Reference
Money is falling from the sky! It's all red fur grandpa! The students ran to the playground with the pots to receive the money. Of course, whoever had the bigger pot was more likely to receive the money.
The location where the money falls is a random location on the playground (every location is possible), and the probability of receiving money is only related to the size of the pot (related to geometric measures), not the shape of the pot. The probability of each student receiving money It is Area (pot)/Area (playground). This is a geometric outline.
Definitions and Formulas
Geometric probability model is a probability model, under this model, the sample space of E is a measurable geometric area (playground), and the occurrence of each sample point has equal probability (each location has the same probability of receiving money). ). The equal probability here is the same as mentioned in the previous chapter. Objectively, when you cannot determine which event is more likely to happen, you have to think it is an equal probability.
The main difference between classical and geometrical models is that the results of experiments are infinite. The dice has only 6 faces, so the number of points on the dice is limited; the landing point of the dice can be any position on the floor of the room, so there are infinite landing points.
Regarding the definition of geometric profiles, there is also a textbook statement, which is probably: the probability that a sample point falls into a measurable region A in the sample space Ω is proportional to the geometric measure of A, and is proportional to the position of A, Shape doesn't matter. This leads to the formula:
The concepts in textbooks are usually very rigorous, but they are not easy to understand. Whether you can remember them or not, don’t worry. Knowing the geometric outlines is just like dropping money in the sky.
Typical problems
Many problems can be converted into geometric measures. For example, the time it takes for a person to arrive at a unit may be any time between 8:00 and 9:00 (converting time into a one-dimensional line segment); throw a pebble into a square, pebble Falls on any point in the grid (converts the grid to 2D coordinates).
Example 1
Two people, A and B, set off from A and B respectively between 9:00 and 10:00 in the morning. They both travel at the same speed and can walk the entire distance in 10 minutes. What is the probability of the two meeting each other?
This is a typical geometric probability. There are infinitely many times between 9:00 and 10:00, and the time points of the two departures are equally likely. Convert the departure time of A to a line segment in minutes:
Add the departure time of B to form two-dimensional coordinates:
The square is the sample space Ω, and each point in it represents the departure time of AB, that is, a sample point, and there are innumerable sample points.
Both A and B complete the journey within 10 minutes. Assuming that A starts first and wants to meet, the departure time of B must be within 10 minutes after the departure of A. If the departure time is T, then T B – T A < 10. Since B may start before A, add the absolute value |TB – TA| < 10. Converting this to a geometric measure, the eligible points all fall within the green area:
Example 2
Example 1 There is another vest. A pair of young people agree to meet on a blind date somewhere from 9:00 to 10:00. If one of them waits for 10 minutes and leaves, what are the chances of them meeting successfully?
Comparative Example 1:
The solution process and results are exactly the same. Such problems also include ship encounters, car encounters, and so on.
Example 3
The distance between the bright street lights of AB is 30 meters. The relevant department wants to add two identical street lights C and D between AB. The probability that the distance between AC and BD is not less than 10 meters?
Street lights can be placed anywhere between AB, so the metric of the sample space Ω is 30.
As shown in the figure above, only when the CD falls within the range of 10~20 at the same time can it meet the requirements. Both C and D fall into the interval with probability 1/3:
Splicing the two pictures together is equivalent to accumulating all points between AB once, and the final result is 1/3:
Author: I am 8 bit
Source: http://www.cnblogs.com/bigmonkey
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