Geometry

Definition

If the probability of each event occurs only with the length (area or volume or degree of degrees) constituting the event area, the probability model is called the geometric probability model, Referred to as geometric.

For example: For a random trial, we understand each basic event to randomly take a point in a particular geometric area, and the opportunity to be taken in the region is the same; An occurrence of a random event is understood to take the point in a certain specified area in the above area. The area here can be a line segment, a planar pattern, a stereoscopic pattern, and the like. The random trial was used to handle the random trial, called geometric sectors.

Geometric profile is relatively classical, and the concept of possible events will extend from a limited direction unlimited. This concept begins in my country's junior high school mathematics.

The main difference between classical profile and geometry is that the geometry is another class, which is possible, which is the difference between the classical profile that the results of the test are unlimited.

Features

Geometric profile features the following two:

(1) Unlimited: The basic events in the experiment (the result) Unlimited multiple. Possibility of

(2): The possibility of each basic event is equal.

Calculating Formula

is set in space g , region g is included in the area g (as shown in Figure 1), and region g and g is a measured (can be available), and now throws a bit of M in g , assuming point m The probability that must be in g and point m in any part of the area g in the area g Only the metric (length, area, volume, etc.) of g is proportional to the position and shape of g . Random test (throw points) with this nature, called geometric sectors. Regarding the geometric random event "Throw a point g to the area g , point m fall in g The probability of the inner section g P is defined as: g to measure the measure of g

p = g Measure / g

Geometric Summary The probability formula of the event A: Generally, in the geometric area D randomly takes a point, the event "This point falls within a region D in its internal" as an event A, the probability of event A For:

p (a) = region length (area or volume) of the area of ​​the regional length (area or volume) / experiment, the region length (area or volume), which is constituted by the area length (area or volume)

Here to point out : The measure of D cannot be 0, wherein "measurement" is determined. When D is a line segment, a planar pattern, a stereoscopic pattern, the corresponding "measurement" is length, area, volume, etc., respectively.

Example 1 Tape problem

Joe Square made a conversation on the activities conducted on the day before. However, the conversation was listened to the recorder recorded, and the tape was 30 minutes. The Federal Investigation Bureau gathers a tape and found that a piece of content in which the long-length content contains the information of their crime, but later discovered that some of this conversation was wiped by a staff member of the Federal Bureau, the staff I claim that she is completely inadvertently wrong, and all the contents immediately afterwards have been wiped off, ask if this 10 second long talk record begins at the half-minute of the tape record, then the crime content How big will the conversation are partially or all of the probability of wipe off?

solution: The 30-minute tape is expressed as a line segment r of the length of 30, and the interval of the conversation related to the 10 seconds and criminal activities is R , 10 seconds of conversations are accidentally wiped off part or all events only in this section or begins any point on the left side of the interval. Therefore, the event r begins with r .

The left end point length of the line segment is 1/2 + 1/6 = 2/3. Therefore, there is

p ( r ) = ( r length) / ( r Length) = (2/3) / 30 = 2/90 = 1/45

A: The probability that the conversation containing the crime is partially or all the probability that the probability will be 1/45.

Example 2 CB walkie-talkie problem

two CB (English abbreviation of Citizen Band) walkie-talkie holders, Lily and Hoy are working for Carl Freight Company, their The receiving range of the walkie-talkie is 25 kilometers. At 3:00 pm, Lili is driving at a place within 30 kilometers of the base, and Hoy is 40 kilometers from the base from 3:00 pm. Save the base to the base, ask how big is the probability of talking through walkie-talkie at 3:00 pm?

solution: x and y represent the distance from Lily and Huoyi, so 0 ≤ x ≤ 30, 0 ≤ Y ≤ 40. Then all possible data of all possible distances constitute a sequence point pair ( x , y ), here x , y Both of their respective restrictions, all such ordered numbers have a set of geometric areas corresponding to the basic event group, and the points in each geometric area represent a specific location of Lily and Hoy. They can happen through the events conversation only when they do not exceed 25 kilometers, so that the point constituting the event is satisfied ( x ^ 2 + Y ^ 2) ≤25 ^ 2 The number of components, this inequality is equivalent to x ^ 2 + y ^ 2 ≤ 625.

Figure 3 represents the basic event group, the shadow part represents the event, the area of ​​the square area is 1 200 square kilometers, and the area of ​​the event is (1/4) π (25) ^ 2 = 625π / 4.

The P = (625π / 4) / 1 200 = 625π / 4 800 = 0.41.

A: At 3:00 pm, the probability they can talk through the walkie-talkie is 0.41