Random events and their probability knowledge points I
Definition of random events:
In a random experiment, events that may or may not appear, but have certain regularity in a large number of repeated experiments, are called random events, which are usually represented by capital letters A, B, C, etc.
Definition of inevitable events:
The inevitable events are called inevitable events;
Impossible events:
An event that will definitely not happen is called an impossible event;
Definition of probability:
The frequency of event a when a large number of repeated tests are performed.
Always close to a constant and swing around it. At this point, this constant is called the probability of event A, and is recorded as P(A).
Significance of m and n: Event A occurred m times in n trials.
Because of 0? m？ N, so, 0? P(A)？ 1, the probability of inevitable events is 1, and the probability of impossible events is 0.
Definition of random event probability:
For a given random event A, with the increase of test times, the frequency of event A increases.
It is always close to a constant in the interval [0, 1], so we call this constant the probability of event A and write it as P(A).
Stability of frequency:
That is, when a large number of experiments are repeated, the frequency of any result (event) is random, but? Stable? Near a certain constant, the more experiments, the smaller the possibility that the frequency deviates from this constant, and this constant becomes the probability of an event;
? Frequency? And then what? Probability? The difference between these two concepts is that:
Frequency is random, which reflects the frequency and possibility of a random event. Probability is an objective constant, which reflects the nature of random events.
Random events and their probability knowledge points II
1, the concept of random events
Under certain conditions, a certain result is called an event.
(1) Random event: an event that may or may not occur under certain conditions;
(2) Inevitable events: events that must occur under certain conditions;
(3) Impossible event: an event that cannot happen under certain conditions.
2. Probability of random events
Probability of Event A: The frequency of Event A when the same experiment is repeated a lot.
Always close to a constant and swing around it. At this point, this constant is called the probability of event A, and is recorded as P(A).
By definition, 0? P(A)？ 1, obviously the probability of inevitable events is 1, and the probability of impossible events is 0.
3, the relationship between events
(1) mutually exclusive events: Two events that cannot happen at the same time are called mutually exclusive events;
(2) Opposing events: two events that cannot happen at the same time, but one of them must happen, are called mutually exclusive events;
4. Operation between events
(1) joint event (and event)
If the occurrence of an event is event A or event B, then this event is called the merger of event A and event B. ..
Note: When A and B are mutually exclusive, the probability of event A+B satisfies the addition formula:
P(A+B)=P(A)+P(B)(A and B are mutually exclusive); And P(A+)
)=P(A)+P(
)= 1。
(2) Cross events (product events)
If an event occurs at the same time as event A and event B, it is called the intersection of event A and event B. ..
5. Classical probability
(1) Two characteristics of classical probability: 1) There are only a limited number of possible basic events in the experiment; 2) The possibility of each basic event is equal;
(2) Probability calculation formula of classical probability: P(A)= 1
;
An experiment with every possible result is called a basic event. Usually, event A in this experiment consists of several basic events. If an experiment has n possible results, that is, the experiment consists of n basic events, and the probability of all the results is equal, then the probability of each basic event is
. If event A contains m results, the probability of event A is P(A)= 1
.
6. The concept of random number
A random number is a number generated randomly within a certain range, and the chances of getting any number within this range are equal.
7. Random number generation method
(1) You can get random numbers from 0 to 1 by using the function calculator;
(2) In Scilab language, different functions can be used to generate random numbers between 0~ 1 or a ~ b.
8, the concept of geometric probability
If the probability of each event is only proportional to the length (area or volume) of the event area, such a probability model is called geometric probability model;
9. Probability formula of geometric probability:
P(A)=
.
10, several common geometric probabilities
(1) Let the line segment L be a part of the line segment L, and any point falls on the line segment L. If the number of points falling on the line segment L is proportional to the length of the line segment L, it has nothing to do with the relative position of the line segment L, then the probability that the point falls on the line segment L is:
P = length /l length
(2) Let the plane area G be a part of the plane area G, and throw any point on the area G. If the number of points falling on the area G is proportional to the area of the area G, it has nothing to do with the relative position of the area G on the area G, then the probability that the point falls on the area G is: area of p =/area of g..
(3) Let the space region V be a part of the space region V, and throw any point into the region V. If the number of points falling on the region V is in direct proportion to the volume of the region V, it has nothing to do with the relative position of the region V, then the probability of a point falling on the region V is: p = volume/volume of V.