Geometric scheme for determining the probability. Geometric definition of the probability of an event
As shown in the Classical Definition of Probability section, in random experiments with a finite number of equally possible elementary outcomes applied classical definition of probability.
To introduce the probability of events in random experiments, the possible outcomes of which (elementary outcomes) are also equally possible and fill the gap completely straight line, figure on the plane or region in space, applied geometric definition of probability. In such experiments, the number of elementary outcomes is not final, and therefore the classical definition of probability cannot be applied to them.
Let us illustrate the introduction of the geometric definition of probability with examples.
Example 1 . A point is randomly thrown on a segment of a number line. Find the probability that the point fell on the segment (Fig. 1).
Answer:
Example 2 . The diagonals KM and LN of the square KLMN intersect the circle inscribed in the square at points E and F, point O is the center of the circle (Fig. 2).
A dot is randomly thrown into a KLMN square. Find the probability that the point will fall into the EOF sector marked in pink in Figure 2.
Answer:
Example 3 . A point is randomly thrown into a cone with vertex S and base center O. Find the probability that the point will fall into the truncated cone obtained by cutting the cone with a plane passing through the midpoint O "of the height of the cone and parallel to the base of the cone (Fig. 3).
Decision . The set of elementary outcomes Ω of a random experiment on throwing a point is the set of all points of the cone with vertex S and base center O .
The hit of a point in a truncated cone is one of the random events, which we will denote by the letter A.
At geometric definition event probability A is calculated by the formula
Let R be the radius of the base of the cone with vertex S and base center O, and let H be the height of this cone. Then the radius of the base and the height of the cone with the vertex S and the center of the base O" will be equal to
respectively.
The volume of a cone with vertex S and base center O is
The classical definition of probability has a limitation in its application. It is assumed that the set of elementary events Ω is finite or countable, i.e., Ω = ( ω 1 , ω 2 , … , ω n , …), and all ω i – equally possible elementary events. However, in practice there are tests for which the set of elementary outcomes is infinite. For example, when manufacturing a certain part on a machine, it is necessary to maintain a certain size. Here, the accuracy of manufacturing a part depends on the skill of the worker, the quality of the cutting tool, the perfection of the machine, etc. If a test is understood as the manufacture of a part, then as a result of such a test, an infinite number of outcomes are possible, in this case obtaining parts of the required size.
To overcome the shortcoming of the classical definition of probability, some concepts of geometry are sometimes used (if, of course, the circumstances of the test allow). In all such cases, the possibility of conducting (at least theoretically) any number of tests is assumed, and the concept equal opportunity also play a major role.
Let us consider a test with a space of events, the elementary outcomes of which are represented as points filling some area Ω (in the three-dimensional space R 3). Let the event BUT consists in hitting a randomly thrown point in the subdomain D domain Ω. event BUT favor elementary events in which the point falls into some subdomain D. Then under probability events BUT we will understand the ratio of the volume of the subdomain D(highlighted area in Fig. 1.11) to the volume of the area Ω, R(BUT) = V(D) / V(Ω).
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Rice.1. 11 |
Here, by analogy with the concept of a favorable outcome, the area D will be called favorable to the appearance of the event BUT. The probability of an event is defined similarly BUT, when the set Ω is a certain area on a plane or a segment on a straight line. In these cases, the volumes of the regions are replaced by the areas of the figures or the lengths of the segments, respectively.
Thus, we come to a new definition - geometric probability for tests with an infinite uncountable set of elementary events, which is formulated as follows.
The geometric probability of an event A is the ratio of the measure of the subdomain that favors the occurrence of this event to the measure of the entire area, i.e.
p(A) =mesD / mesΩ,
where mes– measure of areas D and Ω , D Ì Ω.
The geometric probability of an event has all the properties inherent in the classical definition of probability. For example, the 4th property would be: R(BUT+ AT) = R(BUT) + R(AT).
Classical definition of probability
The basic concept of probability theory is the concept of a random event. A random event is usually called an event, ĸᴏᴛᴏᴩᴏᴇ, under certain conditions, it may or may not occur. For example, hitting or missing an object when shooting at this object with a given weapon is a random event.
An event is usually called reliable if, as a result of the test, it necessarily occurs. It is customary to call an event impossible, ĸᴏᴛᴏᴩᴏᴇ cannot happen as a result of a test.
Random events are said to be inconsistent in a given trial if no two of them can appear together.
Random events form a complete group if any of them can appear at each trial and no other event inconsistent with them can appear.
Consider the complete group of equally possible incompatible random events. Such events will be called outcomes. An outcome is said to be favorable to the occurrence of event A if the occurrence of this event entails the occurrence of event A.
Geometric definition of probability
Let a random test be thought of as throwing a point at random into some geometric region G (on a line, plane, or space). Elementary outcomes are ϶ᴛᴏ separate points of G, any event is a ϶ᴛᴏ subset of this area, the space of elementary outcomes G. We can assume that all points of G are ʼʼequalʼʼ and then the probability of a point falling into one of the ĸᴏᴛᴏᴩᴏᴇ subset is proportional to its measure (length, area , volume) and does not depend on its location and shape.
geometric probability event A is determined by the relation: , where m(G), m(A) are geometric measures (lengths, areas or volumes) of the entire space of elementary outcomes and event A.
Example. A circle of radius r () is randomly thrown onto a plane, delimited by parallel stripes of width 2d, the distance between the axial lines of which is 2D. Find the probability that the circle intersects some strip.
Decision. As an elementary outcome of this test, we will consider the distance x from the center of the circle to the center line of the strip closest to the circle. Then the entire space of elementary outcomes - ϶ᴛᴏ segment. The intersection of the circle with the strip will occur if its center falls into the strip, ᴛ.ᴇ. , or will be located from the edge of the strip at a distance less than the radius, ᴛ.ᴇ. .
For the desired probability, we obtain: .
5. The relative frequency of an event is the ratio of the number of trials in which the event occurred to the total number of practically performed trials. Τᴀᴋᴎᴍ ᴏϬᴩᴀᴈᴏᴍ, the relative frequency A is given by:
(2)where m is the number of occurrences of the event, n is the total number of trials. Comparing the definition of probability and relative frequency, we conclude: the definition of probability does not require that tests be carried out in reality; the definition of the relative frequency assumes that the tests were actually carried out. In other words, the probability is calculated before the experience, and the relative frequency is calculated after the experience.
Example 2. Out of 80 randomly selected employees, 3 people have serious cardiac disorders. Relative frequency of people with heart disease
The relative frequency or a number close to it is taken as a static probability.
DEFINITION (statistical definition of probability). The number to which the stable relative frequency tends is commonly called the statistical probability of this event.
6. sum A+B two events A and B name an event consisting in the occurrence of event A, or event B, or both of these events. For example, if two shots were fired from the gun and A - hit on the first shot, B - hit on the second shot, then A + B - hit on the first shot, or on the second, or in both shots .
In particular, if two events A and B are incompatible, then A + B is an event consisting in the appearance of one of these events, no matter which one. The sum of several events called an event, ĸᴏᴛᴏᴩᴏᴇ consists in the occurrence of at least one of these events. For example, the event A + B + C consists in the occurrence of one of the following events: A, B, C, A and B, A and C, B and C, A and B and C. Let the events A and B be incompatible, and the probabilities of these events are known. How to find the probability that either event A or event B will occur? The answer to this question is given by the addition theorem. Theorem. The probability of occurrence of one of two incompatible events, no matter which one, is equal to the sum of the probabilities of these events:
P (A + B) = P (A) + P (B). Proof
Corollary. The probability of occurrence of one of several pairwise incompatible events, no matter which one, is equal to the sum of the probabilities of these events:
P (A 1 + A 2 + ... + A n) \u003d P (A 1) + P (A 2) + ... + P (A n).
Geometric definition of probability - concept and types. Classification and features of the category "Geometric definition of probability" 2017, 2018.
In practice, such trials are very often encountered, the number of possible outcomes of which is infinite. Sometimes in such cases it is possible to use the method of calculating the probability, in which the concept of the equiprobability of certain events still plays the main role .... .
In a certain square, a point is randomly selected, what is the probability that this point will be inside the region D., where SD is the area of \u200b\u200bthe region D, S is the area of the entire square. Under the classical, a certain zero probability had ... .
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, geometric probabilities are introduced - the probabilities of a point falling into an area. Let a flat figure g (segment or body)... .
Classical definition of probability LECTURE 1. PROBABILITY THEORIES. HISTORY OF ORIGIN. CLASSICAL DEFINITION OF PROBABILITY A.A. Khalafyan BIBLIOGRAPHICAL REFERENCES 1. Kolemaev V.A., Staroverov O.V., Turundaevsky V.B. Theory ... .[read more] .
This definition is used when an experience has an uncountable set of equally possible outcomes. In this case, the space of elementary events can be represented as a certain region G. Each point of this region corresponds to an elementary event. Hit... .
The geometric definition of probability is an extension of the concept of classical probability to the case of an uncountable set of elementary events. In the case when is an uncountable set, the probability is determined not on elementary events, but on their sets.... .
Classical Definition of Probability PROBABILITY OF A RANDOM EVENT Set-Theoretic Interpretation of Operations on Events Let some experiment with a random outcome be carried out. A bunch of &... .
The formula P(A)=m/n loses its meaning if the number of all equally possible incompatible cases is unlimited (forms an infinite set). However, it is sometimes possible to give a quantitative characteristic S in some measures of length, area, volume, time, and so on, to the entire set of infinite equally possible incompatible cases, and to give a part of this set that favors the onset of the event A under consideration, to give a characteristic S b in the same measures. Then the probability of occurrence of event A is determined by the relation:
Example #1. Two numbers x and y are chosen at random from the interval. Find the probability that these numbers satisfy the inequalities x 2 ≤ 4y ≤ 4x.
Decision. The test consists in random selection of a pair of numbers x and y from the interval. We will interpret this as a random choice of a point M(x;y) from the set of all points of a square whose side is equal to two. Let us consider the figure Ф, which is the set of all points of the square whose coordinates satisfy the system of inequalities x 2 ≤ 4y ≤ 4x. The event of interest occurs if and only if the selected point M(x;y) belongs to the figure Ф. 
According to formula (8), the desired probability is equal to the ratio of the area of \u200b\u200bthe figure Ф to the area of the square: 
Example #2. The two agreed to meet at a certain place. Each of them arrives at the appointed place independently of each other at a random moment of time from and waits for no more than time . What is the probability of meeting under such conditions? 
Decision. Let us denote by x the time of arrival of the first person at the agreed place, and by y the time of arrival of the second person there. It follows from the condition that x and y independently run through the time interval . The test consists in fixing the time of arrival of the indicated persons at the meeting place. Then the space of elementary outcomes of this trial is interpreted as the set of all points M(x;y) of the square Ω=((x;y) : 0 ≤ x ≤ T, 0 ≤ y ≤ T). The event A of interest to us - “the meeting happened” occurs if and only if the selected point M(x; y) is inside the figure Ф, which is the set of all points of the square whose coordinates satisfy the inequality |x – y| ≤ t. According to formula (8), the desired probability
is the ratio of the area of the figure Ф to the area of the square Ω:
Analyzing the result obtained in this problem, we see that the probability of meeting increases with increasing. Let, for example, T = 1 hour, t = 20 minutes, then
, that is, more often than in half of the cases, meetings will occur if repeatedly negotiated on the above conditions. Example #3. Two points are chosen at random on the segment l.
P(0 
Example #4. A point is randomly thrown into a circle of radius r in such a way that any location in the circle is equally possible. Find the probability that it will be inside a square with side a located in a circle.
Decision. The probability that a point will be inside a square lying in a circle with side a equals the ratio of the area of the square to the area of the circle.
Square area: Skv \u003d a 2.
Circle area: S = πr 2
Then the probability will be: p \u003d Skv / S \u003d a 2 / πr 2
Example number 5. Two real numbers are chosen at random from the interval. Find the probability that their sum is greater than 4 and their product is less than 4.
Decision.
There are 5 numbers in total: 0,1,2,3,4. The probability of their occurrence p=1/5 = 0.2
a) the probability that their sum will be greater than 4
The total number of such outcomes is 8:
1+4, 2+3, 2+4, 3+4 and 4+1, 3+2, 4+2, 4+3
P = 0.2*0.2*8 = 0.32
b) the product is less than 4.
The total number of such outcomes is 13:
0*1, 0*2, 0*3, 0*4, 1*1, 1*2.1*3 and 1*0, 2*0, 3*0, 4*0, 2*1, 3* one
P = 0.2*0.2*13 = 0.52
Tasks for independent solution
4.3. After the storm, a wire break occurred on the section between the 40th and 70th kilometers of the telephone line. What is the probability that the break occurred between the 45th and 50th kilometer of the line? (The probability of a wire break in any place is assumed to be the same).
Answer: 1/6.
4.4. A point is randomly thrown into a circle of radius r. Find the probability that this point is inside a regular triangle inscribed in the given circle.
Answer:
4.5. Find the probability that the sum of two randomly selected numbers from the interval [-1; 1] is greater than zero, and their product is negative.
Answer: 0;25.
4.6. During combat training, the n-th bomber squadron received the task of attacking the “enemy” oil depot. On the territory of the oil depot, which has the shape of a rectangle with sides of 30 and 50 m, there are four round oil tanks with a diameter of 10 m each. Find the probability of a direct hit of the oil tanks by a bomb that hit the territory of the oil depot, if the bomb hits any point of this base with equal probability.
Answer: π/15.
4.7. Two real numbers x and y are chosen at random so that the sum of their squares is less than 100. What is the probability that the sum of the squares of these numbers is greater than 64?
Answer: 0;36.
4.8. The two friends agreed to meet between 13:00 and 14:00. The first person to arrive waits for the second person for 20 minutes and then leaves. Determine the probability of meeting friends if the moments of their arrival in the specified time interval are equally probable.
Answer: 5/9.
4.9. Two steamboats must come to the same pier. The time of arrival of both ships is equally possible during the given day. Determine the probability that one of the steamers will have to wait for the berth to be released if the first steamer stays for one hour and the second for two hours.
Answer: ≈ 0;121.
4.10. Two positive numbers x and y are taken at random, each of which does not exceed two. Find the probability that the product x y is at most one and the quotient y/x is at most two.
Answer: ≈ 0;38.
4.11. In the region G bounded by the ellipsoid 
, a point is fixed at random. What is the probability that the coordinates (x; y; z) of this point will satisfy the inequality x 2 + y 2 + z 2 ≤4?
Answer: 1/3.
4.12. A point is thrown into a rectangle with vertices R(-2;0), L(-2;9), M (4;9), N (4;0). Find the probability that its coordinates will satisfy the inequalities 0 ≤ y ≤ 2x – x 2 +8.
Answer: 2/3.
4.13. The region G is bounded by the circle x 2 + y 2 = 25, and the region g is bounded by this circle and the parabola 16x - 3y 2 > 0. Find the probability of falling into the region g.
Answer: ≈ 0;346.
4.14. Two positive numbers x and y are taken at random, each of which does not exceed one. Find the probability that the sum x + y does not exceed 1 and the product x · y is not less than 0.09.
Answer: ≈ 0;198.
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