called relative frequency ( or frequency) events A in the series of experiments under consideration.

The relative frequency of the event has the following properties:

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

2. The frequency of an impossible event is zero, i.e.

3. The frequency of a reliable event is 1, i.e.

4. The frequency of the sum of two incompatible events is equal to the sum of the frequency
these events, i.e. if , then

Frequency has another fundamental property called property of statistical stability: with increasing number of experiments (i.e. n) it takes values ​​close to some constant number (they say: the frequency stabilizes, approaching a certain number, the frequency fluctuates around a certain number, or its values ​​are grouped around a certain number).

So, for example, in the experiment (K. Pearson) tossing a coin - the relative frequency of the appearance of the coat of arms with 12,000 and 24,000 tosses turned out to be equal to 0.5015 and 0.5005, respectively, i.e. the frequency approaches the number. The frequency of having a boy, as observations show, fluctuates around the number 0.515.

Note that probability theory studies only those mass random phenomena with an uncertain outcome for which stability of the relative frequency is assumed.

Statistical definition of probability

To study a random event mathematically, it is necessary to introduce some quantitative assessment of the event. It is clear that some events are more likely (“more likely”) to occur than others. This assessment is probability of an event, those. a number expressing the degree of possibility of its occurrence in the experience under consideration. There are several mathematical definitions of probability; they all complement and generalize each other.

Consider an experiment that can be repeated any number of times (they say: “repeated tests are carried out”), in which some event is observed A.

Statistical probability events A is the number around which the relative frequency of the event A fluctuates for a sufficiently large number of trials (experiments).

Probability of event A indicated by the symbol R(A). According to this definition:

.

(1.2)

Mathematical justification for the proximity of relative frequency and probability R(A) of some event A serves as the theorem of J. Bernoulli.

Probabilities R(A) properties of 1-4 relative frequencies are attributed:

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

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

3. The statistical probability of a reliable event is equal to 1, i.e.

4. The statistical probability of the sum of two incompatible events is equal to the sum of the frequency of these events, i.e. if , then

The statistical method of determining probability, based on real experience, quite fully reveals the content of this concept. The disadvantage of the statistical definition is the ambiguity of the statistical probability; So in the example of tossing a coin, you can take as the probability not only the number 0.5, but also 0.49 or 0.51, etc. To reliably determine the probability, you need to do a large number of tests, which is not always easy or cheap.

Classic definition of probability

There is a simple way to determine the probability of an event, based on the equality of any of a finite number of outcomes of the experiment. Let the experiment be carried out with n outcomes that can be represented as complete group of incompatible equally possible events. Such outcomes are called cases, chances, elementary events, experience - classic. They say about such an experience that it boils down to case diagram or urn scheme(since the probabilistic problem for such an experiment can be replaced by an equivalent problem with urns containing balls of different colors).

Case w, which leads to the occurrence of the event A, called favorable(or favorable) to him, i.e. the case w entails the event A: .

Probability of the event A is called the number ratio m cases favorable to this event, to the total number n cases, i.e.

Along with the designation R(A) for the probability of an event A the notation used is r, i.e. p=P(A).

The following follows from the classical definition of probability: properties:

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

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

3. The probability of a reliable event is 1, i.e.

4. The probability of the sum of incompatible events is equal to the sum of the frequency of these events, i.e. if , then

Example 1.3. An urn contains 12 white and 8 black balls. What is the probability that a ball drawn at random will be white?

Solution:

Let A– an event consisting in the fact that a white ball is drawn. It is clear that is the number of all equally possible cases. Number of cases favoring the event A, equals 12, i.e. . Consequently, according to formula (1.3) we have: , i.e. .

Geometric definition of probabilities

The geometric definition of probability is used in the case when the outcomes of the experiment are equally possible, and the PES is an infinite uncountable set. Let us consider on the plane some region Ω having area , and inside the region Ω , region D with area S D(see Fig. 6).

A point is randomly selected in the region Ω X. This choice can be interpreted as throwing a point X to the regionΩ. In this case, the entry of a point into the region Ω is a reliable event, in D- random. It is assumed that all points of the region Ω are equal (all elementary events are equally possible), i.e. that a thrown point can hit any point in the region Ω and the probability of getting into the region D is proportional to the area of ​​this area and does not depend on its location and shape. Let the event, i.e. the thrown point will fall into the area D.

With the classical definition, the probability of an event is determined by the equality P(A)=m/n, where m is the number of elementary test outcomes that favor the occurrence of event A; n is the total number of possible elementary test outcomes.

It is assumed that the elementary outcomes form a complete group and are equally possible.

Relative frequency of event A: W(A)=m/n, where m is the number of trials in which event A occurred; n is the total number of tests performed.

When determining statistically, the probability of an event is taken to be its relative frequency.

Example: two dice are thrown. Find the probability that the sum of points on the rolled sides is even, and a six appears on the side of at least one of the dice.

Solution: on the dropped side of the “first” dice, one point,..., six points may appear. similar six elementary outcomes are possible when throwing the “second” die. Each of the outcomes of throwing the “first” can be combined with each of the outcomes of throwing the “second”. Thus. the total number of elementary test outcomes is 6*6=36. These outcomes form a complete group and, due to the symmetry of the bones, are equally possible. 5 moves are favorable for the event: 1)6,2;2)6,4;3)6,6;4)2,6;5)4,6;

Required probability: P(A)=5/36

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More on topic 3. Relative frequency. Stability of relative frequencies. Statistical definition of probability:

1. 4. Classic definition of probability. Relative frequency of occurrence of an event. Statistical probability. Geometric probability.
2. 27. Statistical determination of the sample. Variation series and their graphic representation. Polygon and histogram of frequencies (relative frequencies).
3. 39. Construction of an interval variation series. Histogram of frequencies and relative frequencies.
4. 4. Probability of deviation of the relative frequency from a constant probability in independent tests

Relative frequency, along with probability, belongs to the basic concepts of probability theory.

Relative frequency events is the ratio of the number of trials in which the event occurred to the total number of trials actually performed. Thus, the relative frequency of the event A is determined by the formula

W(A) = m/n,

Where m– number of occurrences of the event, n– total number of tests.

Comparing the definitions of probability and relative frequency, we conclude: the definition of probability does not require that the tests be actually carried out; determination of the relative frequency assumes that the tests were actually carried out. In other words, the probability is calculated before the experiment, the relative frequency - after the experiment.

Example 1. The inspection department found 3 non-standard parts in a batch of 80 randomly selected parts. Relative frequency of occurrence of non-standard parts

W(A) =3/80.

Example 2. 24 shots were fired at the target, with 19 hits recorded. Relative target hit rate

W(A) =19/24.

Long-term observations have shown that if experiments are carried out under identical conditions, in each of which the number of tests is sufficiently large, then the relative frequency exhibits the property of stability. This property is that in different experiments the relative frequency changes little(the less, the more tests performed), fluctuating around some constant number. It turned out that this constant number is the probability of the event occurring.

Thus, if the relative frequency is established experimentally, then the resulting number can be taken as an approximate probability value.

The relationship between relative frequency and probability will be described in more detail and more precisely below. Now let us illustrate the property of stability with examples.

Example 3. According to Swedish statistics, the relative frequency of births of girls in 1935. By month it is characterized by the following numbers (the numbers are arranged in the order of months starting from January): 0.486; 0.489; 0.490; 0.471; 0.478; 0.482; 0.462; 0.484; 0.485; 0.491; 0.482; 0.473

The relative frequency fluctuates around the number 0.482, which can be taken as an approximate value for the probability of having girls.

Note that statistical data from different countries give approximately the same relative frequency value.

Example 4. Coin tossing experiments were carried out many times, in which the number of appearances of the “coat of arms” was counted. The results of several experiments are given in Table 1.

Here the relative frequencies deviate slightly from the number 0.5, and the smaller the greater the number of tests. For example, with 4040 trials the deviation is 0.0069, and with 24000 trials it is only 0.0005. Taking into account that the probability of the appearance of a “coat of arms” when tossing a coin is 0.5, we again see that the relative frequency fluctuates around the probability.

§ 7. Limitations of the classical definition of probability. Statistical probability

The classical definition of probability assumes that the number of elementary outcomes of a trial is finite. In practice, it is very common to encounter tests in which the number of possible outcomes is infinite. In such cases, the classical definition is not applicable. This circumstance alone indicates the limitations of the classical definition. The noted disadvantage can be overcome, in particular, by introducing geometric probabilities (see § 8) and, of course, using axiomatic probability (see § 3, remark).

The weakest side of the classical definition is that very often it is impossible to represent the result of a test in the form of a set of elementary events. It is even more difficult to indicate the reasons for considering elementary events to be equally possible. Usually, the equipossibility of elementary test outcomes is said to be based on considerations of symmetry. For example, it is assumed that the die has the shape of a regular polyhedron (cube) and is made of a homogeneous material. However, problems in which symmetry considerations can be used are very rare in practice. For this reason, along with the classical definition of probability, other definitions are used, in particular the statistical definition: The relative frequency or a number close to it is taken as the statistical probability of an event. For example, if, as a result of a sufficiently large number of trials, it turns out that the relative frequency is very close to the number 0.4, then this number can be taken as the statistical probability of the event.

It is easy to verify that the properties of probability arising from the classical definition (see § 3) are also preserved in the statistical definition of probability. Indeed, if the event is reliable, then m =n and relative frequency

m/n = n/n = 1,

those. the statistical probability of a reliable event (as in the case of the classical definition) is equal to one.

If the event is impossible, then m= 0 and therefore the relative frequency

0/n = 0,

those. the statistical probability of an impossible event is zero.

For any event 0 m n and therefore the relative frequency

0 m/n 1,

those. the statistical probability of any event lies between zero and one.

For the existence of a statistical probability of an event A required:

a) the possibility, at least in principle, to carry out an unlimited number of tests, in each of which an event A occurs or does not occur;

b) stability of relative frequencies of occurrence A in various series of a sufficiently large number of tests.

The disadvantage of the statistical definition is the ambiguity of the statistical probability; So, in the above example, not only 0.4, but also 0.39 can be taken as the probability of an event; 0.41, etc.

Geometric probabilities

To overcome the disadvantage of the classical definition of probability, which is that it is not applicable to trials with an infinite number of outcomes, we introduce geometric probabilities– the probability of a point hitting an area (segment, part of a plane, etc.).

Let the segment l forms part of a segment L. For a segment L a point was made at random. This means fulfilling the following assumptions: the set point can be at any point on the segment L, the probability of a point falling on a segment l is proportional to the length of this segment and does not depend on its location relative to the segment L. Under these assumptions, the probability of a point falling on a segment l is determined by equality

P= Length l/ Length L.

Example 1. For a segment O.A. length L number axis Ox a dot was placed at random B(x). Find the probability that the smaller of the segments O.B. And B.A. has a length greater L

Solution. Let's split the segment O.A. dots C And D into 3 equal parts. The task requirement will be fulfilled if the point B(x) falls on the segment CD length L/3. Required probability

P = (L /3)/L = 1/3.

Let the flat figure g forms part of a flat figure G. Fit G A dot is thrown at random. This means making the following assumptions: the thrown point can end up at any point on the figure G, the probability of a thrown point hitting a figure g is proportional to the area of ​​this figure and does not depend on its location relative to G, neither from the form g. Under these assumptions, the probability of a point hitting a figure is g is determined by equality

P= Area g/ Square G.

Example 2. Two concentric circles are drawn on the plane, the radii of which are 5 and 10 cm, respectively. Find the probability that a point thrown at random into a large circle falls into the ring formed by the constructed circles. It is assumed that the probability of a point falling into a flat figure is proportional to the area of ​​this figure and does not depend on its location relative to the great circle.

Solution. Area of ​​the ring (figure g)

S g= p(10 2 - 5 2) = 75 p.

Area of ​​a great circle (figure G)

S G= p10 2 = 100 p.

Required probability

P= 75 p/(100 p) = 0.75.

Example 3. The signaling device receives signals from two devices, and the receipt of each of the signals is equally possible at any moment of the time period lasting T. The moments of signal arrival are independent of one another. The alarm is triggered if the difference between the moments of signal receipt is less t(t<T). Find the probability that the alarm will go off in time T,if each device sends one signal.

Solution. Let us denote the moments of arrival of signals from the first and second devices, respectively, by x And y. Due to the conditions of the problem, double inequalities must be satisfied: 0 x T, 0 y T Let us introduce into consideration the rectangular coordinate system xOy. In this system, the double inequalities are satisfied by the coordinates of any point of the square OTAT(Fig. 1).

Thus, this square can be considered as a figure G, the coordinates of the points of which represent all possible values ​​of the moments of signal arrival.

The alarm is triggered if the difference between the moments of signal receipt is less t, i.e. If y-x<t at y>x And x-y<t at x>y, or, what is the same,

y<x+t at y>x, (*)

y >x-t at y<x. (**)

Inequality (*) holds for those points of the figure G, which lie above the line y = x and below the line y = x+t;inequality (**) holds for points located below the line y= x and above the straight line y = x-t.

As can be seen from Fig. 1. all points whose coordinates satisfy inequalities (*) and (**) belong to the shaded hexagon. So this hexagon can be considered as a figure g, the coordinates of the points of which are favorable moments of time x And y.

Required probability

P= Pl. g/ Pl. G = (T 2 - (T - t) 2)/T 2 = (t(2T - t))/T 2 .

Note1. The given definitions are special cases of the general definition of geometric probability. If we denote the measure (length, area, volume) of a region by mes, then the probability of a point thrown at random (in the above sense) falling into the region g– part of the region G, is equal

P=mes g/mes G.

Remark 2. In the case of the classical definition, the probability of a reliable (impossible) event is equal to one (zero); Converse statements are also true (for example, if the probability of an event is zero, then the event is impossible). In the case of a geometric definition of probability, the converse statements do not hold. For example, the probability of a thrown point hitting one specific point in the area G is zero, but this event can happen and is therefore not impossible.

Tasks

1. The box contains 50 identical parts, 5 of which are painted. One piece is taken out at random. Find the probability that the extracted part will be painted

Reply. p = 0,1.

2. A die is thrown. Find the probability of getting an even number of points.

Reply. p = 0,5.

3. Participants in the draw draw tokens with numbers from 1 to 100 from the box. Find the probability that the number of the first token drawn at random does not contain the number 5.

Reply. p = 0,81.

4. There are 5 identical cubes in the bag. One of the following letters is written on all faces of each cube: o, p, p, s, t. Find the probability that the word “sport” can be read on the cubes stretched out one at a time and arranged in one line.

Reply. p = 1/120.

5. Each of the six identical cards has one of the following letters printed on them: a, t, m, p, s, o. The cards are thoroughly mixed. Find the probability that the word “cable” can be read on four cards drawn out one at a time and arranged “in one line”.

Reply. p = 1/ = 1/360.

6. A cube, all edges of which are colored, is sawn into a thousand cubes of the same size, which are then thoroughly mixed. Find the probability that a cube drawn at random will have colored faces: a) one; b) two; c) three.

Reply. a)0.384; b)0.096; c)0.008.

7. From a thoroughly mixed complete set of 28 dominoes, a tile is drawn at random. Find the probability that the second bone drawn at random can be placed next to the first if the first bone: a) turns out to be a double; b) there is no double.

Reply. a)2/9; b)4/9.

8. The lock has five discs on a common axis. Each disk is divided into six sectors, on which different letters are written. The lock opens only if each disc occupies one specific position relative to the lock body. Find the probability that if the disks are installed randomly, the lock can be opened.

Reply. p = 1/6 5 .

9. Eight different books are placed at random on one shelf. Find the probability that two specific books will be placed next to each other.

Reply. p= 7*2!*6!/8! = ¼.

10. The library consists of ten different books, with five books costing 4 rubles each, three books costing one ruble each, and two books costing 3 rubles each. Find the probability that two books taken at random cost 5 rubles.

Reply. p =

11. In a batch of 100 parts, the technical control department discovered 5 non-standard parts. What is the relative frequency of occurrence of non-standard parts?

Reply. w = 0,05.

12. When shooting from a rifle, the relative frequency of hitting the target was equal to 0.85. Find the number of hits if a total of 120 shots were fired.

Reply. 102 hits.

13. For a segment O.A. length L number axis Ox a dot was placed at random B(x).Find the probability that the smaller of the segments O.B. And B.A. has a length less than L/3. It is assumed that the probability of a point falling on a segment is proportional to the length of the segment and does not depend on its location on the number axis.

Reply. p = 2/3.

14. Inside the radius circle R A dot is thrown at random. Find the probability that a point will be inside a square inscribed in a circle. It is assumed that the probability of a point falling into a square is proportional to the area of ​​the square and does not depend on its location relative to the circle.

P = 7/16.

Chapter two

There are several definitions of the concept of probability. Let's give the classic definition. It is associated with the concept of a favorable outcome. Those elementary outcomes (e.i.), in cat. the event we are interested in occurs, we will call it favorable for this event. Def.: I believe the event A is called. the ratio of the number of outcomes favorable to this event to the total number of all equally possible incompatible e. i., forming a complete group. P(A) = m/n, where m is the number of e. i., favorable to event A; n – number of all possible e. And. tests. From the definition of probability follows its properties:1) ver.(c) of a reliable event is always equal to 1. Because. the event is reliable, then everything is e. And. trials favor this event, i.e. m=n. P(A)=n/n = 1; 2) V. impossible personal. is equal to 0. Because event is impossible, then there is no e. i., favorable to this event, means m=0. P(A) = 0/n = 0; 3) The value of a random event is a non-negative value contained between 0 and 1, i.e. 0

4. Relative frequency. Relative frequency stability.

The relative frequency (RF) of an event is the ratio of the number of trials in which the event occurred to the total number of trials actually performed. (NOT omega!!!). W(A) = m/n, where m is the number of occurrences of event A, n is the total number of trials. The determination of probability does not require that the tests be actually carried out. The definition of OC assumes that the tests were actually carried out, i.e. ver. calculated before the experiment, and OC after the experiment. If experiments are carried out under the same conditions, in each of the cat. the number of tests is large enough, then the OC exhibits stability. This property lies in the fact that in various experiments the OC changes little, the less the more tests are performed, fluctuating around a certain constant number. This number is ver. occurrence of the event. That. It has been experimentally established that the OR can be taken as an approximate probability value.

5.Statistical probability.

The classical definition of probability assumes that the number of elementary outcomes of a trial is finite. In practice, there are often tests with a number of possible outcomes. endlessly. In such cases, the classical definition is not applicable. Along with the classic def. use statistics. Def: stat. ver. (st.v.) events - relative frequency (RF) or a number close to it. Holy probabilities arising from the classical. definitions are also preserved in statistical cases. If the event is reliable, then its PR = 1, i.e. st.v. also =1. If the event is impossible, then OCH = 0, i.e. st.v. also = 0. For any event 0W(A) 1, next. st.v. is contained between 0 and 1. For the existence of st.v. required: 1) the ability to carry out, at least in principle, is unlimited. number of tests in each cat. the event occurs or does not occur; 2) stability of the frequency of occurrence of an event in various series of a sufficiently large number of tests. The disadvantage of statistical definition is the ambiguity of Art. For example, if, as a result of a sufficiently large number of tests, it turns out that the OC is very close to 0.6, then this number can be taken as st.v. But as the probability of an event, you can take not only 0.6, but also 0.59 and 0.61.

Relative frequency. Relative frequency stability

Relative frequency, along with probability, belongs to the basic concepts of probability theory.

Relative frequency events is the ratio of the number of trials in which the event occurred to the total number of trials actually performed. Thus, the relative frequency of event A is determined by the formula

where m is the number of occurrences of the event, n is the total number of trials.

Comparing the definitions of probability and relative frequency, we conclude: the definition of probability does not require that the tests be actually carried out; determination of the relative frequency assumes that the tests were actually carried out. In other words, the probability is calculated before the experiment, and the relative frequency after the experiment.

Example 1. The inspection department found 3 non-standard parts in a batch of 80 randomly selected parts. Relative frequency of occurrence of non-standard parts

Example 2. 24 shots were fired at the target, with 19 hits recorded. Relative target hit rate

Long-term observations have shown that if experiments are carried out under identical conditions, in each of which the number of tests is sufficiently large, then the relative frequency exhibits the property of stability. This property is that in different experiments the relative frequency changes little (the less, the more tests are performed), fluctuating around a certain constant number. It turned out that this constant number is the probability of the event occurring.

Thus, if the relative frequency is established experimentally, then the resulting number can be taken as an approximate probability value.

The relationship between relative frequency and probability will be described in more detail and more precisely below. Now let us illustrate the property of stability with examples.

Example 3. According to Swedish statistics, the relative frequency of births of girls for 1935 by month is characterized by the following numbers (the numbers are arranged in order of months, starting from January): 0.486; 0.489; 0.490; 0.471; 0.478; 0.482; 0.462; 0.484; 0.485; 0.491; 0.482; 0.473.

The relative frequency fluctuates around the number 0.482, which can be taken as an approximate value for the probability of having girls.

Note that statistical data from different countries give approximately the same relative frequency value.

Example 4. Coin tossing experiments were carried out many times, and the number of times the “coat of arms” appeared was counted. The results of several experiments are given in table. 1.

Here the relative frequencies deviate slightly from the number 0.5, and the current is less, the greater the number of tests. For example, with 4040 trials the deviation is 0.0069, and with 24,000 trials it is only 0.0005. Taking into account that the probability of the appearance of a “coat of arms” when tossing a coin is 0.5, we again see that the relative frequency fluctuates around the probability .