Geometric possibilities. Geometrical appearance of the shape under the contact

Another scheme for describing experiments with ambiguously predicted consequences, as it allows you to simply introduce a succinct characteristic of the nature of the situation and the other way around - this is the scheme of geometric effects, as, for example, the scheme of vipadkіv, exploitation of the result is considered. In the same way as it was developed in the scheme of vipadkivs, the characteristic characteristic of the imposition of the subdivision - iogo imovirnist - is shown as a normalized value, proportional to the stock of results, which is to accept the imposition of the subdivision. Let the impersonal results of the previous experiment be described as an impersonal point of some geometric continuum - the skin result shows the same point and the skin point shows the same result. In the capacity of the “geometric continuum” Q can act as a straight line, an arc straightens as a curve on a plane, or in space, impersonal squares on a plane (tricutnik, rectocut, colo, elips, etc.) єm near the space (a bagatohedron - a prism, a pyramid, a sump, an elіpsoїd, etc.) podіyu, but only so, like a world (dovzhina, flat, hug) we can die. Taking into account the equal possibility of the last, we call the imovirnistyu subdivision And the number, proportional to the world, submultiple A multiplier P: Geometrical imovirnosti Yakscho 0 - podіya, impossible in this experiment, and Q - reliable, then P(0) = О, = 1. podії And bude with whom it is laid mіzh zero - imovirnіstyu podії unimpossible, and loneliness - imovіrnіstyu podії veracious4*. Umov's normalization allows you to set the constant up to - the coefficient of proportion, which sets the possibility. Він виявляється дорівнює Таким чином, у схемі геометричних ймовірностей ймовірність будь-якої події визначається як відношення міри підмножини А, що описує подію, до міри безлічі il, що описує експеримент в цілому: Відзначимо деякі властивості так певної ймовірності: Властивість очевидно випливає з тієї обставини , what is impersonal, what is avenged in the middle of the other, cannot be more than the rest. Like in the scheme of deviations, under the scheme of geometrical ambiences, it is possible to unite, subdue and be on their basis protilezhn - when it comes out, vzagali seeming, vіdminnі vіd vihіdnih podії. The coming power is more important. 3. If it’s not crazy, then, zokrema, the fair principle of dodatkovostі: The power of power, called the rule of folding omnivornosti, obviously vyplyvaє z additivnosti measures5 *. At the end, it is significant that the immovability of the creation of any result in the scheme of geometrical immutability is always equal to zero, so it is the same as the zero imovirnity of whether it is the same, which is described by the “thinner” of the faceless points, that is. impersonal, like a world (vіdpovіdno - dovzhina, area, obsjag) to zero. Let's take a look at a few examples that illustrate the calculation of the possibilities in the scheme of geometrical possibilities. Example 1. The experiment favors the vertical choice of the point of the intrusion [а, 6|. To know the immovability of what the point is taken, what lies at the left half of the wind, what is being looked at. 4 As a matter of fact, the ability to choose a point, whether it be multiplied by a score, is greater than zero, and its turn is negative.
Value: 0;25.

4.6. Under the hour of combat training, the squadron of bombers took off the task of attacking the oil depot of the "enemy". On the territory of the oil base, which has the shape of a rectangle with sides of 30 and 50 m, there are chotiri round oil tanks with a diameter of 10 m leather. Know the immovability of a direct attack on the oil tanks with a bomb that wasted on the territory of the oil depot, as if the bomb had hit the point of origin of the base, even if it was equal.
Testimony: π/15.

4.7. Two effective numbers x and y are chosen so that the sum of their squares is less than 100. How is it possible that the sum of the squares of these numbers is greater than 64?
Value: 0;36.

4.8. Two friends lived together between 13 and 14 years old. Priyshovshi the first check of another with a stretch of 20 hvilin, after which go. Vznachte ymovіrnіst zustrіchі druzіv, yakscho moments of the arrival of the appointed hour at the appointed time is equally possible.
Response: 5/9.

4.9. Two steamships are to go to one and the same berth. The hour of the arrival of both steamboats is equally powerful with a stretch of doby. It is important that one of the steamships happens to check the berth, as the hour of the first steamboat's stay is one year old, and the other - for two years.
Vidpovid: ≈ 0;121.

4.10. Navmannya take two positive numbers x and y, skin of them do not outweigh the two. Find out the possibility that the increase x · y will be no more than one, and the private y/x will be no more than two.
Vidpovid: ≈ 0;38.

4.11. In area G, surrounded by ellipsoid , the point is fixed. What is the possibility that the coordinates (x; y; z) of the points satisfy the irregularities x 2 + y 2 + z 2 ≤4?
Verdict: 1/3.

4.12. A rectangle with vertices R(-2;0), L(-2;9), M(4;9), N(4;0) has a point. Find the flexibility of what її coordinates satisfy the irregularities 0 ≤ y ≤ 2x – x 2 +8.
Verdict: 2/3.

4.13. The area G is surrounded by a stake x 2 + y 2 = 25, and the area g is surrounded by a stake and a parabola 16x - 3y 2 > 0.
Vidpovid: ≈ 0;346.

4.14. Navmannya taken two positive numbers x and y, skin of them do not outweigh the ones. Find the flexibility of the fact that the sum x + y does not exceed 1, and the total x · y is not less than 0.09.
Vidpovid: ≈ 0;198.

Let's look at how geometrical imovirnist (geometrically designed imovirnosti) is on the butts of deyaky zavdans. Let me give you a vіdrіzok zavdovka. Let's divide the yogo navpіl (for unambiguity, the point under the line is taken to the left half). Navmannya vkladaetsya speck on tsey vіdrіzok. There are two possibilities: "the point has spent half of it on the left" - a podia; "the point has spent half on the right" - podia. So, as a point is put navmannya, then it is important to take into account that they are equally able. To di ymovіrnіst podії, so go out by itself and z.

Now we divide the vіdrіzok into 10 equal parts (skin dozhina). Vipadkovo we throw a speck on the whole tree. Possible vipadki: "a speck drank on the first windrow" - a podia, "a speck drank on another vine" - a podia, "a speck drank on the third breeze" - a podia and so on until the tenth windfall -. In respect of qi podії equally possible, it is important that the skin condition is 0.1. Tobto, .

Let the pod_ya polagaє in the fact that the point has been squandered in a vipadkovo way, for example, on a vіdrіzok. Shards of the undergrowth can be taken from possible vipadkіv, the same ymovіrnіst can be manifested:

Sounds are screaming:

- Imovirnist vpadkovy hit point on the vіdrіzok dovzhina, which is to avenge on the vіdrіzku dovzhina.

Guidance higher pidhіd can be used for flat figures (div. Fig. 1), as well as for space for bodies.

Rice. one

Let the figure, the area of ​​\u200b\u200bwhich is more beautiful, be placed near the figure, the square of which, then the same imovirnist podії, as if it’s a lie in the fact that the point is left to spend at the figure, the better in the area of ​​\u200b\u200btheir figures, tobto:

For formulas (1) and (2) it is possible to avoid the "equal possibility" of the vertical hit of the point in the right point of the vіdrіzka or figure.

Geometric model

With the method of accuracy, we look at the following model:

Let the figure be a rectangle of the world x (yogo square), descriptions of the figure painted on the asphalt. Instead of a point, as if they were navmannya at a straight cut, vvazhatimeme drop a board, which only starts. After the singing hour, the square cutter was closed on the board and it was spared a lot of streaks, like they squandered on the whole rectangle, and also a lot of streaks, like they squandered on the figure. Let's calculate the visible frequency. We already know that behind the formula (2) one can know the dynamism of the undersides, as if the point of the figure is right for the vypadkovy choice. At the same time spіvvіdnoshnennia area, from the other side. That is why jealousy is coming into us, for the help of which you can know the area of ​​\u200b\u200bthe figure,

I realized that this butt was pointed for accuracy, but it’s really more accurate to count for an additional computer program. However, the technique and not zavzhd can be under the hand. I would like to show a few examples, so that you could understand the theme of geometrical immovability more clearly.

Task on the topic: “Geometric design of imovirnosti”

butt 1

manager

Two students then took up the house and sat down at the entrance to the building. So, as a skin z they could show up unperturbed furnishing, the stench of the house, that the sound will be from 14:00 to 15:00. In this order, the first to arrive before the month of recognition, that one checks 15 hvilin (but not later than 15:00) and go. Know the imovirnіst zustrіchі, so take the year of ochіkuvannya:

a) 15 khvilin;

b) 20 quilin;

c) 30 min.

Solution

Let it be - the year of the arrival of the first student in the field of zustrіch - another.

Zustrich thinks for the mind that:

Anonymous rozv'yazkіv nerіvnosti is shown in fig. 2.

The area of ​​a square. Figure area.

Therefore, the imovirnіst zustrіchі (podіya):

When hvilin may;

for whilin:;

for whilin: .

Rice. 2

Vidpovid

Possibility of grafting when scoring 15 quills is approximately 0.44;

Imovirnist zustrіchі when scoring 20 quills = 0.55;

Imovirnist zustrіchі when scoring 30 quills is close to 0.75.

butt 2

manager

Find the area of ​​the parabolic segment, which tasks are equal:

Solution

The parabolic segment is shown in fig. 3.

Rice. 3

Krapki peretina z vіssyu - i.

Qiu area can be calculated for the help of the sing integral or for the help of the formula:

de is the coefficient for the parity of the parabola.

It will be shown how to know the need for space, vicory geometrically defined imovirnosti. Let's describe the parabolic segment as a square with a side of 4 units. Area sq. one. (Div. Fig. 3). For the help of the standard function of generating yaw points, as if they are taken from the square, among the points that are in the parabolic segment, we know the frequency of yielding points from the parabolic segment. Same for formula (3) is known.

In table 1, the results of the analysis of the closest values ​​of the area of ​​the parabolic segment for different values ​​of i are shown. So, from Fig. 3 it can be seen that the square has consumed 10 specks, and the segment - 6, so the first approach to the area is to enter:

; what is recorded in the first row of the table. one:

Table 1

Vidpovid

The area is known for the first time of the geometric designation of mobility, as it is written down in the table.

Geometrical design of the ymovirnosti vpadkovo podії updated: fall 22, 2019 by: Statti.Ru

Outline of breakdowns

Geometrical imovirnist

Objectives and tasks: 1) Get to know learners in one of the possible ways of learning

imovirnosti;

2) Repetition of what has been passed and consolidation of the skill of formalization

textual imovirnіsnyh tasks for additional geometric shapes.

Training results:

1) Know the purpose of the geometrical flexibility of the choice of a point

middle figures on the flat and straight;

2) Consider the simplest tasks for geometrical imovirnist,

knowing the areas of the figures and counting them together.

I. Select the points of the figure on the plane.

example 1. Let's look at the experiment: throw a point on the square, the side of which is closer.

Whose manager has to go about so called geometrical imovirnosti.

Throw a speck at the figure F on the flat. What is the imovirnist of the fact that the dot squanders a figure into a figure g, how to take revenge on the figure F.

It is necessary to lie down because of what kind of sense we put into viraz “throw a speck of navmannya”.

Sound this whistle is interpreted as follows:

1. Abandoned point can be spent in a part of the figure F.

2. Immortality of the fact that the speck is eating a figure from the deak G in the middle of the figure F, directly proportional to the area of ​​the figure G.

Pіdіb'єmo pіdbag: let's go and - square figures Fі G. Imovirnіst podії BUT"point X lie on the figure g, how to take revenge on the figure F”, Rivna

Respectfully, what area figure G no more, lower figure area F, to that

Let's turn to our boss. figure F whose butt has a square with a side of 1. To that = 1.

Krapka is farther away from the square between the square, no more than lower, as if she had squandered it into the figure shaded on the little one G. To know the area, it is necessary to know the figure area F see the area of ​​the inner square from the side.

Todi imovirnist of what the speck ate at the figure g, dorivnyuє

butt 2. From the tricoutnik ABC, the point X is selected vipadkovo.

Solution: The middle lines of the tricot are divided into 4 equal tricots. To mean,

The possibility that point X belongs to the KMN tricutnik is more advanced:

Visnovok. The movability of a point in a figure is directly proportional to the area of ​​the figure.

Manager. Impatient duelists.

Duels near the town of Oberezhnost rarely end with a total result. On the right, in the fact that the skin duellant arrives at the midst of the morning at the peak of the moment, at the hour between 5 and 6 years old, the wound і, looking for the supernik 5 hvilin, can be seen. At the time of the arrival of the rest in q 5, a duel will be held. What part of the duel really ends in a duel?

Solution: Come on Xі at signify the hour of the arrival of the 1st and 2nd duelists, vidpovidno, died in parts of the year, starting from 5 years.

Duelists are chattering, like, tobto. x - < y< x + .

Let's picture it on the armchair.

A part of the square is shaded in a vіdpovіdaє vpadka, if the duelists are sharpened.

The area of ​​the whole square 1, the area of ​​the shaded part:

.

Otzhe, chances for a duel equal.

II. Choose a point from a vіdrіzka and an arc of a stake.

Let's take a look at the experiment, which shows the choice of one point X from the other side of MN.

You can think like this, don’t throw point X in a vipadkovo “throw” on the wind. Elementary podієyu tsgogo dosvіdu can become vybіr whether it's a point of vіdrіzka.

Let the CD vdrіzok mіstsya near the MN vіdrіzku. We squawk podia BUT , which means that the point X is chosen to lie under CD.

The method of calculating the value of the size of the same figure that is on the plane: the value of the proportion of the length of the cross CD.

Father, imovirnіst podії BUT “Point X lies next to the CD, which is to be moved by the MN” earlier, .

example 1. In the middle of the MN point, point X is selected vertically. Find out whether point X is closer to point N, lower to M.

Solution: Let point O be the middle of the MN section. Our pod_ya will be the same, if the point X lies in the middle of the line ON.

Todi .

Nothing is changed, as the point X is selected from the rail, that as if from the arcs of a curved line.

butt 2. On the number of points A and B, moreover, the qi points are not diametrically opposite. On which point C is chosen.

Solution: Let Dovzhina Cola Dorovnyu L. Tsikava is a Podia for Us Before “Vdrіzok VS change the diameter DA” now, only just so.

.

example 3. When point A is taken. Point B is “thrown” at when.

Solution: Come on r is the radius of the stake.

In order for the chord AB to be short for the radius of the stake, the point should be spent on the arc B1AB2, which is more expensive than the stake.

Imovirnist of the fact that the length of the chord AB will be less for the radius of the stake, more:

III. Pick a point from a numeric line

Geometrical imovirnіst can be fixed up to numerical intervals. It is acceptable that the number X is selected vipadkovo, which pleases the mind. This point can be replaced by a point, for which a point with the X coordinate is selected from the point on the number line.

Let's take a look at the way, as if we are saying that the point with the X coordinate is taken from the vіdrіzka, scho to go near the vіdrіzka. Qiu podіyu significant. Yogo ymovіrnіst dorіvnyuє vіdnoshenyu dovzhin vіdrіzkіv і.

.

example 1. To know the imovirnist of what the point, like vipadkovo is chosen from the vіdrіzka, to lay the vіdrіzka.

Solution: Behind the formula of geometrical imovirnosti we know:

.

butt 2. According to the rules of the road traffic, a pishohid can cross the street at an unidentified city, as there are no pishohid passages in the boundaries of visibility. Near the city of Mirgorod there is a bridge between pedestrian crossings on the streets of Sonyachniy Dorіvnyu 1 km. Pishohid cross the Sonyachnu street here between two crossings. Vіn mozhe bachiti the sign of the transition was not given lower for 100 m from oneself. Know the immovability of the one that pishohid does not break the rules.

Solution: Accelerated by the geometric method. Raztashuyemo numerically straight so that the village of the street between the transitions appears in a vіdrіzkom. Let the pishohid go to the street at the deaky point with the X coordinate.

.

example 3. The train should pass the platform beyond pivkhvilini. Like a mit, gazing out of his compartment at the window, Ivan Ivanovich sighed, as the train went up the platform. Ivan Ivanovich marveled at the window for exactly ten seconds, and then turned back. Know the immovability of the fact that Ivan Nikiforovich’s bachelor is the one standing in the middle of the platform.

Solution: Accelerated by the geometric method. We will conduct a review in seconds. For 0 seconds, we can take the moment when Ivan Ivanovich rises from the cob of the platform. The same end of the platform was reached at the moment of 30 seconds. For X sec. Significant moment, if Ivan Ivanovich looked at the window. Also, the number X is vipadkovo chosen from vіdrіzka. I called Ivan for the moment of 15 seconds. Winning Ivan Nikiforovich, only just looking at the window not later than this moment, but not earlier, lower 10 seconds before the next. In this rank, it is necessary to know the geometrical imovirnіst podії. Behind the formula we know

.

"Ymovirnіsne pіdґruntya"

On the very cob, sing "Dead Souls" two peasants oppose how far the wheel in Chichikov's carriage reaches:

“... two Russian peasants, who stood at the door of the tavern against the guest, raised the deacons of respect, that they hung around together, more to the crew, lower to the point where they were sitting at the new one. “Bach ti”, saying one to one: “Axis like a wheel! What do you think, you get the wheel, the yakbi trapilos, to Moscow, why don’t you get there? - “Doїde”, vіdpovіdav іnshiy. “And to Kazan, I think, it won’t reach?” “We won’t get to Kazan,” having said another.

Order for cherry blossom.

1. To know the impersonality of the fact that the point is vertically thrown into the square ABCD with side 4 and tucked away at the square A1B1C1D1 with side 3, which is located in the middle of the square ABCD.

Vidpovid. 9/16.

2. Two individuals A and B got together in the singing area in the interim hour from 900 to 1000. The skin from them comes in the morning (at the time of the hour), independently in the interim and checks 10 hvilin. What is the imovirnist of what the stench will stink?

Vidpovid. 11/36.

3. Point C appears in the opposite direction of AB at the point 3. It is important to note the possibility of moving from the point to B, moving 1.

Vidpovid. 2/3.

4. A tricot of the largest area is inscribed near the circle with a radius of 5. Vznachte imovіrnіst hit in trikutnik points vipadkovo thrown into the colo.

5. Pinocchio planted on a straight leaf with a size of 20 cm by 25 cm a round bodice with a radius of 1 cm. To know the immovability of the fact that two flops do not stick together.

6. The square ABCD is inscribed at the ring. On which point M is selected vipadkovo. b) a larger arc AB.

Vidpovid. a) 1/4; b) 3/4.

7. Point X is thrown on the vіdrіzok vіpadkovo. b); in)?

Vidpovid. a) 1/3; b) 1/3; c) 1/3.

8. About the village of Ivanovo, it is less likely to be found here on the highway between Mirgorod and Stargorod. Dovzhina highway is 200 km. Find out the meaning of what:

a) from Mirgorod to Ivanovo highway less than 20 km;

b) from Stargorod to Ivanovo Highway more than 130 km;

c) Ivanovo is located less than 5 km from the middle of the road between the towns.

Vidpovid. a) 0.1; b) 0.35; c) 0.05.

Additive material

Geometrical pіdkhіd to ymovіrnostі podії not lie in the form of vimіryuvan geometric expanse: it is important only, schob elementary podіy F і impersonal G, which represents podіyu A, but would be of the same kind and same vimiryuvan.

2. Vipadkova point X is evenly divided near the square . Know the possibility that the square with the center X and the sides of the length b, parallel to the coordinate axes, is entirely located near the square A.

Literature:

1. Theory of dynamics and statistics / , . - 2nd view., Recycled. - M.: MTsNMO: assistants ", 2008. - 256 p.: Іl.

2. Theory of imaginings and mathematical statistics in applications and tasks from Excel / , . - View. 4th. - Rostov n / D: Phoenix, 2006. - 475 p.: Іl. - (Vishcha osvita).

3. Fifty-fifty cіkavih imovіrnіsnyh orders from decisions. Prov. from English / Ed. . 3rd view. - M.: Nauka, Head edition of physical and mathematical literature, 1985. - 88 p.

4. Collection of the head of the theory of immovability: Navch. Help for universities. /, - 2nd species., Vipr. I add. - M.: Science. Goal. ed. Phys.-Math. Lit. - 1989. - 320s.

5. Optional course in mathematics: Theory of imovirnosti: Navch. Posіbnik for 9-11 cells. middle shk./ – 3rd kind. rework - M.: Prosvitnitstvo, 1990. - 160 p.