Classical and statistical definitions of probability. Classical and statistical probability

The randomness of the occurrence of events is associated with the impossibility of predicting in advance the outcome of a particular test. However, if we consider, for example, the test: multiple tossing of a coin, ω 1 , ω 2 , … , ω n , then it turns out that in approximately half of the outcomes ( n / 2) a certain pattern is found that corresponds to the concept of probability.

Under probability events BUT some numerical characteristic of the possibility of the occurrence of an event is understood BUT. We denote this numerical characteristic R(BUT). There are several approaches to determining probability. The main ones are statistical, classical and geometric.

Let produced n tests and at the same time some event BUT came n A times. Number n A is called absolute frequency(or just the frequency) of the event BUT, and the relation is called the relative frequency of occurrence of event A. Relative frequency of any event characterized by the following properties:

The basis for applying the methods of probability theory to the study of real processes is the objective existence of random events that have the property of frequency stability. Numerous trials of the event under study BUT show that for large n relative frequency ( BUT) remains approximately constant.

The statistical definition of probability lies in the fact that the probability of an event A is taken to be a constant value p(A), around which the values of the relative frequencies fluctuate (BUT) with an unlimited increase in the number of trialsn.

Remark 1. Note that the limits of change in the probability of a random event from zero to one are chosen by B. Pascal for the convenience of its calculation and application. In correspondence with P. Fermat, Pascal pointed out that any interval could be chosen as the indicated interval, for example, from zero to one hundred and other intervals. In the problems below in this tutorial, the probabilities are sometimes given as percentages, i.e. from zero to one hundred. In this case, the percentages given in the tasks must be converted into shares, i.e. divide by 100.

Example 1 Conducted 10 series of coin tosses, 1000 tosses in each. Value ( BUT) in each of the series is 0.501; 0.485; 0.509; 0.536; 0.485; 0.488; 0.500; 0.497; 0.494; 0.484. These frequencies cluster around R(BUT) = 0,5.

This example confirms that the relative frequency ( BUT) is approximately equal to R(BUT), i.e.

The classical definition of probability assumes that all elementary outcomes equally possible. Equivalence of the outcomes of the experiment is concluded due to symmetry considerations (as in the case of a coin or a dice). Problems in which symmetry considerations can be used are rare in practice. In many cases it is difficult to give grounds for believing that all elementary outcomes are equally probable. In this regard, it became necessary to introduce another definition of probability, called statistical. To give this definition, we first introduce the concept of the relative frequency of an event.

Relative event frequency, or frequency, is the ratio of the number of experiments in which this event appeared to the number of all experiments performed. Let us denote the frequency of the event by , then by definition

(1.4.1)
where is the number of experiments in which the event appeared and is the number of all experiments performed.

An event frequency has the following properties.

Observations made it possible to establish that the relative frequency has the properties of statistical stability: in various series of polynomial tests (in each of which this event may or may not appear), it takes values close enough to some constant. This constant, which is an objective numerical characteristic of the phenomenon, is considered the probability of this event.

Probability event is called the number around which the values are grouped, the frequency of this event in various series of a large number of tests.

This definition of probability is called statistical.

In the case of a statistical definition, a probability has the following properties:
1) the probability of a certain event is equal to one;
2) the probability of an impossible event is zero;
3) the probability of a random event is between zero and one;
4) the probability of the sum of two incompatible events is equal to the sum of the probabilities of these events.

Example 1 Of the 500 parts taken at random, 8 were defective. Find the frequency of defective parts.

Decision. Since in this case = 8, = 500, then in accordance with formula (1.4.1) we find

Example 2. A dice is rolled 60 times six appeared 10 times. What is the frequency of occurrence sixes?

Decision. It follows from the conditions of the problem that = 60, = 10, therefore

Example 3 Among 1000 newborns there were 515 boys. What is the birth rate of boys?
Decision. Since in this case , , then .

Example 4 As a result of 20 shots at the target, 15 hits were received. What is the hit frequency?

Decision. Since = 20, = 15, then

Example 5 When shooting at a target, hit frequency = 0.75. Find the number of hits with 40 shots.

Decision. From formula (1.4.1) it follows that . Since \u003d 0.75, \u003d 40, then . Thus, 30 hits were received.

Example 6 www.. 970 seeds sprouted from the seeds sown. How many seeds were sown?

Decision. From formula (1.4.1) it follows that . Since , , then . So, 1000 seeds were sown.

Example 7 On a segment of the natural series from 1 to 20, find the frequency of prime numbers.

Decision. On the indicated segment of the natural series of numbers there are the following prime numbers: 2, 3, 5, 7, 11, 13, 17, 19; there are 8 in total. Since = 20, = 8, then the desired frequency

Example 8 Three series of repeated tossings of a symmetrical coin were carried out, the number of appearances of the coat of arms was calculated: 1) = 4040, =2048, 2) = 12000, = 6019; 3) = 24000, = 12012. Find the frequency of the appearance of the coat of arms in each test series.

Decision. In accordance with formula (1.4.1) we find:

Comment. These examples show that, in repeated trials, the frequency of an event differs little from its probability. The probability of the appearance of the coat of arms when tossing a coin is p \u003d 1/2 \u003d 0.5, since in this case n \u003d 2, m \u003d 1.

Example 9 Among the 300 parts made on an automatic machine, there were 15 that did not meet the standard. Find the frequency of occurrence of non-standard parts.

Decision. In this case, n = 300, m = 15, so

Example 10 The controller, checking the quality of 400 products, found that 20 of them belong to the second grade, and the rest - to the first. Find the frequency of products of the first grade, the frequency of products of the second grade.

Decision. First of all, we find the number of products of the first grade: 400 - 20 = 380. Since n = 400, = 380, then the frequency of products of the first grade

Similarly, we find the frequency of products of the second grade:

Tasks

The technical control department found 10 non-standard items in a batch of 1000 items. Find the frequency of manufacturing defective products.
To determine the quality of seeds, 100 seeds were selected and sown in laboratory conditions. 95 seeds gave a normal shoot. What is the frequency of normal seed germination?
Find the frequency of occurrence of prime numbers in the following segments of the natural series: a) from 21 to 40; b) from 41 to 50; c) from 51 to 70.
Find the frequency of occurrence of the number in 100 tosses of a symmetrical coin. (Experiment on your own).
Find the frequency of occurrence of a six in 90 rolls of a dice.
By interviewing all the students in your course, determine the frequency of birthdays that fall in each month of the year.
Find the frequency of five-letter words in any newspaper text.

Answers

0.01. 2. 0.95; 0.05. 3. a) 0.2; b) 0.3; c) 0.2.

Questions

What is the frequency of an event?
What is the frequency of a certain event?
What is the frequency of an impossible event?
What is the frequency range of a random event?
What is the frequency of the sum of two disjoint events?
What is the statistical definition of probability?
What are the properties of statistical probability?

Tags . Look .

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

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

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

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

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

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

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

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

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

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

Examples:

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

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

BUT- a red ball drawn at random:

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

B- two red balls drawn at random:

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

1) The probability of an impossible event is 0;

2) The probability of a certain event is 1;

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

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

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

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

where is the probability of occurrence of event A;

Relative frequency of occurrence of event A;

The number of trials in which event A appeared;

The total number of trials.

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

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

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

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

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

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

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

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

Rice. 1.1 Fig 1.2

Example 1.2. Two students agreed to meet at a certain place between 10 and 11 o'clock in the afternoon. The first person to arrive waits for the second for 15 minutes, after which he leaves. Find the probability that the meeting will take place if each student randomly chooses the time of his arrival between 10 and 11 o'clock.

Decision. Let us denote the moments of arrival at a certain place of the first and second students, respectively, through x and y. In a rectangular coordinate system Oxy Let's take 10 hours as the starting point, and 1 hour as the unit of measurement. By condition 0 ≤ x ≤ 1, 0 ≤ y≤ 1. These inequalities are satisfied by the coordinates of any point belonging to the square OKLM with a side equal to 1 (Fig. 1.2). Event BUT– a meeting of two students – will happen if the difference between x and not y will exceed 1/4 hour (in absolute value), i.e. | y – x| ≤ 0,25.

The solution to this inequality is the strip x – 0,25 ≤ y ≤ x+ 0.25 which is inside the square G represents the shaded area g. By formula (1.3)

As mentioned above, the classical definition of probability assumes that all elementary outcomes are equally likely. Equivalence of outcomes of the experiment is concluded due to considerations of symmetry. Problems in which symmetry considerations can be used are rare in practice. In many cases it is difficult to give grounds for believing that all elementary outcomes are equally probable. In this regard, it became necessary to introduce another definition of probability, called statistical. Let us first introduce the concept of relative frequency.

Relative event frequency, or frequency, is the ratio of the number of experiments in which this event occurred to the number of all experiments made. Denote the frequency of the event BUT through W(A), then

where n is the total number of experiments; m is the number of experiments in which the event occurred BUT.

With a small number of experiments, the frequency of the event is largely random and can vary markedly from one group of experiments to another. For example, with some ten tosses of a coin, it is quite possible that the coat of arms will appear 2 times (frequency 0.2), with other ten tosses, we may well get 8 coats of arms (frequency 0.8). However, as the number of experiments increases, the frequency of the event loses its random character more and more; the accidental circumstances inherent in each individual experience cancel each other out in a mass, and the frequency tends to stabilize, approaching, with slight fluctuations, some average constant value. This constant, which is an objective numerical characteristic of the phenomenon, is considered the probability of this event.

Statistical definition of probability: probability events is called a number around which the values of the frequency of a given event in various series of a large number of tests are grouped.

The frequency stability property, repeatedly verified experimentally and confirmed by the experience of mankind, is one of the most characteristic regularities observed in random phenomena. There is a deep connection between the frequency of an event and its probability, which can be expressed as follows: when we estimate the degree of possibility of an event, we associate this assessment with a greater or lesser frequency of occurrence of similar events in practice.

geometric probability

The classical definition of probability assumes that the number of elementary outcomes is finite. In practice, there are experiments for which the set of such outcomes is infinite. In order to overcome this shortcoming of the classical definition of probability, which is that it is not applicable to trials with an infinite number of outcomes, one introduces geometric probabilities - the probabilities of a point falling into an area.

Let us assume that a squaring region is given on the plane G, i.e. area having an area S G. In area G contains area g area Sg. To area G a dot is thrown at random. We will assume that the thrown point can fall into some part of the area G with a probability proportional to the area of this part and independent of its shape and location. Let the event BUT- "hitting the thrown point in the area g”, then the geometric probability of this event is determined by the formula:

In the general case, the concept of geometric probability is introduced as follows. Denote the measure of the area g(length, area, volume) through mes g, and the measure of the area G- through mes G ; let also BUT– event “thrown point hits the area g, which is contained in the area G". Area Hit Chance g point thrown into the area G, is determined by the formula

Task. A square is inscribed in a circle. A dot is randomly thrown into the circle. What is the probability that the point will fall into the square?

Decision. Let the radius of the circle be R, then the area of the circle is . The diagonal of a square is , then the side of the square is , and the area of the square is . The probability of the desired event is defined as the ratio of the area of the square to the area of the circle, i.e. .

test questions

1. What is called a test (experiment)?

2. What is called an event?

3. What event is called a) reliable? b) random? c) impossible?

4. What events are called a) incompatible? b) joint?

5. What events are called opposite? Are they a) incompatible b) joint is random?

6. What is called a complete group of random events?

7. If the events cannot all happen together as a result of the test, will they be pairwise incompatible?

8. Do events form BUT and the whole group?

9. What elementary outcomes favor this event?

10. What definition of probability is called classical?

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

12. Under what conditions is classical probability applied?

13. Under what conditions is geometric probability applied?

14. What definition of probability is called geometric?

15. What is the frequency of an event?

16. What definition of probability is called statistical?

Control tasks

1. From the letters of the word "conservatory" one letter is randomly extracted. Find the probability that this letter is a vowel. Find the probability that this is the letter "o".

2. The letters “o”, “p”, “s”, “t” are written on identical cards. Find the probability that the word "rope" will appear on cards laid out at random in a row.

3. There are 4 women and 3 men in the team. Among the members of the brigade, 4 tickets to the theater are raffled off. What is the probability that there will be 2 women and 2 men among the ticket holders?

4. Two dice are rolled. Find the probability that the sum of the points on both dice is greater than 6.

5. The letters l, m, o, o, t are written on five identical cards. What is the probability that by removing the cards one at a time at random, we will get the word “hammer” in the order of their release?

6. Out of 10 tickets, 2 are winning. What is the probability that among five tickets taken at random, one is winning?

7. What is the probability that in a randomly chosen two-digit number the digits are such that their product is equal to zero.

8. A number not exceeding 30 is chosen at random. Find the probability that this number is a divisor of 30.

9. A number not exceeding 30 is chosen at random. Find the probability that this number is a multiple of 3.

10. A number not exceeding 50 is chosen at random. Find the probability that this number is prime.