Classical and statistical definition of probability

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

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

Classic definition probability is related to the notion of a favorable outcome. Let's dwell on this in more detail.

Let the outcomes of some test form a complete group of events and be equally probable, i.e. are uniquely possible, inconsistent and equally possible. Such outcomes are called elementary outcomes, or cases. It is said that the test is reduced to case chart or " urn scheme”, because any probabilistic problem for such a test can be replaced by an equivalent problem with urns and balls of different colors.

Exodus is called favorable event BUT if the occurrence of this case entails the occurrence of the event BUT.

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

,

(1.1)

where P(A)- the probability of an event BUT; m- the number of cases favorable to the event BUT; n is the total number of cases.

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

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

P(A) = = . ◄

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

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

0 ≤ R(BUT) ≤ 1.

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

3. The probability of an impossible event is zero.

As mentioned earlier, the classical definition of probability is applicable only for those events that can appear as a result of trials that have symmetry of possible outcomes, i.e. reducible to the scheme of cases. However, there is a large class of events whose probabilities cannot be calculated using the classical definition.

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

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

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

,

(1.2)

where R * (A) is the statistical probability of an event BUT; w(A) is the relative frequency of the event BUT; m is the number of trials in which the event occurred BUT; n is the total number of trials.

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

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

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

Let a flat figure g forms part of a flat figure G(Fig. 1.1). On the figure G a dot is thrown at random. This means that all points in the area G"equal" in relation to hitting it with a thrown random point. Assuming that the probability of an event BUT- hitting a thrown point on a figure g- proportional to the area of ​​\u200b\u200bthis figure and does not depend on its location relative to G, neither from the form g, find

Probability event is the ratio of the number of elementary outcomes that favor a given event to the number of all equally possible outcomes of experience in which this event may occur. The probability of an event A is denoted by P(A) (here P is the first letter of the French word probabilite - probability). According to the definition
(1.2.1)
where is the number of elementary outcomes favoring event A; - the number of all equally possible elementary outcomes of experience, forming a complete group of events.
This definition of probability is called classical. It arose at the initial stage of the development of probability theory.

The probability of an event has the following properties:
1. The probability of a certain event is equal to one. Let's designate a certain event by the letter . For a certain event, therefore
(1.2.2)
2. The probability of an impossible event is zero. We denote the impossible event by the letter . For an impossible event, therefore
(1.2.3)
3. The probability of a random event is expressed as a positive number less than one. Since the inequalities , or are satisfied for a random event, then
(1.2.4)
4. The probability of any event satisfies the inequalities
(1.2.5)
This follows from relations (1.2.2) -(1.2.4).

Example 1 An urn contains 10 balls of the same size and weight, of which 4 are red and 6 are blue. One ball is drawn from the urn. What is the probability that the drawn ball is blue?

Decision. The event "the drawn ball turned out to be blue" will be denoted by the letter A. This test has 10 equally possible elementary outcomes, of which 6 favor the event A. In accordance with formula (1.2.1), we obtain

Example 2 All natural numbers from 1 to 30 are written on identical cards and placed in an urn. After thoroughly mixing the cards, one card is removed from the urn. What is the probability that the number on the card drawn is a multiple of 5?

Decision. Denote by A the event "the number on the taken card is a multiple of 5". In this trial, there are 30 equally possible elementary outcomes, of which 6 outcomes favor event A (numbers 5, 10, 15, 20, 25, 30). Hence,

Example 3 Two dice are thrown, the sum of points on the upper faces is calculated. Find the probability of the event B, consisting in the fact that the top faces of the cubes will have a total of 9 points.

Decision. There are 6 2 = 36 equally possible elementary outcomes in this trial. Event B is favored by 4 outcomes: (3;6), (4;5), (5;4), (6;3), so

Example 4. A natural number not exceeding 10 is chosen at random. What is the probability that this number is prime?

Decision. Denote by the letter C the event "the chosen number is prime". In this case, n = 10, m = 4 (primes 2, 3, 5, 7). Therefore, the desired probability

Example 5 Two symmetrical coins are tossed. What is the probability that both coins have digits on the top sides?

Decision. Let's denote by the letter D the event "there was a number on the top side of each coin". There are 4 equally possible elementary outcomes in this test: (G, G), (G, C), (C, G), (C, C). (The notation (G, C) means that on the first coin there is a coat of arms, on the second - a number). Event D is favored by one elementary outcome (C, C). Since m = 1, n = 4, then

Example 6 What is the probability that the digits in a randomly chosen two-digit number are the same?

Decision. Two-digit numbers are numbers from 10 to 99; there are 90 such numbers in total. 9 numbers have the same digits (these are the numbers 11, 22, 33, 44, 55, 66, 77, 88, 99). Since in this case m = 9, n = 90, then
,
where A is the "number with the same digits" event.

Example 7 From the letters of the word differential one letter is chosen at random. What is the probability that this letter will be: a) a vowel b) a consonant c) a letter h?

Decision. There are 12 letters in the word differential, of which 5 are vowels and 7 are consonants. Letters h this word does not. Let's denote the events: A - "vowel", B - "consonant", C - "letter h". The number of favorable elementary outcomes: - for event A, - for event B, - for event C. Since n \u003d 12, then
, and .

Example 8 Two dice are tossed, the number of points on the top face of each dice is noted. Find the probability that both dice have the same number of points.

Decision. Let us denote this event by the letter A. Event A is favored by 6 elementary outcomes: (1;]), (2;2), (3;3), (4;4), (5;5), (6;6). In total there are equally possible elementary outcomes that form a complete group of events, in this case n=6 2 =36. So the desired probability

Example 9 The book has 300 pages. What is the probability that a randomly opened page will have a sequence number that is a multiple of 5?

Decision. It follows from the conditions of the problem that there will be n = 300 of all equally possible elementary outcomes that form a complete group of events. Of these, m = 60 favor the occurrence of the specified event. Indeed, a number that is a multiple of 5 has the form 5k, where k is a natural number, and , whence . Hence,
, where A - the "page" event has a sequence number that is a multiple of 5".

Example 10. Two dice are thrown, the sum of points on the upper faces is calculated. What is more likely to get a total of 7 or 8?

Decision. Let's designate the events: A - "7 points fell out", B - "8 points fell out". Event A is favored by 6 elementary outcomes: (1; 6), (2; 5), (3; 4), (4; 3), (5; 2), (6; 1), and event B - by 5 outcomes: (2; 6), (3; 5), (4; 4), (5; 3), (6; 2). There are n = 6 2 = 36 of all equally possible elementary outcomes. Hence, and .

So, P(A)>P(B), that is, getting a total of 7 points is a more likely event than getting a total of 8 points.

Tasks

1. A natural number not exceeding 30 is chosen at random. What is the probability that this number is a multiple of 3?
2. In the urn a red and b blue balls of the same size and weight. What is the probability that a randomly drawn ball from this urn is blue?
3. A number not exceeding 30 is chosen at random. What is the probability that this number is a divisor of zo?
4. In the urn a blue and b red balls of the same size and weight. One ball is drawn from this urn and set aside. This ball is red. Then another ball is drawn from the urn. Find the probability that the second ball is also red.
5. A natural number not exceeding 50 is chosen at random. What is the probability that this number is prime?
6. Three dice are thrown, the sum of points on the upper faces is calculated. What is more likely - to get a total of 9 or 10 points?
7. Three dice are tossed, the sum of the dropped points is calculated. What is more likely to get a total of 11 (event A) or 12 points (event B)?

Answers

1. 1/3. 2 . b/(a+b). 3 . 0,2. 4 . (b-1)/(a+b-1). 5 .0,3.6 . p 1 \u003d 25/216 - the probability of getting 9 points in total; p 2 \u003d 27/216 - the probability of getting 10 points in total; p2 > p1 7 . P(A) = 27/216, P(B) = 25/216, P(A) > P(B).

Questions

1. What is called the probability of an event?
2. What is the probability of a certain event?
3. What is the probability of an impossible event?
4. What are the limits of the probability of a random event?
5. What are the limits of the probability of any event?
6. What definition of probability is called classical?

The probability of an event is understood as some numerical characteristic of the possibility of the occurrence of this event. There are several approaches to determining probability.

Probability of an event BUT is the ratio of the number of outcomes favorable to this event to the total number of all equally possible incompatible elementary outcomes that form a complete group. So the probability of an event BUT is determined by the formula

where m is the number of elementary outcomes favoring BUT, n- the number of all possible elementary outcomes of the test.

Example 3.1. In the experiment with throwing a dice, the number of all outcomes n is 6 and they are all equally possible. Let the event BUT means the appearance of an even number. Then for this event, favorable outcomes will be the appearance of numbers 2, 4, 6. Their number is 3. Therefore, the probability of the event BUT is equal to

Example 3.2. What is the probability that the digits in a randomly chosen two-digit number are the same?

Two-digit numbers are numbers from 10 to 99, there are 90 such numbers in total. 9 numbers have the same numbers (these are the numbers 11, 22, ..., 99). Since in this case m=9, n=90, then

where BUT- event, "a number with the same digits."

Example 3.3. There are 7 standard parts in a lot of 10 parts. Find the probability that there are 4 standard parts among six randomly selected parts.

The total number of possible elementary outcomes of the test is equal to the number of ways in which 6 parts can be extracted from 10, i.e., the number of combinations of 10 elements of 6 elements. Determine the number of outcomes that favor the event of interest to us BUT(among the six parts taken, 4 are standard). Four standard parts can be taken from seven standard parts in ways; at the same time, the remaining 6-4=2 parts must be non-standard, but you can take two non-standard parts from 10-7=3 non-standard parts in different ways. Therefore, the number of favorable outcomes is .

Then the desired probability is equal to

The following properties follow from the definition of probability:

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

Indeed, if the event is reliable, then each elementary outcome of the test favors the event. In this case m=n, hence

2. The probability of an impossible event is zero.

Indeed, if the event is impossible, then none of the elementary outcomes of the trial favors the event. In this case it means

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

Indeed, only a part of the total number of elementary outcomes of the test favors a random event. In this case< m< n, means 0 < m/n < 1, i.e. 0< P(A) < 1. Итак, вероятность любого события удовлетворяет двойному неравенству

The construction of a logically complete probability theory is based on the axiomatic definition of a random event and its probability. In the system of axioms proposed by A. N. Kolmogorov, undefined concepts are an elementary event and probability. Here are the axioms that define the probability:

1. Every event BUT assigned a non-negative real number P(A). This number is called the probability of the event. BUT.

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

3. The probability of occurrence of at least one of the pairwise incompatible events is equal to the sum of the probabilities of these events.

Based on these axioms, the properties of probabilities and the relationships between them are derived as theorems.

Questions for self-examination

1. What is the name of the numerical characteristic of the possibility of an event?

2. What is called the probability of an event?

3. What is the probability of a certain event?

4. What is the probability of an impossible event?

5. What are the limits of the probability of a random event?

6. What are the limits of the probability of any event?

7. What definition of probability is called classical?

MUNICIPAL EDUCATIONAL INSTITUTION

GYMNASIUM No. 6

on the topic "Classical definition of probability".

Completed by a student of the 8th "B" class

Klimantova Alexandra.

Mathematics teacher: Videnkina V. A.

Voronezh, 2008

Many games use a dice. The die has 6 faces, on each face a different number of points is marked - from 1 to 6. The player throws the die and looks at how many points there are on the dropped face (on the face that is located on top). Quite often, the dots on the edge of the die are replaced by the corresponding number and then they talk about a roll of 1, 2 or 6. Throwing a die can be considered an experience, experiment, test, and the result obtained is the outcome of the test or an elementary event. People are interested in guessing the onset of an event, predicting its outcome. What predictions can they make when a dice is rolled? For example, these:

1. event A - the number 1, 2, 3, 4, 5 or 6 falls out;
2. event B - the number 7, 8 or 9 falls out;
3. event C - the number 1 falls out.

Event A, predicted in the first case, will definitely come. In general, an event that is sure to occur in a given experience is called certain event.

Event B, predicted in the second case, will never occur, it is simply impossible. In general, an event that cannot occur in a given experiment is called impossible event.

Will the event C, predicted in the third case, happen or not? We are not able to answer this question with complete certainty, since 1 may or may not fall out. An event that in a given experience may or may not occur is called random event.

Thinking about the onset of a certain event, we most likely will not use the word “probably”. For example, if today is Wednesday, then tomorrow is Thursday, this is a certain event. On Wednesday we will not say: "Probably tomorrow is Thursday", we will say briefly and clearly: "Tomorrow is Thursday." True, if we are prone to beautiful phrases, then we can say this: "With one hundred percent probability I say that tomorrow is Thursday." On the contrary, if today is Wednesday, then the coming of tomorrow is Friday—an impossible event. Evaluating this event on Wednesday, we can say this: "I'm sure that tomorrow is not Friday." Or like this: "It's unbelievable that tomorrow is Friday." Well, if we are prone to beautiful phrases, then we can say this: “The probability that tomorrow is Friday is zero.” So, a certain event is an event that occurs under given conditions. with 100% certainty(i.e. coming in 10 cases out of 10, in 100 cases out of 100, etc.). An impossible event is an event that never occurs under given conditions, an event with zero probability.

But, unfortunately (and perhaps fortunately), not everything in life is so clear and clear: it will always be (certain event), this will never happen (impossible event). Most often, we are faced with random events, some of which are more likely, others less likely. Usually people use the words "more likely" or "less likely", as they say, on a whim, relying on what is called common sense. But very often such estimates turn out to be insufficient, since it is important to know how much percent likely a random event or how many times one random event is more likely than another. In other words, we need exact quantitative characteristics, you need to be able to characterize the probability by a number.

We have already taken the first steps in this direction. We said that the probability of a certain event occurring is characterized as one hundred percent, and the probability of an impossible event occurring as zero. Given that 100% equals 1, people have agreed on the following:

1. the probability of a certain event is considered to be equal to 1;
2. the probability of an impossible event is considered equal to 0.

How do you calculate the probability of a random event? After all, it happened by chance, which means that it does not obey laws, algorithms, formulas. It turns out that certain laws operate in the world of randomness, allowing you to calculate probabilities. This is the branch of mathematics that is called- probability theory.

Mathematics deals with model some phenomenon of the reality around us. Of all the models used in probability theory, we will limit ourselves to the simplest.

Classical probabilistic scheme

To find the probability of an event A during some experiment, one should:

1) find the number N of all possible outcomes of this experiment;

2) accept the assumption that all these outcomes are equally probable (equally possible);

3) find the number N(A) of those outcomes of the experience in which the event A occurs;

4) find a private ; it will be equal to the probability of event A.

It is customary to designate the probability of an event A as P(A). The explanation for this designation is very simple: the word "probability" in French is probability, in English- probability.The designation uses the first letter of the word.

Using this notation, the probability of an event A according to the classical scheme can be found using the formula

P(A)=.

Often all the points of the given classical probabilistic scheme are expressed in one rather long phrase.

The classical definition of probability

The probability of an event A during a certain test is the ratio of the number of outcomes, as a result of which the event A occurs, to the total number of all equally possible outcomes of this test.

Example 1. Find the probability that in one throw of a dice: a) 4; b) 5; c) an even number of points; d) the number of points greater than 4; e) number of points not a multiple of three.

Decision. In total, there are N=6 possible outcomes: dropping a face of a cube with a number of points equal to 1, 2, 3, 4, 5, or 6. We believe that none of them has any advantages over the others, i.e., we accept the assumption of the similarity of these outcomes.

a) Exactly in one of the outcomes, the event of interest to us A will occur - the loss of the number 4. Hence, N (A) \u003d 1 and

P(A)= =.

b) The solution and the answer are the same as in the previous paragraph.

c) The event B of interest to us will occur exactly in three cases when the number of points is 2, 4 or 6. Hence,

N(B)=3 andP(B)==.

d) The event C of interest to us will occur exactly in two cases when the number of points is 5 or 6. Hence,

N(C) =2 and P(C)=.

e) Of the six possible numbers drawn, four (1, 2, 4 and 5) are not multiples of three, and the remaining two (3 and 6) are divisible by three. This means that the event of interest to us occurs exactly in four out of six possible and equally probable among themselves and equally probable among themselves outcomes of the experience. So the answer is .

Answer: a); b) ; in) ; G) ; e).

A real playing dice may well differ from an ideal (model) dice, therefore, to describe its behavior, a more accurate and detailed model is required, taking into account the advantages of one face over another, the possible presence of magnets, etc. But “the devil is in the details”, and more accuracy tends to lead to more complexity, and getting an answer becomes a problem. We confine ourselves to considering the simplest probabilistic model, where all possible outcomes are equally probable.

Remark 1. Let's consider another example. The question was asked: "What is the probability of getting a three on one roll of the die?" The student answered like this: "The probability is 0.5." And he explained his answer: “The three will either fall out or not. This means that there are two outcomes in total, and in exactly one event the event of interest to us occurs. According to the classical probabilistic scheme, we get the answer 0.5. Is there an error in this reasoning? At first glance, no. However, it is still there, and in a fundamental moment. Yes, indeed, the triple will either fall out or not, that is, with such a definition of the outcome of the throw, N = 2. It is also true that N(A)=1 and, of course, it is true that =0, 5, i.e., three points of the probabilistic scheme are taken into account, but the fulfillment of point 2) is doubtful. Of course, from a purely legal point of view, we have the right to believe that the loss of a triple is equally likely to fail. But can we think so without violating our own natural assumptions about the "sameness" of the faces? Of course not! Here we are dealing with correct reasoning within some model. Only this model itself is “wrong”, not corresponding to the real phenomenon.

Remark 2. When discussing probability, do not lose sight of the following important circumstance. If we say that when throwing a die, the probability of getting one point is equal to , this does not mean at all that by rolling the die 6 times, you will get one point exactly once, by throwing the die 12 times, you will get one point exactly twice, by rolling the die 18 times, you get one point exactly three times, and so on. The word is probably speculative. We assume that is likely to happen. Probably if we roll the die 600 times, one point will come up 100 times, or about 100.

Probability theory arose in the 17th century when analyzing various gambling games. It is not surprising, therefore, that the first examples are of a playful nature. From the dice examples, let's move on to the random drawing of playing cards from the deck.

Example 2. From a deck of 36 cards, 3 cards are randomly drawn at the same time. What is the probability that there is no Queen of Spades among them?

Decision. We have a set of 36 elements. We select three elements, the order of which is not important. Hence, it is possible to obtain N=C outcomes. We will act according to the classical probabilistic scheme, that is, we will assume that all these outcomes are equally probable.

It remains to calculate the required probability according to the classical definition:

And what is the probability that among the chosen three cards there is a Queen of Spades? The number of all such outcomes is not difficult to calculate, you just need to subtract from all outcomes N all those outcomes in which there is no queen of spades, that is, subtract the number N(A) found in Example 3. Then this difference N - N (A) in accordance with the classical probabilistic scheme should be divided by N. This is what we get:

We see that there is a certain relationship between the probabilities of the two events. If event A consists in the absence of the Queen of Spades, and event B consists in her presence among the chosen three cards, then

P (B) \u003d 1 - P (A),

P(A)+P(B)=1.

Unfortunately, in the equality P(A)+P(B)=1 there is no information about the relationship between events A and B; we have to keep this connection in mind. It would be more convenient to give the event B a name and designation in advance, clearly indicating its connection with A.

Definition 1. Event B called opposite to event A and denote B=Ā if event B occurs if and only if event A does not occur.

TTheorem 1. To find the probability of the opposite event, subtract the probability of the event itself from unity: Р(Ā)= 1—Р(А). Indeed,

In practice, they calculate what is easier to find: either P(A) or P(Ā). After that, they use the formula from the theorem and find, respectively, either P(Ā)= 1-P(A), or P(A)= 1-P(Ā).

Often used is the method of solving a particular problem by "enumeration of cases", when the conditions of the problem are divided into mutually exclusive cases, each of which is considered separately. For example, “if you go to the right, you will lose your horse, if you go straight, you will solve a problem according to probability theory, if you go to the left…”. Or when plotting the function y=│x+1│—│2x—5│, consider the cases of x

Example 3. Of the 50 dots, 17 are shaded blue and 13 are orange. Find the probability that a randomly selected point will be shaded.

Decision. In total, 30 points out of 50 are shaded. Hence, the probability is = 0.6.

Answer: 0.6.

Let's take a closer look at this simple example, however. Let event A be that the selected point is blue, and event B be that the selected point is orange. By convention, events A and B cannot happen at the same time.

We denote by the letter C the event of interest to us. Event C occurs if and only if it occurs at least one of the events A or B. It is clear that N(C)= N(A)+N(B).

Let us divide both sides of this equality by N, the number of all possible outcomes of the given experiment; we get

We have analyzed an important and frequently occurring situation using a simple example. There is a special name for her.

Definition 2. Events A and B are called incompatible if they cannot occur at the same time.

Theorem 2. The probability of occurrence of at least one of two incompatible events is equal to the sum of their probabilities.

When translating this theorem into mathematical language, it becomes necessary to somehow name and designate an event consisting in the occurrence of at least one of the two given events A and B. Such an event is called the sum of events A and B and denoted by A+B.

If A and B are incompatible, then P(A+B)= P(A)+P(B).

Indeed,

The incompatibility of events A and B can be conveniently illustrated by a figure. If all the outcomes of the experience are some set of points in the figure, then the events A and B are some subsets of a given set. The incompatibility of A and B means that these two subsets do not intersect. A typical example of incompatible events is any event A and the opposite event Ā.

Of course, this theorem is true for three, four, and for any finite number of pairwise incompatible events. The probability of the sum of any number of pairwise incompatible events is equal to the sum of the probabilities of these events. This important statement exactly corresponds to the method of solving problems by "enumeration of cases".

Between the events that occur as a result of some experience, and between the probabilities of these events, there may be some relationships, dependencies, connections, etc. For example, events can be “added”, and the probability of the sum of incompatible events is equal to the sum of their probabilities.

In conclusion, we discuss the following fundamental question: is it possible to prove, that the probability of getting "tails" in one toss of a coin is equal to

The answer is negative. Generally speaking, the question itself is not correct, the exact meaning of the word "prove" is not clear. After all, we always prove something within the framework of some models, in which the rules, laws, axioms, formulas, theorems, etc. are already known. If we are talking about an imaginary, “ideal” coin, then that is why it is considered ideal because, a-priory, the probability of getting heads is equal to the probability of getting heads. And, in principle, we can consider a model in which the probability of falling “tails” is twice the probability of falling “heads”, or three times less, etc. Then the question arises: for what reason from the various possible models for tossing a coin do we choose one in which both outcomes of the toss are equally likely?

A completely frontal answer is: “But it’s easier, clearer and more natural for us!” But there are more substantive arguments as well. They come from practice. The vast majority of textbooks on probability theory give examples of the French naturalist J. Buffon (18th century) and the English mathematician-statistician C. Pearson (late 19th century), who threw a coin 4040 and 24000 times, respectively, and counted the number of falling “eagles ” or “tails”. Their “tails” fell out, respectively, 1992 and 11998 times. If you count drop frequency“tails”, then you get = = 0.493069 ... for Buffon and = 0.4995 for Pearson. Arises naturally assumption that with an unlimited increase in the number of tosses of a coin, the frequency of falling "tails", as well as the frequency of falling "eagles", will more and more approach 0.5. It is this assumption, based on practical data, that is the basis for choosing a model with equiprobable outcomes.

Now we can sum up. The basic concept is probability of a random event, which is calculated within the framework of the simplest model— classical probabilistic scheme. The concept is important both in theory and in practice. opposite event and the formula Р(Ā)= 1—Р(А) for finding the probability of such an event.

Finally, we met incompatible events and with formulas.

P (A + B) \u003d P (A) + P (B),

P (A + B + C) \u003d P (A) + P (B) + P (C),

allowing to find probabilities amounts such events.

Bibliography

1. Events. Probabilities. Statistical data processing: Add. paragraphs to the course of algebra 7-9 cells. educational institutions / A. G. Mordkovich, P. V. Semenov.—4th ed.—M.: Mnemozina, 2006.—112 p.: ill.

2.Yu. N. Makarychev, N. G. Mindyuk “Algebra. Elements of statistics and probability theory.—Moscow, Enlightenment, 2006.

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The basic concept of probability theory is the concept of a random event. random event An event is called an event that, under certain conditions, may or may not occur. For example, hitting or missing an object when firing at this object with a given weapon is a random event.

The event is called reliable if, as a result of the test, it necessarily occurs. Impossible An event is called an event that cannot occur as a result of the test.

Random events are called incompatible in a given trial if no two of them can appear together.

Random events form full group, if at each trial any of them can appear and no other event incompatible with them can appear.

Consider the complete group of equally possible incompatible random events. Such events will be called outcomes or elementary events. Exodus is called favorable occurrence of the event $A$, if the occurrence of this outcome entails the appearance of the event $A$.

Example. An urn contains 8 numbered balls (each ball has one number from 1 to 8). Balls with numbers 1, 2, 3 are red, the rest are black. The appearance of the ball with the number 1 (or the number 2 or the number 3) is an event favorable for the appearance of the red ball. The appearance of a ball with the number 4 (or the number 5, 6, 7, 8) is an event that favors the appearance of a black ball.

Probability of an event$A$ is the ratio of the number $m$ of outcomes favoring this event to the total number $n$ of all equally possible incompatible elementary outcomes that form the complete group $$P(A)=\frac(m)(n). \quad(1)$$

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.

So, the probability of any event satisfies the double inequality $0 \le P(A) \le 1$ .

Online calculators

A large layer of problems solved using formula (1) relates to the topic of hypergeometric probability. Below the links you can find descriptions of popular tasks and online calculators for their solutions:

• Problem about balls (an urn contains $k$ white and $n$ black balls, $m$ balls are taken out...)
• Parts problem (a box contains $k$ standard and $n$ defective parts, $m$ parts are taken out...)
• Problem about lottery tickets ($k$ winning and $n$ losing tickets participate in the lottery, $m$ tickets are bought...)

Examples of solutions to problems on classical probability

Example. There are 10 numbered balls in the urn with numbers from 1 to 10. One ball is taken out. What is the probability that the number of the drawn ball does not exceed 10?

Decision. Let the event BUT= (The number of the drawn ball does not exceed 10). Number of occurrences of favorable events BUT equals the number of all possible cases m=n=10. Hence, R(BUT)=1. Event A reliable.

Example. There are 10 balls in an urn: 6 white and 4 black. Pulled out two balls. What is the probability that both balls are white?

Decision. You can take out two balls out of ten in the following number of ways: .
The number of times when there are two white balls among these two is .
Desired probability
.

Example. There are 15 balls in an urn: 5 white and 10 black. What is the probability of drawing a blue ball from the urn?

Decision. Since there are no blue balls in the urn, m=0, n=15. Therefore, the desired probability R=0. The event of drawing a blue ball impossible.

Example. One card is drawn from a deck of 36 cards. What is the probability of a card of hearts appearing?

Decision. Number of elementary outcomes (number of cards) n=36. Event BUT= (Appearance of a heart suit card). Number of times favorable for the occurrence of the event BUT, m=9. Hence,
.

Example. There are 6 men and 4 women in the office. 7 people were randomly selected for the move. Find the probability that there are three women among the selected persons.