you can do is toss that die n times (where n is a big enough number, say n=1000 or n=5000), count the number of rolls of three n 3 and calculate the probability of a three roll outcome as n 3 /n - the relative frequency of getting three points. Similarly, you can determine the probabilities of the remaining elementary outcomes - ones, twos, fours, etc. Theoretically, this course of action can be justified by introducing statistical definition of probability.

The probability P(M i ) is defined as the limit of the relative frequency of occurrence of the outcome M i in the process of an unlimited increase in the number of random experiments n , that is

P i = P (M i ) = lim m n (M i ) , n ®¥n

where m n (M i ) is the number of random experiments (out of the total number n of random experiments performed) in which the occurrence of an elementary outcome M i is registered.

Since no evidence is given here, we can only hope that the limit in the last formula exists, substantiating the hope with life experience and intuition.

geometric probability

In one special case, let us define the probability of an event for a random experiment with an uncountable set of outcomes.

If a one-to-one correspondence can be established between the set W of elementary outcomes of a random experiment and the set of points of some flat figure S (large sigma), and a one-to-one correspondence can also be established between the set of elementary outcomes that favor the event A and the set of points of a flat figure I (small sigma) , which is part of the figure S , then

P(A) = S ,

where s is the area of ​​figure s, S is the area of ​​figure S.

Example. Two people have lunch in the dining room, which is open from 12 to 13 hours. Each of them comes at a random time and has lunch for 10 minutes. What is the probability of their meeting?

Let x be the arrival time of the first in the canteen, аy be the arrival time of the second

£12 x £13; £12y £13.

You can establish a one-to-one correspondence between all pairs of numbers (x ;y ) (or a set of outcomes) and the set of points of a square with side equal to 1 on the coordinate plane, where the origin corresponds to the number 12 on the x-axis and on the y-axis, as shown in the figure 6. Here, for example, point A corresponds to the outcome, which consists in the fact that the first came at 12.30, and the second - at 13.00. In this case, obviously

the meeting did not take place.

If the first one arrived no later than the second one (y ³ x), then

the meeting will occur under the condition 0 £ y - x £ 1/6

(10 minutes is 1/6 hour).

If the second one arrived no later than the first one (x ³ y ), then

the meeting will occur under the condition 0 £ x - y £ 1/6..

Between many favorable outcomes

meeting, and the set of points of the area s depicted on

Figure 7 in shaded form, you can install

one-to-one correspondence.

The desired probability p is equal to the ratio of the area

area s to the area of ​​the whole square.. Area of ​​the square

equals unity, and the area of ​​the region s can be defined as

the difference between a unit and the total area of ​​two

triangles shown in Figure 7. It follows from this:

p=1 -

Continuous probability space.

As mentioned earlier, the set of elementary outcomes can be more than countable (that is, uncountable). In this case, any subset of the set W cannot be considered an event.

To introduce the definition of a random event, consider a system (finite or countable) of subsets A 1 , A 2 ,... A n of the space of elementary outcomes W .

If three conditions are met: 1) W belongs to this system;

2) membership of A in this system implies membership of A in this system;

3) membership of A i and A j in this system implies membership of A i U A j in this system

such a system of subsets is called an algebra.

Let W be some space of elementary outcomes. Make sure that the two subset systems are:

1) W ,Æ ; 2) W , A , A , Æ (here A is a subset of W ) are algebras.

Let A 1 and A 2 belong to some algebra. Prove that A 1 \A 2 and A 1 ∩ A 2 belong to this algebra.

A subset A of an uncountable set of elementary outcomes 9 is an event if it belongs to some algebra.

Let us formulate an axiom called A.N. Kolmogorov.

Each event corresponds to a non-negative number P(A) not exceeding one, called the probability of the event A , and the function P(A) has the following properties:

1) P (9)=1

2) if the events A 1 ,A 2 ,...,A n are incompatible, then

P (A 1 U A 2 U ... U A n) \u003d P (A 1) + P (A 2) + ... + P (A n)

If the space of elementary outcomes W is given, the algebra of events and the function P defined on it that satisfies the conditions of the above axiom, then we say that probability space.

This definition of a probability space can be extended to the case of a finite space of elementary outcomes W . Then, as an algebra, we can take the system of all subsets of the set W .

Probability addition formulas.

From point 2 of the above axiom it follows that if A 1 and A2 are incompatible events, then

P (A 1 U A 2) \u003d P (A 1) + P (A 2)

If A 1 and A 2 are joint events, then A 1 U A 2 =(A 1 \A 2 )U A 2 , and it is obvious that A 1 \A 2 and A 2 are incompatible events. This implies:

P (A 1 U A 2 ) =P (A1 \A 2 ) +P (A2 )

Further, it is obvious: A 1 = (A1 \A 2 )U (A 1 ∩ A 2 ), and A1 \A 2 and A 1 ∩ A 2 are incompatible events, whence follows: P (A 1 ) =P (A1 \A 2 ) +P (A 1 ∩ A 2 ) Find an expression for P (A1 \A 2 ) from this formula and substitute it into the right side of the formula (*). As a result, we obtain the formula for adding probabilities:

P (A 1 U A 2 ) =P (A 1 ) +P (A 2 ) –P (A 1 ∩ A 2 )

From the last formula, it is easy to obtain a formula for adding probabilities for incompatible events by setting A 1 ∩ A 2 =Æ .

Example. Find the probability of drawing an ace or a suit of hearts by randomly selecting one card from a deck of 32 sheets.

P (ACE) \u003d 4/32 \u003d 1/8; P (HEART SUIT) \u003d 8/32 \u003d 1/4;

P (ACE OF HEARTS) = 1/32;

P ((ACE) U (HEART SUIT)) \u003d 1/8 + 1/4 - 1/32 \u003d 11/32

The same result could be achieved using the classical definition of probability by counting the number of favorable outcomes.

Conditional probabilities.

Let's consider the problem. Before the exam, a student learned from 30 tickets tickets with numbers from 1 to 5 and from 26 to 30. It is known that a student pulled out a ticket with a number not exceeding 20 during the exam. What is the probability that the student pulled out a learned ticket?

Let's define the space of elementary outcomes: W =(1,2,3,...,28,29,30). Let event A be that the student pulled out a learned ticket: A = (1,...,5,25,...,30,), and event B is that the student pulled out a ticket from the first twenty: B = ( 1,2,3,...,20)

The event A ∩ B consists of five outcomes: (1,2,3,4,5), and its probability is 5/30. This number can be represented as the product of 5/20 and 20/30. The number 20/30 is the probability of event B . The number 5/20 can be considered as the probability of event A, provided that event B happened (let's denote it as P (A / B)). Thus, the solution to the problem is determined by the formula

P (A ∩ B) \u003d P (A / B) P (B)

This formula is called the probability multiplication formula, and the probability P (A / B) is the conditional probability of the event A.

Example .. From an urn containing 7 white and 3 black balls, two balls are randomly drawn one after the other (without replacement). What is the probability that the first ball is white and the second black?

Let X be the event that the first draw is a white ball, and Y be the event that the second draw is a black ball. Then X ∩ Y is the event that the first ball is white and the second is black. P (Y /X ) =3/9 =1/3 is the conditional probability that the second ball will draw a black ball, if the white ball was drawn first. Considering that P (X ) = 7/10, according to the probability multiplication formula we get: P (X ∩ Y ) = 7/30

Event A is called independent of event B (in other words: events A and B are called independent) if P (A / B) = P (A ). For the definition of independent events, we can take the consequence of the last formula and the multiplication formula

P (A ∩ B) \u003d P (A) P (B)

Prove for yourself that if A and B are independent events, then A and B are also independent events.

Example. Consider a problem similar to the previous one, but with one additional condition: after drawing the first ball, remember its color and return the ball to the urn, after which we mix all the balls. In this case, the result of the second extraction does not depend in any way on which ball - black or white - appeared during the first extraction. The probability of a white ball appearing first (event A) is 7/10. The probability of event B - the appearance of the second black ball - is 3/10. Now the multiplication formula gives: P (A ∩ B) = 21/100.

Extracting balls in the manner described in this example is called fetch with return or return sampling.

It should be noted that if in the last two examples we set the initial numbers of white and black balls equal to 7000 and 3000, respectively, then the results of calculating the same probabilities will differ negligibly small for the return and irrevocable samples.

The classic definition of probability.

As mentioned above, with a large number n test frequency P*(A)=m/ n occurrence of an event A is stable and gives an approximate value of the probability of an event A , i.e. .

This circumstance allows us to find approximately the probability of an event empirically. In practice, this method of finding the probability of an event is not always convenient. After all, we need to know in advance the probability of some event, even before the experience. This is the heuristic, predictive role of science. In a number of cases, the probability of an event can be determined prior to experiment using the concept of equiprobability of events (or equiprobability).

The two events are called equiprobable (or equally possible ), if there are no objective reasons to believe that one of them may occur more often than the other.

So, for example, the appearance of a coat of arms or an inscription when a coin is tossed are equiprobable events.

Let's consider another example. Let them throw a dice. Due to the symmetry of the cube, we can assume that the appearance of any of the numbers 1, 2, 3, 4, 5 or 6 equally possible (equally probable).

Events in this experience form full group if at least one of them must occur as a result of the experiment. So, in the last example, the complete group of events consists of six events - the appearance of numbers 1, 2, 3, 4, 5 and 6.

Obviously any event A and its opposite event form a complete group.

Event B called favorable event A if the occurrence of the event B triggers an event A . So if A - the appearance of an even number of points when throwing a dice, then the appearance of a number 4 represents an event favorable to the event A.

Let the events in this experiment form a complete group of equally probable and pairwise incompatible events. Let's call them outcomes tests. Let's assume that the event A favor test outcomes. Then the probability of the event A in this experiment is called the ratio. So we come to the next definition.

The probability P(A) of an event in a given experiment is the ratio of the number of outcomes of the experience that favor event A to the total number of possible outcomes of the experience that form a complete group of equally probable pairwise incompatible events: .

This definition of probability is often called classic. It can be shown that the classical definition satisfies the axioms of probability.

Example 1.1. A batch of 1000 bearings. Accidentally got into this batch 30 bearings that do not meet the standard. Determine Probability P(A) the fact that a bearing taken at random will be standard.

Decision: The number of standard bearings is 1000-30=970 . We assume that each bearing has the same probability of being selected. Then the complete group of events consists of equally probable outcomes, of which the event A favor outcomes. So .

Example 1.2. in the urn 10 balls: 3 whites and 7 black. Two balls are taken out of the urn at once. What is the probability R that both balls are white?

Decision: The number of all equally probable outcomes of a trial is equal to the number of ways in which 10 take out two balls, i.e. the number of combinations from 10 elements by 2 (full group of events):

The number of favorable outcomes (in how many ways can 3 balls to choose 2) : . Therefore, the desired probability .

Looking ahead, this problem can be solved in another way.

Decision: The probability that a white ball will be drawn on the first trial (drawing a ball) is equal to (total balls 10 , of them 3 whites). The probability that at the second trial a white ball will be taken out again is equal to (the total number of balls has become 9, because one was taken out, it became white 2, because they took out the white one). Therefore, the probability of combining events is equal to the product of their probabilities, i.e. .

Example 1.3. in the urn 2 green, 7 red, 5 brown and 10 white balls. What is the probability of a colored ball appearing?

Decision: We find, respectively, the probabilities of the appearance of green, red and brown balls: ; ; . Since the events under consideration are obviously incompatible, then, using the addition axiom, we find the probability of the appearance of a colored ball:

Or, in another way. The probability of a white ball appearing is . Then the probability of the appearance of a non-white ball (i.e. colored), i.e. the probability of the opposite event is equal to .

Geometric definition of probability. To overcome the disadvantage of the classical definition of probability (it is not applicable to tests with an infinite number of outcomes), a geometric definition of probability is introduced - the probability of a point falling into an area (a segment, part of a plane, etc.).

Let the segment be part of the segment . A point is randomly placed on the segment, which means that the following assumptions are fulfilled: the set point can be at any point of the segment , the probability of a point falling on the segment is proportional to the length of this segment and does not depend on its location relative to the segment . Under these assumptions, the probability of a point falling on a segment is determined by the equality

Fundamentals of Probability Theory

Plan:

1. Random events

2. Classical definition of probability

3. Calculation of event probabilities and combinatorics

4. Geometric probability

Theoretical information

Random events.

random phenomenon- a phenomenon, the outcome of which is unambiguously determined. This concept can be interpreted in a fairly broad sense. Namely: everything in nature is quite accidental, the appearance and birth of any individual is a random phenomenon, the choice of goods in a store is also a random phenomenon, getting a mark on an exam is a random phenomenon, illness and recovery are random phenomena, etc.

Examples of random phenomena:

~ Shooting is carried out from a gun set at a given angle to the horizon. Hitting it on the target is accidental, but hitting a projectile in a certain "fork" is a pattern. You can specify the distance closer than and beyond which the projectile will not fly. Get some "fork dispersion of shells"

~ The same body is weighed several times. Strictly speaking, different results will be obtained each time, albeit differing by a negligibly small amount, but different.

~ An aircraft flying along the same route has a certain flight corridor within which the aircraft can maneuver, but it will never have exactly the same route

~ An athlete will never be able to run the same distance with the same time. His results will also be within a certain numerical range.

Experience, experiment, observation are tests

Trial- observation or fulfillment of a certain set of conditions that are performed repeatedly, and regularly repeated in this and the same sequence, duration, while observing other identical parameters.

Let's consider performance by the sportsman of a shot on a target. In order for it to be produced, it is necessary to fulfill such conditions as the preparation of the athlete, loading the weapon, aiming, etc. "Hit" and "miss" are events as a result of a shot.

Event– qualitative test result.

An event may or may not occur Events are indicated by capital Latin letters. For example: D ="The shooter hit the target". S="White ball drawn". K="Random lottery ticket without winning.".

Tossing a coin is a test. The fall of her "coat of arms" is one event, the fall of her "number" is the second event.

Any test involves the occurrence of several events. Some of them may be needed at a given time by the researcher, while others may not be needed.

The event is called random, if under the implementation of a certain set of conditions S it can either happen or not happen. In the future, instead of saying "the set of conditions S is implemented", we will say briefly: "the test was carried out." Thus, the event will be considered as the result of the test.

~ The shooter shoots at a target divided into four areas. The shot is a test. Hitting a certain area of ​​the target is an event.

~ There are colored balls in the urn. One ball is drawn at random from the urn. Removing a ball from an urn is a test. The appearance of a ball of a certain color is an event.

Types of random events

1. Events are said to be incompatible if the occurrence of one of them excludes the occurrence of other events in the same trial.

~ A part was taken at random from a box with parts. The appearance of a standard part excludes the appearance of a non-standard part. Events € a standard part appeared" and with a non-standard part appeared" - incompatible.

~ A coin is thrown. The appearance of the "coat of arms" excludes the appearance of the inscription. The events "a coat of arms appeared" and "an inscription appeared" are incompatible.

Several events form full group, if at least one of them appears as a result of the test. In other words, the occurrence of at least one of the events of the complete group is a certain event.

In particular, if the events that form a complete group are pairwise incompatible, then one and only one of these events will appear as a result of the test. This special case is of greatest interest to us, since it is used below.

~ Two tickets of the money and clothing lottery were purchased. One and only one of the following events must occur:

1. "the winnings fell on the first ticket and did not fall on the second",

2. "the winnings did not fall on the first ticket and fell on the second",

3. "the winnings fell on both tickets",

4. "both tickets did not win."

These events form a complete group of pairwise incompatible events,

~ The shooter fired a shot at the target. One of the following two events is sure to occur: hit, miss. These two disjoint events also form a complete group.

2. Events are called equally possible if there is reason to believe that neither is more possible than the other.

~ The appearance of a "coat of arms" and the appearance of an inscription when a coin is tossed are equally possible events. Indeed, it is assumed that the coin is made of a homogeneous material, has a regular cylindrical shape, and the presence of a coinage does not affect the loss of one or another side of the coin.

~ The appearance of one or another number of points on a thrown dice is an equally probable event. Indeed, it is assumed that the die is made of a homogeneous material, has the shape of a regular polyhedron, and the presence of points does not affect the loss of any face.

3. The event is called authentic, if it cannot happen

4. The event is called not reliable if it can't happen.

5. The event is called opposite to some event if it consists of the non-occurrence of the given event. Opposite events are not compatible, but one of them must necessarily occur. Opposite events are commonly referred to as negations, i.e. a dash is written above the letter. The events are opposite: A and Ā; U and Ū, etc. .

The classical definition of probability

Probability is one of the basic concepts of probability theory.

There are several definitions of this concept. Let us give a definition that is called classical. Next, we point out the weaknesses of this definition and give other definitions that make it possible to overcome the shortcomings of the classical definition.

Consider the situation: A box contains 6 identical balls, 2 being red, 3 being blue and 1 being white. Obviously, the possibility of drawing a colored (i.e., red or blue) ball at random from an urn is greater than the possibility of drawing a white ball. This possibility can be characterized by a number, which is called the probability of an event (the appearance of a colored ball).

Probability- a number characterizing the degree of possibility of occurrence of the event.

In the situation under consideration, we denote:

Event A = "Pulling out a colored ball".

Each of the possible outcomes of the test (the test consists in extracting a ball from the urn) is called elementary (possible) outcome and event. Elementary outcomes can be denoted by letters with indexes below, for example: k 1 , k 2 .

In our example, there are 6 balls, so there are 6 possible outcomes: a white ball appeared; a red ball appeared; a blue ball appeared, and so on. It is easy to see that these outcomes form a complete group of pairwise incompatible events (only one ball will necessarily appear) and they are equally probable (the ball is taken out at random, the balls are the same and thoroughly mixed).

Elementary outcomes, in which the event of interest to us occurs, we will call favorable outcomes this event. In our example, the event is favored BUT(the appearance of a colored ball) the following 5 outcomes:

Thus the event BUT observed if one occurs in the test, no matter which, of the elementary outcomes that favor BUT. This is the appearance of any colored ball, of which there are 5 pieces in the box

In the considered example of elementary outcomes 6; of which 5 favor the event BUT. Hence, P(A)= 5/6. This number gives that quantification of the degree of possibility of the appearance of a colored ball.

Probability definition:

Probability of event A 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.

P(A)=m/n or P(A)=m: n, where:

m is the number of elementary outcomes that favor BUT;

P- the number of all possible elementary outcomes of the test.

It is assumed here that the elementary outcomes are incompatible, equally probable and form a complete group.

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 p=1

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 m=0, hence p=0.

3.The probability of a random event is a positive number between zero and one. 0t< n.

In subsequent topics, theorems will be given that allow us to find the probabilities of other events from the known probabilities of some events.

Measurement. There are 6 girls and 4 boys in the group of students. What is the probability that a randomly selected student will be a girl? will it be a young man?

p dev = 6 / 10 = 0.6 p jun = 4 / 10 = 0.4

The concept of "probability" in modern rigorous courses of probability theory is built on a set-theoretic basis. Let's take a look at some of this approach.

Suppose that as a result of the test one and only one of the following events occurs: w i(i=1, 2, .... n). Events w i, is called elementary events (elementary outcomes). O it follows that the elementary events are pairwise incompatible. The set of all elementary events that can appear in a trial is called elementary event spaceΩ (Greek letter omega capital), and the elementary events themselves - points in this space..

Event BUT identified with a subset (of the space Ω) whose elements are elementary outcomes favoring BUT; event AT is a subset Ω whose elements are outcomes that favor AT, etc. Thus, the set of all events that can occur in the test is the set of all subsets of Ω. Ω itself occurs for any outcome of the test, therefore Ω is a certain event; an empty subset of the space Ω is an -impossible event (it does not occur for any outcome of the test).

Elementary events are distinguished from among all events by topics, "each of them contains only one element Ω

To every elementary outcome w i match a positive number p i is the probability of this outcome, and the sum of all p i equal to 1 or with the sign of the sum, this fact will be written as an expression:

By definition, the probability P(A) events BUT is equal to the sum of the probabilities of elementary outcomes favoring BUT. Therefore, the probability of a certain event is equal to one, impossible - to zero, arbitrary - is between zero and one.

Let us consider an important particular case, when all outcomes are equally probable. The number of outcomes is equal to l, the sum of the probabilities of all outcomes is equal to one; hence the probability of each outcome is 1/n. Let the event BUT favors m outcomes.

Event Probability BUT is equal to the sum of the probabilities of outcomes favoring BUT:

P(A)=1/n + 1/n+…+1/n = n 1/n=1

The classical definition of probability is obtained.

There is still axiomatic approach to the concept of "probability". In the system of axioms proposed. Kolmogorov A.N., undefined concepts are elementary event and probability. The construction of a logically complete probability theory is based on the axiomatic definition of a random event and its 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 for the relationship between them are derived as theorems.