Probability Theory. Basic terms and concepts

Mom washed the frame

Under the curtain of lengthy summer holidays it's time to slowly return to higher mathematics and solemnly open an empty Verd file in order to start creating a new section - . I confess that the first lines are not easy, but the first step is half the way, so I suggest everyone to carefully study the introductory article, after which it will be 2 times easier to master the topic! I'm not exaggerating at all. ... On the eve of the next September 1, I remember the first grade and primer .... Letters form syllables, syllables into words, words into short sentences - Mom washed the frame. Mastering terver and mathematical statistics is as easy as learning to read! However, for this it is necessary to know the key terms, concepts and designations, as well as some specific rules, to which this lesson is devoted.

But first, please accept my congratulations on the beginning (continuation, completion, note as necessary) school year and accept the gift. Best gift is a book and independent work I recommend the following literature:

1) Gmurman V.E. Theory of Probability and Mathematical Statistics

A legendary textbook that has gone through more than ten reprints. It differs in intelligibility and the ultimate simple presentation of the material, and the first chapters are completely accessible, I think, already for students in grades 6-7.

2) Gmurman V.E. Guide to Problem Solving in Probability and Mathematical Statistics

Reshebnik of the same Vladimir Efimovich with detailed examples and tasks.

NECESSARILY download both books from the Internet or get their paper originals! A 60s-70s version will do, which is even better for dummies. Although the phrase "probability theory for dummies" sounds rather ridiculous, since almost everything is limited to elementary arithmetic operations. They slip, however, in places derivatives And integrals, but this is only in places.

I will try to achieve the same clarity of presentation, but I must warn you that my course is focused on problem solving and theoretical calculations are kept to a minimum. Thus, if you need a detailed theory, proofs of theorems (theorem-theorem!), please refer to the textbook. Well, who wants learn to solve problems in probability theory and mathematical statistics in the shortest possible time, follow me!

Enough to get started =)

As you read the articles, it is advisable to get acquainted (at least briefly) with additional tasks of the types considered. On the page Ready-made solutions for higher mathematics relevant pdf-ki with examples of solutions will be placed. There will also be significant assistance IDZ 18.1 Ryabushko(easier) and solved IDZ according to the collection of Chudesenko(more difficult).

1) sum two events and is called the event which consists in the fact that or event or event or both events at the same time. In case the events incompatible, the last option disappears, that is, it can occur or event or event .

The rule also applies to large quantity terms, for example, an event is what will happen at least one from events , A if the events are incompatible – that one and only one event from this sum: or event , or event , or event , or event , or event .

Plenty of examples:

The event (when throwing a die does not drop 5 points) is that or 1, or 2, or 3, or 4, or 6 points.

Event (will drop no more two points) is that 1 or 2points.

Event (there will be an even number of points) is that the or 2 or 4 or 6 points.

The event is that a card of red suit (heart) will be drawn from the deck or tambourine), and the event - that the “picture” will be extracted (jack or lady or king or ace).

A little more interesting is the case with joint events:

The event is that a club will be drawn from the deck or seven or seven of clubs According to the above definition, at least something- or any club or any seven or their "crossing" - seven clubs. It is easy to calculate that this event corresponds to 12 elementary outcomes (9 club cards + 3 remaining sevens).

The event is that tomorrow at 12.00 AT LEAST ONE of the summable joint events, namely:

- or there will be only rain / only thunder / only sun;
- or only some pair of events will come (rain + thunderstorm / rain + sun / thunderstorm + sun);
– or all three events will appear at the same time.

That is, the event includes 7 possible outcomes.

The second pillar of the algebra of events:

2) work two events and call the event, which consists in the joint appearance of these events, in other words, multiplication means that under some circumstances there will come And event , And event . A similar statement is true for a larger number of events, for example, the work implies that under certain conditions, there will be And event , And event , And event , …, And event .

Consider a trial in which two coins are tossed and the following events:

- heads will fall on the 1st coin;
- the 1st coin will land tails;
- the 2nd coin will land heads;
- the 2nd coin will come up tails.

Then:
And on the 2nd) an eagle will fall out;
- the event consists in the fact that on both coins (on the 1st And on the 2nd) tails will fall out;
– the event is that the 1st coin will land heads And on the 2nd coin tails;
- the event is that the 1st coin will come up tails And on the 2nd coin an eagle.

It is easy to see that the events incompatible (since it cannot, for example, fall out 2 heads and 2 tails at the same time) and form full group (since taken into account All possible outcomes of tossing two coins). Let's summarize these events: . How to interpret this entry? Very simple - multiplication means logical connection AND, and the addition is OR. Thus, the sum is easy to read in understandable human language: “two eagles will fall or two tails or heads on the 1st coin And on the 2nd tail or heads on the 1st coin And eagle on the 2nd coin »

This was an example when in one test several objects are involved, in this case two coins. Another scheme commonly used in practice is repeated tests when, for example, the same dice is thrown 3 times in a row. As a demonstration, consider the following events:

- in the 1st throw, 4 points will fall out;
- in the 2nd roll, 5 points will fall out;
- in the 3rd throw, 6 points will fall out.

Then the event consists in the fact that in the 1st roll 4 points will fall out And in the 2nd roll will drop 5 points And in the 3rd roll, 6 points will fall. Obviously, in the case of a die, there will be significantly more combinations (outcomes) than if we were tossing a coin.

... I understand that, perhaps, they don’t understand very well interesting examples, but these are things that are often encountered in tasks and cannot be avoided. In addition to a coin, a die and a deck of cards, urns with colorful balls, several anonymous people shooting at the target, and a tireless worker who constantly grinds out some details =)

Event Probability

Event Probability is a central concept in probability theory. ...A deadly logical thing, but you had to start somewhere =) There are several approaches to its definition:

;
Geometric definition of probability ;
Statistical definition of probability .

In this article, I will focus on the classical definition of probabilities, which is most widely used in educational tasks.

Notation. The probability of some event is denoted by a capital Latin letter , and the event itself is taken in brackets, acting as a kind of argument. For example:

Also, a small letter is widely used to represent probability. In particular, one can abandon the cumbersome designations of events and their probabilities in favor of the following style:

is the probability that the toss of a coin will result in heads;
- the probability that 5 points will fall out as a result of throwing a dice;
is the probability that a card of the club suit will be drawn from the deck.

This option is popular in solving practical problems, since it allows you to significantly reduce the solution entry. As in the first case, it is convenient to use “talking” subscripts/superscripts here.

Everyone has long guessed about the numbers that I just wrote down above, and now we will find out how they turned out:

The classical definition of probability:

The probability of an event occurring in some test is the ratio , where:

is the total number of all equally possible, elementary outcomes of this test, which form full group of events;

- quantity elementary outcomes favorable event .

When a coin is tossed, either heads or tails can fall out - these events form full group, thus, the total number of outcomes ; while each of them elementary And equally possible. The event is favored by the outcome (heads). According to the classical definition of probabilities: .

Similarly, as a result of a roll of a die, elementary equally possible outcomes may appear, forming a complete group, and the event is favored by a single outcome (rolling a five). That's why: .THIS IS NOT ACCEPTED TO DO (although it is not forbidden to figure out the percentages in your mind).

It is customary to use fractions of a unit, and, obviously, the probability can vary within . Moreover, if , then the event is impossible, If - reliable, and if , then we are talking about random event.

! If in the course of solving any problem you get some other probability value - look for an error!

In the classical approach to the definition of probability, the extreme values (zero and one) are obtained by exactly the same reasoning. Let 1 ball be drawn at random from an urn containing 10 red balls. Consider the following events:

in a single trial, an unlikely event will not occur.

That is why you will not hit the Jackpot in the lottery if the probability of this event is, say, 0.00000001. Yes, yes, it is you - with the only ticket in a particular circulation. However, more tickets and more draws will not help you much. ... When I tell others about this, I almost always hear in response: "but someone wins." Okay, then let's do the following experiment: please buy any lottery ticket today or tomorrow (don't delay!). And if you win ... well, at least more than 10 kilo rubles, be sure to unsubscribe - I will explain why this happened. For a percentage, of course =) =)

But there is no need to be sad, because there is an opposite principle: if the probability of some event is very close to unity, then in a single test it almost certain will happen. Therefore, before a parachute jump, do not be afraid, on the contrary - smile! After all, absolutely unthinkable and fantastic circumstances must arise for both parachutes to fail.

Although all this is poetry, since, depending on the content of the event, the first principle may turn out to be cheerful, and the second - sad; or even both are parallel.

Probably enough for now, in class Tasks for the classical definition of probability we will squeeze the maximum out of the formula. In the final part of this article, we consider one important theorem:

The sum of the probabilities of events that form a complete group is equal to one. Roughly speaking, if events form a complete group, then with 100% probability one of them will happen. In the simplest case, opposite events form a complete group, for example:

- as a result of a coin toss, an eagle will fall out;
- as a result of tossing a coin, tails will fall out.

According to the theorem:

It is clear that these events are equally likely and their probabilities are the same. .

Because of the equality of probabilities, equally probable events are often called equiprobable . And here is the tongue twister for determining the degree of intoxication turned out =)

Dice example: events are opposite, so .

The theorem under consideration is convenient in that it allows you to quickly find the probability of the opposite event. So, if you know the probability that a five will fall out, it is easy to calculate the probability that it will not fall out:

This is much easier than summing up the probabilities of five elementary outcomes. For elementary outcomes, by the way, this theorem is also valid:
. For example, if is the probability that the shooter will hit the target, then is the probability that he will miss.

! In probability theory, it is undesirable to use the letters and for any other purpose.

In honor of Knowledge Day, I will not ask homework=), but it is very important that you can answer next questions:

What types of events are there?
– What is chance and equal possibility of an event?
– How do you understand the terms compatibility / incompatibility of events?
– What is a complete group of events, opposite events?
What does the addition and multiplication of events mean?
– What is the essence of the classical definition of probability?
– Why is the addition theorem for the probabilities of events forming a complete group useful?

No, you don’t need to cram anything, these are just the basics of probability theory - a kind of primer that will fit in your head pretty quickly. And so that this happens as soon as possible, I suggest that you read the lessons

Nizhny Novgorod State Technical University

them. A.E. Alekseeva

Essay on the discipline theory of probability

Completed by: Ruchina N.A gr 10MENz

Checked: Gladkov V.V.

Nizhny Novgorod, 2011

Probability theory……………………………………

The subject of probability theory…………………………

Basic concepts of probability theory……………

Random events, probabilities of events……………………………………………………

Limit theorems……………………………………

Random processes……………………………………

Historical reference…………………………………

Used Books…………………………………………

Probability theory

Probability theory - a mathematical science that allows, by the probabilities of some random events, to find the probabilities of other random events related in some way to the first.

Statement that an event occurs with probability , equal to, for example, 0.75, does not yet represent in itself the final value, since we are striving for reliable knowledge. The final cognitive value are those results of the theory of probability, which allow us to assert that the probability of the occurrence of any event A very close to unity or (which is the same) the probability of the event not occurring A very small. In accordance with the principle of "neglecting sufficiently small probabilities", such an event is rightly considered practically certain. Conclusions of scientific and practical interest of this kind are usually based on the assumption that the occurrence or non-occurrence of an event A depends on a large number of random, little-related factors . Therefore, we can also say that the theory of probability is a mathematical science that explains the patterns that arise when a large number of random factors interact.

The subject of probability theory

The subject of probability theory. To describe a regular relationship between certain conditions S and event A, the occurrence or non-occurrence of which under given conditions can be precisely established, natural science usually uses one of the following two schemes:

a) each time the conditions are met S an event occurs A. For example, all the laws of classical mechanics have this form, which state that for given initial conditions and forces acting on a body or a system of bodies, the movement will occur in a uniquely defined way.

b) Under the conditions S event A has a certain probability P(A/S), equal to R. So, for example, the laws of radioactive radiation state that for each radioactive substance there is a certain probability that from a given amount of substance a certain number will decay in a given period of time. N atoms.

Let's call the event frequency A in this series of n tests (ie. n re-implementation of conditions S) relation h = m/n numbers m the tests in which A has come, to their total number n. The presence of the event A under conditions S certain probability equal to R, manifests itself in the fact that in almost every sufficiently long series of tests, the frequency of the event A approximately equal to R.

Statistical regularities, that is, regularities described by a scheme of type (b), were first discovered on the example of gambling games like dice. The statistical regularities of birth and death have also been known for a very long time (for example, the probability of a newborn being a boy is 0.515). Late 19th century and 1st half of the 20th century. marked by the discovery of a large number of statistical regularities in physics, chemistry, biology, etc.

The possibility of applying the methods of probability theory to the study of statistical regularities relating to very distant fields of science is based on the fact that the probabilities of events always satisfy certain simple relations. The study of the properties of the probabilities of events on the basis of these simple relations is the subject of probability theory.

Basic concepts of probability theory

Basic concepts of probability theory. The basic concepts of probability theory, as a mathematical discipline, are most simply defined within the framework of the so-called elementary probability theory. Every test T, considered in elementary probability theory is such that it ends with one and only one of the events E 1 , E 2 ,..., E S (one or the other, depending on the case). These events are called trial outcomes. With every outcome E k binds a positive number R To - the likelihood of this outcome. Numbers p k must add up to one. Then events are considered. A, consisting in the fact that "comes or E i , or E j ,..., or E k". outcomes E i , E j ,..., E k are called favorable A, and by definition assume the probability R(A) events A equal to the sum of probabilities of favorable outcomes:

P(A) =p i +p s +… +p k . (1)

special case p 1 =p 2 =...p s= 1/S leads to the formula

R(A) =r/s.(2)

Formula (2) expresses the so-called classical definition of probability, according to which the probability of an event A is equal to the ratio of the number r favorable outcomes A, to the number s all "equally possible" outcomes. The classical definition of probability only reduces the notion of "probability" to the notion of "equipossibility", which remains without a clear definition.

Example. When throwing two dice, each of the 36 possible outcomes can be labeled ( i,j), Where i- the number of points dropped on the first die, j- On the second. The outcomes are assumed to be equally probable. event A -"the sum of the points is 4", three outcomes favor (1; 3), (2; 2), (3; 1). Hence, R(A) = 3/36= 1/12.

Based on any data of events, two new events can be defined: their union (sum) and combination (product).

Event IN is called the union of events A 1 , A 2 ,..., A r ,-, if it looks like: "coming or A 1 , or A 2 ,..., or A r ».

The event C is called the coincidence of events A 1 , A. 2 ,..., A r , if it looks like: "comes and A 1 , And A 2 ,..., And A r » . The combination of events is denoted by the sign , and the combination - by the sign . Thus, they write:

B = A 1  A 2  …  A r , C = A 1  A 2  …  A r .

Events A And IN are called incompatible if their simultaneous implementation is impossible, that is, if there is not a single favorable and A And IN.

Two main theorems of the theory of probability are connected with the introduced operations of combining and combining events - the theorems of addition and multiplication of probabilities.

Probability addition theorem: If events A 1 ,A 2 ,...,A r are such that every two of them are incompatible, then the probability of their union is equal to the sum of their probabilities.

So, in the example above with the throw of two dice, the event IN -"the sum of points does not exceed 4", there is a union of three incompatible events A 2 ,A 3 ,A 4 , consisting in the fact that the sum of points is equal to 2, 3, 4, respectively. The probabilities of these events are 1/36; 2/36; 3/36. By the addition theorem, the probability R(IN) is equal to

1/36 + 2/36 + 3/36 = 6/36 = 1/6.

Events A 1 ,A 2 ,...,A r are called independent if the conditional probability of each of them, provided that any of the others occurred, is equal to its "unconditional" probability.

Probability multiplication theorem: Probability of coincidence of events A 1 ,A 2 ,...,A r is equal to the probability of the event A 1 , multiplied by the probability of the event A 2 taken under the condition that A 1 happened,..., multiplied by the probability of the event A r provided that A 1 ,A 2 ,...,A r-1 have arrived. For independent events, the multiplication theorem leads to the formula:

P(A 1 A 2 …A r) =P(A 1 )P(A 2 )· … · P(A r), (3)

that is, the probability of combining independent events is equal to the product of the probabilities of these events. Formula (3) remains valid if some of the events in both parts of it are replaced by opposite ones.

Example. Fires 4 shots at the target with a hit probability of 0.2 on a single shot. Target hits for different shots are assumed to be independent events. What is the probability of hitting the target exactly three times?

Each test outcome can be indicated by a sequence of four letters [e.g., (y, n, n, y) means that the first and fourth shots were hit (success), and the second and third hits were not (failure)]. In total there will be 2 2 2 2 = 16 outcomes. In accordance with the assumption of the independence of the results of individual shots, formula (3) and a note to it should be used to determine the probabilities of these outcomes. So, the probability of the outcome (y, n. n, n) should be set equal to 0.2 0.8 0.8 0.8 = 0.1024; here 0.8 \u003d 1-0.2 - the probability of a miss with a single shot. The event “the target is hit three times” is favored by the outcomes (y, y, y, n), (y, y, n, y), (y, n, y, y). (n, y, y, y), the probability of each is the same:

0.2 0.2 0.2 0.8 =...... = 0.8 0.2 0.2 0.2 = 0.0064;

therefore, the desired probability is equal to

4 0.0064 = 0.0256.

Generalizing the reasoning of the analyzed example, we can derive one of the basic formulas of probability theory: if the events A 1 , A 2 ,..., A n are independent and each have a probability R, then the probability of exactly m of which is equal to

P n (m)=C n m p m (1-p) n-m ; (4)

Here C n m denotes the number of combinations of n elements by m. At large n calculations by formula (4) become difficult.

Among the basic formulas of elementary probability theory is also the so-called total probability formula: if events A 1 , A 2 ,..., A r are pairwise incompatible and their union is a certain event, then for any event IN its probability is equal to their sum.

The probabilities multiplication theorem is especially useful when considering compound tests. They say the test T made up of trials T 1 , T 2 ,..., T n-1 , T n, If each test outcome T there is a combination of some outcomes A i , B j ,..., X k , Y l related tests T 1 , T 2 ,..., T n-1 , T n. From one reason or another, the probabilities are often known

P(A i), P(B j /A i), …,P(Y l /A i B j …X k). (5)

Probabilities (5) can be used to determine the probabilities R(E) for all outcomes E composite test, and at the same time the probabilities of all events associated with this test. From a practical point of view, two types of composite tests seem to be the most significant:

a) the components of the test are independent, that is, the probabilities (5) are equal to the unconditional probabilities P(A i), P(B j),...,P(Y l);

b) the probabilities of the outcomes of any test are affected by the results of only the immediately preceding test, that is, the probabilities (5) are equal, respectively: P(A i), P(B j /A i),...,P(Y i / X k). In this case, one speaks of tests connected in a Markov chain. The probabilities of all events associated with a composite test are completely determined here by the initial probabilities R(A i) and transition probabilities P(B j /A i),...,P(Y l / X k).

Basic formulas in probability theory

Formulas of probability theory.

1. Basic formulas of combinatorics

a) permutations.

\b) placement

c) combinations .

2. Classical definition of probability.

Where is the number of favorable outcomes for the event, is the number of all elementary equally possible outcomes.

3. Probability of the sum of events

The addition theorem for the probabilities of incompatible events:

The theorem of addition of probabilities of joint events:

4. Probability of producing events

The theorem of multiplication of probabilities of independent events:

The theorem of multiplication of probabilities of dependent events:

The conditional probability of an event given that the event occurred,

The conditional probability of an event given that the event occurred.

Combinatorics is a branch of mathematics that studies questions about how many different combinations, subject to certain conditions, can be made from given objects. The basics of combinatorics are very important for estimating the probabilities of random events, because it is they that make it possible to calculate the fundamentally possible number of different scenarios for the development of events.

Basic combinatorics formula

Let there be k groups of elements, and the i-th group consists of ni elements. Let's choose one element from each group. Then the total number N of ways in which such a choice can be made is determined by the relation N=n1*n2*n3*...*nk.

Example 1 Let's explain this rule with a simple example. Let there be two groups of elements, the first group consisting of n1 elements, and the second group consisting of n2 elements. How many various couples elements can be made up of these two groups, so that the pair contains one element from each group? Suppose we took the first element from the first group and, without changing it, went through all possible pairs, changing only the elements from the second group. There are n2 such pairs for this element. Then we take the second element from the first group and also make all possible pairs for it. There will also be n2 such pairs. Since there are only n1 elements in the first group, there will be n1 * n2 possible options.

Example 2. How many three-digit even numbers can be made from the digits 0, 1, 2, 3, 4, 5, 6 if the digits can be repeated?

Solution: n1=6 (since you can take any digit from 1, 2, 3, 4, 5, 6 as the first digit), n2=7 (because you can take any digit from 0 as the second digit , 1, 2, 3, 4, 5, 6), n3=4 (since you can take any digit from 0, 2, 4, 6 as the third digit).

So, N=n1*n2*n3=6*7*4=168.

In the case when all groups consist of the same number of elements, i.e. n1=n2=...nk=n we can assume that each choice is made from the same group, and the element after the choice is returned to the group again. Then the number of all selection methods is equal to nk. Such a selection method is called sampling with return.

Example. How many four-digit numbers can be made from the numbers 1, 5, 6, 7, 8?

Solution. There are five possibilities for each digit of a four-digit number, so N=5*5*5*5=54=625.

Consider a set consisting of n elements. This set will be called the general population.

Definition 1. An arrangement of n elements by m is any ordered set of m distinct elements selected from population in n elements.

Example. Different arrangements of three elements (1, 2, 3) two by two will be sets (1, 2), (2, 1), (1, 3), (3, 1), (2, 3), (3, 2 ). Placements can differ from each other both in elements and in their order.

The number of placements is denoted by A, m from n and is calculated by the formula:

Note: n!=1*2*3*...*n (read: "en factorial"), besides, it is assumed that 0!=1.

Example 5. How many two-digit numbers are there in which the tens digit and the units digit are different and odd?

Solution: because there are five odd digits, namely 1, 3, 5, 7, 9, then this problem is reduced to choosing and placing two of the five different digits in two different positions, i.e. the given numbers will be:

Definition 2. A combination of n elements by m is any unordered set of m different elements selected from a general population of n elements.

Example 6. For the set (1, 2, 3), the combinations are (1, 2), (1, 3), (2, 3).

The number of combinations is denoted by Cnm and is calculated by the formula:

Definition 3. A permutation of n elements is any ordered set of these elements.

Example 7a. All possible permutations of a set consisting of three elements (1, 2, 3) are: (1, 2, 3), (1, 3, 2), (2, 3, 1), (2, 1, 3), ( 3, 2, 1), (3, 1, 2).

The number of different permutations of n elements is denoted by Pn and is calculated by the formula Pn=n!.

Example 8. In how many ways can seven books by different authors be arranged on a shelf in one row?

Solution: This problem is about the number of permutations of seven different books. There are P7=7!=1*2*3*4*5*6*7=5040 ways to arrange the books.

Discussion. We see that the number of possible combinations can be calculated according to different rules (permutations, combinations, placements), and the result will be different, because the principle of counting and the formulas themselves are different. Looking closely at the definitions, you can see that the result depends on several factors at the same time.

Firstly, from how many elements we can combine their sets (how large is the general population of elements).

Secondly, the result depends on what size sets of elements we need.

Finally, it is important to know whether the order of the elements in the set is significant for us. Let's explain last factor on the next example.

Example. There are 20 people at the parent meeting. How many different options for the composition of the parent committee are there if it should include 5 people?

Solution: In this example, we are not interested in the order of the names on the committee list. If, as a result, the same people appear in its composition, then in terms of meaning for us this is the same option. Therefore, we can use the formula to count the number of combinations of 20 elements by 5.

Things will be different if each member of the committee is initially responsible for a certain area of work. Then, with the same payroll of the committee, 5 are possible inside it! permutation options that matter. The number of different (both in terms of composition and area of responsibility) options is determined in this case by the number of placements of 20 elements by 5.

Geometric definition of probability

Let a random test be thought of as throwing a point at random into some geometric region G (on a line, plane, or space). Elementary outcomes are individual points G, any event is a subset of this area, the space of elementary outcomes G. We can assume that all points G are “equal” and then the probability of a point falling into a certain subset is proportional to its measure (length, area, volume) and independent of its location and shape.

The geometric probability of event A is determined by the relation: , where m(G), m(A) are geometric measures (lengths, areas or volumes) of the entire space of elementary outcomes and event A.

Example. A circle of radius r () is thrown at random onto a plane divided by parallel strips of width 2d, the distance between the axial lines of which is equal to 2D. Find the probability that the circle intersects some strip.

Solution. As an elementary outcome of this test, we will consider the distance x from the center of the circle to the center line of the strip closest to the circle. Then the entire space of elementary outcomes is a segment . The intersection of a circle with a strip will occur if its center falls into the strip, i.e., or is located at a distance less than the radius from the edge of the strip, i.e..

For the desired probability, we obtain: .

Classification of events into possible, probable and random. The concepts of simple and complex elementary events. Operations on events. The classical definition of the probability of a random event and its properties. Elements of combinatorics in probability theory. geometric probability. Axioms of the theory of probability.

1. Classification of events

One of the basic concepts of probability theory is the concept of an event. An event is understood to mean any fact that can occur as a result of an experience or test. Under the experience, or test, is understood the implementation of a certain set of conditions.

Event examples:

- hitting the target when firing from a gun (experience - the product of a shot; event - hitting the target);

- the loss of two coats of arms during a three-time toss of a coin (experience - a three-time toss of a coin; an event - the loss of two coats of arms);

- the appearance of a measurement error within the specified limits when measuring the distance to the target (experiment - distance measurement; event - measurement error).

Countless such examples could be cited. Events are indicated by capital letters of the Latin alphabet, etc.

Distinguish between joint and non-joint events. Events are called joint if the occurrence of one of them does not exclude the occurrence of the other. Otherwise, the events are called incompatible. For example, two dice are tossed. Event - loss of three points on the first dice, event - loss of three points on the second die. and - joint events. Let the store receive a batch of shoes of the same style and size, but different color. An event - a box taken at random will be with black shoes, an event - a box will be with brown shoes, and - incompatible events.

An event is called certain if it will necessarily occur under the conditions of a given experiment.

An event is said to be impossible if it cannot occur under the conditions of the given experience. For example, the event that a standard part is taken from a batch of standard parts is certain, but a non-standard part is impossible.

An event is called possible or random if, as a result of experience, it may or may not occur. An example of a random event is the identification of product defects during the control of a batch of finished products, the discrepancy between the size of the processed product and the given one, the failure of one of the links of the automated control system.

Events are said to be equally likely if, under the conditions of the test, none of these events is objectively more likely than the others. For example, suppose a store is supplied with light bulbs (and in equal quantities) by several manufacturers. Events consisting in buying a light bulb from any of these factories are equally probable.

An important concept is the complete group of events. Several events in a given experiment form a complete group if at least one of them necessarily appears as a result of the experiment. For example, there are ten balls in an urn, of which six are red and four are white, five of which are numbered. - the appearance of a red ball with one drawing, - the appearance of a white ball, - the appearance of a ball with a number. Events form a complete group of joint events.

Let us introduce the concept of the opposite, or additional, event. An opposite event is an event that must necessarily occur if some event has not occurred. Opposite events are incompatible and the only possible ones. They form a complete group of events. For example, if a batch of manufactured items consists of good and defective items, then when one item is removed, it can turn out to be either good - an event, or defective - an event.

2. Operations on events

When developing the apparatus and methodology for studying random events in probability theory, the concept of the sum and product of events is very important.

"Randomness is not accidental"... It sounds like a philosopher said, but in fact, the study of accidents is the destiny of the great science of mathematics. In mathematics, chance is the theory of probability. Formulas and examples of tasks, as well as the main definitions of this science will be presented in the article.

What is Probability Theory?

Probability theory is one of the mathematical disciplines that studies random events.

To make it a little clearer, let's give a small example: if you toss a coin up, it can fall heads or tails. As long as the coin is in the air, both of these possibilities are possible. That is, the probability possible consequences the ratio is 1:1. If one is drawn from a deck with 36 cards, then the probability will be indicated as 1:36. It would seem that there is nothing to explore and predict, especially with the help of mathematical formulas. Nevertheless, if you repeat a certain action many times, then you can identify a certain pattern and, on its basis, predict the outcome of events in other conditions.

To summarize all of the above, the theory of probability in the classical sense studies the possibility of the occurrence of one of the possible events in a numerical sense.

From the pages of history

The theory of probability, formulas and examples of the first tasks appeared in the distant Middle Ages, when attempts to predict the outcome of card games first arose.

Initially, the theory of probability had nothing to do with mathematics. It was justified by empirical facts or properties of an event that could be reproduced in practice. The first work in this area as in mathematical discipline appeared in the 17th century. The founders were Blaise Pascal and Pierre Fermat. For a long time they studied gambling and saw certain patterns, which they decided to tell the public about.

The same technique was invented by Christian Huygens, although he was not familiar with the results of the research of Pascal and Fermat. The concept of "probability theory", formulas and examples, which are considered the first in the history of the discipline, were introduced by him.

Of no small importance are the works of Jacob Bernoulli, Laplace's and Poisson's theorems. They made probability theory more like a mathematical discipline. Probability theory, formulas and examples of basic tasks got their present form thanks to Kolmogorov's axioms. As a result of all the changes, the theory of probability has become one of the mathematical branches.

Basic concepts of probability theory. Events

The main concept of this discipline is "event". Events are of three types:

Reliable. Those that will happen anyway (the coin will fall).
Impossible. Events that will not happen in any scenario (the coin will remain hanging in the air).
Random. The ones that will or won't happen. They can be influenced by various factors that are very difficult to predict. If we talk about a coin, then random factors that can affect the result: the physical characteristics of the coin, its shape, initial position, throw force, etc.

All events in the examples are indicated by capital letters. with Latin letters, with the exception of P, which has a different role. For example:

A = "students came to the lecture."
Ā = "students didn't come to the lecture".

IN practical tasks Events are written down in words.

One of the most important characteristics events - their equivalence. That is, if you toss a coin, all variants of the initial fall are possible until it falls. But events are also not equally probable. This happens when someone deliberately influences the outcome. For example, "labeled" playing cards or dice, in which off center gravity.

Events are also compatible and incompatible. Compatible events do not exclude the occurrence of each other. For example:

A = "the student came to the lecture."
B = "the student came to the lecture."

These events are independent of each other, and the appearance of one of them does not affect the appearance of the other. Incompatible events are defined by the fact that the occurrence of one precludes the occurrence of the other. If we talk about the same coin, then the loss of "tails" makes it impossible for the appearance of "heads" in the same experiment.

Actions on events

Events can be multiplied and added, respectively, logical connectives "AND" and "OR" are introduced in the discipline.

The amount is determined by the fact that either event A, or B, or both can occur at the same time. In the case when they are incompatible, the last option is impossible, either A or B will drop out.

The multiplication of events consists in the appearance of A and B at the same time.

Now you can give a few examples to better remember the basics, probability theory and formulas. Examples of problem solving below.

Exercise 1: The firm is bidding for contracts for three types of work. Possible events that may occur:

A = "the firm will receive the first contract."
A 1 = "the firm will not receive the first contract."
B = "the firm will receive a second contract."
B 1 = "the firm will not receive a second contract"
C = "the firm will receive a third contract."
C 1 = "the firm will not receive a third contract."

Let's try to express the following situations using actions on events:

K = "the firm will receive all contracts."

In mathematical form, the equation will look like this: K = ABC.

M = "the firm will not receive a single contract."

M \u003d A 1 B 1 C 1.

We complicate the task: H = "the firm will receive one contract." Since it is not known which contract the firm will receive (the first, second or third), it is necessary to record the entire range of possible events:

H \u003d A 1 BC 1 υ AB 1 C 1 υ A 1 B 1 C.

And 1 BC 1 is a series of events where the firm does not receive the first and third contract, but receives the second one. Other possible events are also recorded by the corresponding method. The symbol υ in the discipline denotes a bunch of "OR". If we translate the above example into human language, then the company will receive either the third contract, or the second, or the first. Similarly, you can write other conditions in the discipline "Probability Theory". The formulas and examples of solving problems presented above will help you do it yourself.

Actually, the probability

Perhaps, in this mathematical discipline, the probability of an event is a central concept. There are 3 definitions of probability:

classical;
statistical;
geometric.

Each has its place in the study of probabilities. Probability theory, formulas and examples (Grade 9) mostly use the classic definition, which sounds like this:

The probability of situation A is equal to the ratio of the number of outcomes that favor its occurrence to the number of all possible outcomes.

The formula looks like this: P (A) \u003d m / n.

And, actually, an event. If the opposite of A occurs, it can be written as Ā or A 1 .

m is the number of possible favorable cases.

n - all events that can happen.

For example, A \u003d "pull out a heart suit card." There are 36 cards in a standard deck, 9 of them are of hearts. Accordingly, the formula for solving the problem will look like:

P(A)=9/36=0.25.

As a result, the probability that a heart-suited card will be drawn from the deck will be 0.25.

to higher mathematics

Now it has become a little known what the theory of probability is, formulas and examples of solving problems that come across in school curriculum. However, the theory of probability is also found in higher mathematics, which is taught in universities. Most often, they operate with geometric and statistical definitions of the theory and complex formulas.

The theory of probability is very interesting. Formulas and examples (higher mathematics) are better to start learning from a small one - from a statistical (or frequency) definition of probability.

The statistical approach does not contradict the classical one, but slightly expands it. If in the first case it was necessary to determine with what degree of probability an event will happen, then in this method you need to specify how often it will occur. Here a new concept of “relative frequency” is introduced, which can be denoted by W n (A). The formula is no different from the classic:

If the classical formula is calculated for forecasting, then the statistical one is calculated according to the results of the experiment. Take, for example, a small task.

The department of technological control checks products for quality. Among 100 products, 3 were found to be of poor quality. How to find the frequency probability of a quality product?

A = "the appearance of a quality product."

W n (A)=97/100=0.97

Thus, the frequency of a quality product is 0.97. Where did you get 97 from? Of the 100 products that were checked, 3 turned out to be of poor quality. We subtract 3 from 100, we get 97, this is the quantity of a quality product.

A bit about combinatorics

Another method of probability theory is called combinatorics. Its main principle is that if a certain choice A can be made m different ways, and the choice of B - n different ways, then the choice of A and B can be done by multiplication.

For example, there are 5 roads from city A to city B. There are 4 routes from city B to city C. How many ways are there to get from city A to city C?

It's simple: 5x4 = 20, that is, there are twenty different ways to get from point A to point C.

Let's make the task harder. How many ways are there to play cards in solitaire? In a deck of 36 cards, this is the starting point. To find out the number of ways, you need to “subtract” one card from the starting point and multiply.

That is, 36x35x34x33x32…x2x1= the result does not fit on the calculator screen, so it can simply be denoted as 36!. Sign "!" next to the number indicates that the entire series of numbers is multiplied among themselves.

In combinatorics, there are such concepts as permutation, placement and combination. Each of them has its own formula.

An ordered set of set elements is called a layout. Placements can be repetitive, meaning one element can be used multiple times. And without repetition, when the elements are not repeated. n is all elements, m is the elements that participate in the placement. The formula for placement without repetitions will look like:

A n m =n!/(n-m)!

Connections of n elements that differ only in the order of placement are called permutations. In mathematics, this looks like: P n = n!

Combinations of n elements by m are such compounds in which it is important which elements they were and what their total. The formula will look like:

A n m =n!/m!(n-m)!

Bernoulli formula

In the theory of probability, as well as in every discipline, there are works of outstanding researchers in their field who have taken it to a new level. One of these works is the Bernoulli formula, which allows you to determine the probability of a certain event occurring under independent conditions. This suggests that the appearance of A in an experiment does not depend on the appearance or non-occurrence of the same event in previous or subsequent tests.

Bernoulli equation:

P n (m) = C n m ×p m ×q n-m .

The probability (p) of the occurrence of the event (A) is unchanged for each trial. The probability that the situation will happen exactly m times in n number of experiments will be calculated by the formula that is presented above. Accordingly, the question arises of how to find out the number q.

If event A occurs p number of times, accordingly, it may not occur. A unit is a number that is used to designate all outcomes of a situation in a discipline. Therefore, q is a number that indicates the possibility of the event not occurring.

Now you know the Bernoulli formula (probability theory). Examples of problem solving (the first level) will be considered below.

Task 2: A store visitor will make a purchase with a probability of 0.2. 6 visitors entered the store independently. What is the probability that a visitor will make a purchase?

Solution: Since it is not known how many visitors should make a purchase, one or all six, it is necessary to calculate all possible probabilities using the Bernoulli formula.

A = "the visitor will make a purchase."

In this case: p = 0.2 (as indicated in the task). Accordingly, q=1-0.2 = 0.8.

n = 6 (because there are 6 customers in the store). The number m will change from 0 (no customer will make a purchase) to 6 (all store visitors will purchase something). As a result, we get the solution:

P 6 (0) \u003d C 0 6 × p 0 × q 6 \u003d q 6 \u003d (0.8) 6 \u003d 0.2621.

None of the buyers will make a purchase with a probability of 0.2621.

How else is the Bernoulli formula (probability theory) used? Examples of problem solving (second level) below.

After the above example, questions arise about where C and p have gone. With respect to p, a number to the power of 0 will be equal to one. As for C, it can be found by the formula:

C n m = n! /m!(n-m)!

Since in the first example m = 0, respectively, C=1, which in principle does not affect the result. Using the new formula, let's try to find out what is the probability of buying goods by two visitors.

P 6 (2) = C 6 2 ×p 2 ×q 4 = (6×5×4×3×2×1) / (2×1×4×3×2×1) × (0.2) 2 × (0.8) 4 = 15 × 0.04 × 0.4096 = 0.246.

The theory of probability is not so complicated. The Bernoulli formula, examples of which are presented above, is a direct proof of this.

Poisson formula

The Poisson equation is used to calculate unlikely random situations.

Basic formula:

P n (m)=λ m /m! × e (-λ) .

In this case, λ = n x p. Here is such a simple Poisson formula (probability theory). Examples of problem solving will be considered below.

Task 3 A: The factory produced 100,000 parts. The appearance of a defective part = 0.0001. What is the probability that there will be 5 defective parts in a batch?

As you can see, marriage is an unlikely event, and therefore the Poisson formula (probability theory) is used for calculation. Examples of solving problems of this kind are no different from other tasks of the discipline, we substitute the necessary data into the above formula:

A = "a randomly selected part will be defective."

p = 0.0001 (according to the assignment condition).

n = 100000 (number of parts).

m = 5 (defective parts). We substitute the data in the formula and get:

R 100000 (5) = 10 5 / 5! X e -10 = 0.0375.

Just like the Bernoulli formula (probability theory), examples of solutions using which are written above, the Poisson equation has an unknown e. In essence, it can be found by the formula:

e -λ = lim n ->∞ (1-λ/n) n .

However, there are special tables that contain almost all the values of e.

De Moivre-Laplace theorem

If in the Bernoulli scheme the number of trials is sufficiently large, and the probability of the occurrence of event A in all schemes is the same, then the probability of the occurrence of event A a certain number of times in a series of trials can be found by the Laplace formula:

Р n (m)= 1/√npq x ϕ(X m).

Xm = m-np/√npq.

To better remember the Laplace formula (probability theory), examples of tasks to help below.

First we find X m , we substitute the data (they are all indicated above) into the formula and get 0.025. Using tables, we find the number ϕ (0.025), the value of which is 0.3988. Now you can substitute all the data in the formula:

P 800 (267) \u003d 1 / √ (800 x 1/3 x 2/3) x 0.3988 \u003d 3/40 x 0.3988 \u003d 0.03.

So the probability that the flyer will hit exactly 267 times is 0.03.

Bayes formula

The Bayes formula (probability theory), examples of solving tasks with the help of which will be given below, is an equation that describes the probability of an event, based on the circumstances that could be associated with it. The main formula is as follows:

P (A|B) = P (B|A) x P (A) / P (B).

A and B are definite events.

P(A|B) - conditional probability, that is, event A can occur, provided that event B is true.

Р (В|А) - conditional probability of event В.

So, the final part of the short course "Theory of Probability" is the Bayes formula, examples of solving problems with which are below.

Task 5: Phones from three companies were brought to the warehouse. At the same time, part of the phones that are manufactured at the first plant is 25%, at the second - 60%, at the third - 15%. It is also known that the average percentage of defective products at the first factory is 2%, at the second - 4%, and at the third - 1%. It is necessary to find the probability that a randomly selected phone will be defective.

A = "randomly taken phone."

B 1 - the phone that the first factory made. Accordingly, introductory B 2 and B 3 will appear (for the second and third factories).

As a result, we get:

P (B 1) \u003d 25% / 100% \u003d 0.25; P (B 2) \u003d 0.6; P (B 3) \u003d 0.15 - so we found the probability of each option.

Now you need to find the conditional probabilities of the desired event, that is, the probability of defective products in firms:

P (A / B 1) \u003d 2% / 100% \u003d 0.02;

P (A / B 2) \u003d 0.04;

P (A / B 3) \u003d 0.01.

Now we substitute the data into the Bayes formula and get:

P (A) \u003d 0.25 x 0.2 + 0.6 x 0.4 + 0.15 x 0.01 \u003d 0.0305.

The article presents the theory of probability, formulas and examples of problem solving, but this is only the tip of the iceberg of a vast discipline. And after all that has been written, it will be logical to ask the question of whether the theory of probability is needed in life. To the common man difficult to answer, it is better to ask someone who has hit the jackpot more than once with it.

What is Probability Theory?

Probability theory is one of the mathematical disciplines that studies random events.

To make it a little clearer, let's give a small example: if you toss a coin up, it can fall heads or tails. As long as the coin is in the air, both of these possibilities are possible. That is, the probability of possible consequences correlates 1:1. If one is drawn from a deck with 36 cards, then the probability will be indicated as 1:36. It would seem that there is nothing to explore and predict, especially with the help of mathematical formulas. Nevertheless, if you repeat a certain action many times, then you can identify a certain pattern and, on its basis, predict the outcome of events in other conditions.

To summarize all of the above, the theory of probability in the classical sense studies the possibility of the occurrence of one of the possible events in a numerical sense.

From the pages of history

The theory of probability, formulas and examples of the first tasks appeared in the distant Middle Ages, when attempts to predict the outcome of card games first arose.

Initially, the theory of probability had nothing to do with mathematics. It was justified by empirical facts or properties of an event that could be reproduced in practice. The first works in this area as a mathematical discipline appeared in the 17th century. The founders were Blaise Pascal and Pierre Fermat. For a long time they studied gambling and saw certain patterns, which they decided to tell the public about.

Basic concepts of probability theory. Events

The main concept of this discipline is "event". Events are of three types:

Reliable. Those that will happen anyway (the coin will fall).
Impossible. Events that will not happen in any scenario (the coin will remain hanging in the air).
Random. The ones that will or won't happen. They can be influenced by various factors that are very difficult to predict. If we talk about a coin, then random factors that can affect the result: the physical characteristics of the coin, its shape, initial position, throw force, etc.

All events in the examples are denoted by capital Latin letters, with the exception of R, which has a different role. For example:

A = "students came to the lecture."
Ā = "students didn't come to the lecture".

In practical tasks, events are usually recorded in words.

One of the most important characteristics of events is their equal possibility. That is, if you toss a coin, all variants of the initial fall are possible until it falls. But events are also not equally probable. This happens when someone deliberately influences the outcome. For example, "marked" playing cards or dice, in which the center of gravity is shifted.

Events are also compatible and incompatible. Compatible events do not exclude the occurrence of each other. For example:

A = "the student came to the lecture."
B = "the student came to the lecture."

Actions on events

Events can be multiplied and added, respectively, logical connectives "AND" and "OR" are introduced in the discipline.

The amount is determined by the fact that either event A, or B, or both can occur at the same time. In the case when they are incompatible, the last option is impossible, either A or B will drop out.

The multiplication of events consists in the appearance of A and B at the same time.

Now you can give a few examples to better remember the basics, probability theory and formulas. Examples of problem solving below.

Exercise 1: The firm is bidding for contracts for three types of work. Possible events that may occur:

A = "the firm will receive the first contract."
A 1 = "the firm will not receive the first contract."
B = "the firm will receive a second contract."
B 1 = "the firm will not receive a second contract"
C = "the firm will receive a third contract."
C 1 = "the firm will not receive a third contract."

Let's try to express the following situations using actions on events:

K = "the firm will receive all contracts."

In mathematical form, the equation will look like this: K = ABC.

M = "the firm will not receive a single contract."

M \u003d A 1 B 1 C 1.

H \u003d A 1 BC 1 υ AB 1 C 1 υ A 1 B 1 C.

Actually, the probability

Perhaps, in this mathematical discipline, the probability of an event is a central concept. There are 3 definitions of probability:

classical;
statistical;
geometric.

Each has its place in the study of probabilities. Probability theory, formulas and examples (Grade 9) mostly use the classic definition, which sounds like this:

The probability of situation A is equal to the ratio of the number of outcomes that favor its occurrence to the number of all possible outcomes.

The formula looks like this: P (A) \u003d m / n.

And, actually, an event. If the opposite of A occurs, it can be written as Ā or A 1 .

m is the number of possible favorable cases.

n - all events that can happen.

For example, A \u003d "pull out a heart suit card." There are 36 cards in a standard deck, 9 of them are of hearts. Accordingly, the formula for solving the problem will look like:

P(A)=9/36=0.25.

As a result, the probability that a heart-suited card will be drawn from the deck will be 0.25.

to higher mathematics

Now it has become a little known what the theory of probability is, formulas and examples of solving tasks that come across in the school curriculum. However, the theory of probability is also found in higher mathematics, which is taught in universities. Most often, they operate with geometric and statistical definitions of the theory and complex formulas.

The theory of probability is very interesting. Formulas and examples (higher mathematics) are better to start learning from a small one - from a statistical (or frequency) definition of probability.

The statistical approach does not contradict the classical one, but slightly expands it. If in the first case it was necessary to determine with what degree of probability an event will occur, then in this method it is necessary to indicate how often it will occur. Here a new concept of “relative frequency” is introduced, which can be denoted by W n (A). The formula is no different from the classic:

If the classical formula is calculated for forecasting, then the statistical one is calculated according to the results of the experiment. Take, for example, a small task.

The department of technological control checks products for quality. Among 100 products, 3 were found to be of poor quality. How to find the frequency probability of a quality product?

A = "the appearance of a quality product."

W n (A)=97/100=0.97

A bit about combinatorics

Another method of probability theory is called combinatorics. Its basic principle is that if a certain choice A can be made in m different ways, and a choice B in n different ways, then the choice of A and B can be made by multiplying.

For example, there are 5 roads from city A to city B. There are 4 routes from city B to city C. How many ways are there to get from city A to city C?

It's simple: 5x4 = 20, that is, there are twenty different ways to get from point A to point C.

In combinatorics, there are such concepts as permutation, placement and combination. Each of them has its own formula.

A n m =n!/(n-m)!

Connections of n elements that differ only in the order of placement are called permutations. In mathematics, this looks like: P n = n!

Combinations of n elements by m are such compounds in which it is important which elements they were and what is their total number. The formula will look like:

A n m =n!/m!(n-m)!

Bernoulli formula

Bernoulli equation:

P n (m) = C n m ×p m ×q n-m .

Now you know the Bernoulli formula (probability theory). Examples of problem solving (the first level) will be considered below.

Task 2: A store visitor will make a purchase with a probability of 0.2. 6 visitors entered the store independently. What is the probability that a visitor will make a purchase?

Solution: Since it is not known how many visitors should make a purchase, one or all six, it is necessary to calculate all possible probabilities using the Bernoulli formula.

A = "the visitor will make a purchase."

In this case: p = 0.2 (as indicated in the task). Accordingly, q=1-0.2 = 0.8.

P 6 (0) \u003d C 0 6 × p 0 × q 6 \u003d q 6 \u003d (0.8) 6 \u003d 0.2621.

None of the buyers will make a purchase with a probability of 0.2621.

How else is the Bernoulli formula (probability theory) used? Examples of problem solving (second level) below.

After the above example, questions arise about where C and p have gone. With respect to p, a number to the power of 0 will be equal to one. As for C, it can be found by the formula:

C n m = n! /m!(n-m)!

P 6 (2) = C 6 2 ×p 2 ×q 4 = (6×5×4×3×2×1) / (2×1×4×3×2×1) × (0.2) 2 × (0.8) 4 = 15 × 0.04 × 0.4096 = 0.246.

The theory of probability is not so complicated. The Bernoulli formula, examples of which are presented above, is a direct proof of this.

Poisson formula

The Poisson equation is used to calculate unlikely random situations.

Basic formula:

P n (m)=λ m /m! × e (-λ) .

In this case, λ = n x p. Here is such a simple Poisson formula (probability theory). Examples of problem solving will be considered below.

Task 3 A: The factory produced 100,000 parts. The appearance of a defective part = 0.0001. What is the probability that there will be 5 defective parts in a batch?

A = "a randomly selected part will be defective."

p = 0.0001 (according to the assignment condition).

n = 100000 (number of parts).

m = 5 (defective parts). We substitute the data in the formula and get:

R 100000 (5) = 10 5 / 5! X e -10 = 0.0375.

Just like the Bernoulli formula (probability theory), examples of solutions using which are written above, the Poisson equation has an unknown e. In essence, it can be found by the formula:

e -λ = lim n ->∞ (1-λ/n) n .

However, there are special tables that contain almost all the values of e.

De Moivre-Laplace theorem

Р n (m)= 1/√npq x ϕ(X m).

Xm = m-np/√npq.

To better remember the Laplace formula (probability theory), examples of tasks to help below.

P 800 (267) \u003d 1 / √ (800 x 1/3 x 2/3) x 0.3988 \u003d 3/40 x 0.3988 \u003d 0.03.

So the probability that the flyer will hit exactly 267 times is 0.03.

Bayes formula

P (A|B) = P (B|A) x P (A) / P (B).

A and B are definite events.

P(A|B) - conditional probability, that is, event A can occur, provided that event B is true.

Р (В|А) - conditional probability of event В.

So, the final part of the short course "Theory of Probability" is the Bayes formula, examples of solving problems with which are below.

A = "randomly taken phone."

B 1 - the phone that the first factory made. Accordingly, introductory B 2 and B 3 will appear (for the second and third factories).

As a result, we get:

P (B 1) \u003d 25% / 100% \u003d 0.25; P (B 2) \u003d 0.6; P (B 3) \u003d 0.15 - so we found the probability of each option.

Now you need to find the conditional probabilities of the desired event, that is, the probability of defective products in firms:

P (A / B 1) \u003d 2% / 100% \u003d 0.02;

P (A / B 2) \u003d 0.04;

P (A / B 3) \u003d 0.01.

Now we substitute the data into the Bayes formula and get:

P (A) \u003d 0.25 x 0.2 + 0.6 x 0.4 + 0.15 x 0.01 \u003d 0.0305.

The article presents the theory of probability, formulas and examples of problem solving, but this is only the tip of the iceberg of a vast discipline. And after all that has been written, it will be logical to ask the question of whether the theory of probability is needed in life. It is difficult for a simple person to answer, it is better to ask someone who has hit the jackpot more than once with her help.

Probability theory - a mathematical science that studies the patterns of random phenomena. Random phenomena are understood as phenomena with an uncertain outcome that occur when a certain set of conditions is repeatedly reproduced.

For example, when you toss a coin, you cannot predict which side it will fall on. The result of tossing a coin is random. But with a sufficiently large number of coin tosses, there is a certain pattern (the coat of arms and the lattice will fall out approximately the same number of times).

Basic concepts of probability theory

test (experiment, experiment) - the implementation of a certain set of conditions in which this or that phenomenon is observed, this or that result is fixed.

For example: tossing a dice with a loss of points; air temperature difference; method of treating the disease; some period of a person's life.

Random event (or just an event) - outcome of the test.

Examples of random events:

dropping one point when throwing a dice;

exacerbation coronary disease hearts with a sharp increase in air temperature in summer;

the development of complications of the disease with the wrong choice of treatment method;

admission to a university with successful study at school.

Events denote capital letters Latin Alpha Vita: A , B , C , …

The event is called reliable if as a result of the test it must necessarily occur.

The event is called impossible if, as a result of the test, it cannot occur at all.

For example, if all products in a batch are standard, then the extraction of a standard product from it is a reliable event, and the extraction of a defective product under the same conditions is an impossible event.

CLASSICAL DEFINITION OF PROBABILITY

Probability is one of the basic concepts of probability theory.

The classical probability of an event is the ratio of the number of cases favorable to the event , to the total number of cases, i.e.

, (5.1)

Where
- event probability ,

- number of favorable events ,

is the total number of cases.

Event Probability Properties

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

The probability of a certain event is equal to one, i.e.

The probability of an impossible event is zero, i.e.

(Suggest to solve a few simple tasks orally).

STATISTICAL DEFINITION OF PROBABILITY

In practice, often when evaluating the probabilities of events, they are based on how often a given event will occur in the tests performed. In this case, the statistical definition of probability is used.

Statistical probability of an event is called the limit of relative frequency (the ratio of the number of cases m, favorable to the occurrence of the event , to the total number performed tests), when the number of tests tends to infinity, i.e.

Where
- statistical probability events ,
- number of trials in which the event appeared , - total number of trials.

Unlike classical probability, statistical probability is a characteristic of an experimental one. Classical probability is used to theoretically calculate the probability of an event under given conditions and does not require that tests be carried out in reality. The statistical probability formula is used to experimentally determine the probability of an event, i.e. it is assumed that the tests were actually carried out.

The statistical probability is approximately equal to the relative frequency of a random event, therefore, in practice, the relative frequency is taken as the statistical probability, since statistical probability is almost impossible to find.

The statistical definition of probability applies to random events that have the following properties:

Theorems of addition and multiplication of probabilities

Basic concepts

a) The only possible events

Events
are called the only possible ones if, as a result of each test, at least one of them will surely occur.

These events form a complete group of events.

For example, when rolling a dice, the only possible events are the face rolls with one, two, three, four, five, and six points. They form a complete group of events.

b) Events are called incompatible if the occurrence of one of them excludes the occurrence of other events in the same trial. Otherwise, they are called joint.

c) Opposite name two uniquely possible events that form a complete group. designate And .

G) Events are called independent, if the probability of occurrence of one of them does not depend on the commission or non-completion of others.

Actions on events

The sum of several events is an event consisting in the occurrence of at least one of these events.

If And are joint events, then their sum
or
denotes the occurrence of either event A, or event B, or both events together.

If And are incompatible events, then their sum
means occurrence or event , or events .

Amount events are:

The product (intersection) of several events is an event consisting in the joint occurrence of all these events.

The product of two events is
or
.

Work events denote

The addition theorem for the probabilities of incompatible events

The probability of the sum of two or more incompatible events is equal to the sum of the probabilities of these events:

For two events;

- For events.

Consequences:

a) The sum of the probabilities of opposite events And is equal to one:

The probability of the opposite event is denoted :
.

b) Sum of probabilities events that form a complete group of events is equal to one: or
.

Addition theorem for joint event probabilities

The probability of the sum of two joint events is equal to the sum of the probabilities of these events without the probabilities of their intersection, i.e.

Probability multiplication theorem

a) For two independent events:

b) For two dependent events

Where
is the conditional probability of the event , i.e. event probability , calculated under the condition that the event happened.

c) For independent events:

d) The probability of the occurrence of at least one of the events , forming a complete group of independent events:

Conditional Probability

Event Probability , calculated assuming that an event has occurred , is called the conditional probability of the event and denoted
or
.

When calculating conditional probability according to the classical probability formula, the number of outcomes And
is calculated taking into account the fact that before the event an event happened .