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The word "probability" comes from the Latin word "probability itas" and can also be interpreted as "integrity". Integrity means "integrity and honesty". In Europe, it is used to express the authority of witness testimony in court cases, which is usually related to the reputation of witnesses. In short, it is different from the meaning of probability "possibility" in the modern sense.
Classical definition
If the test meets two requirements:
(1) The experiment has only a limited number of basic results;
(2) The possibility of each basic result of the test is the same.
Such an experiment is a classic experiment.
For event A in the classical experiment, its probability is defined as: P(A)=, where n represents the total number of all possible basic results in the experiment. M represents the number of basic test results contained in event A. This method of defining probability is called the classical definition of probability. ?
Frequency definition
As the problems people encounter become more and more complicated, the equal possibility gradually exposes its weakness, especially for the same event, different probabilities can be calculated from different equal possibility angles, resulting in various paradoxes. On the other hand, with the accumulation of experience, people gradually realize that when doing a large number of repeated experiments, with the increase of the number of experiments, the frequency of an event always swings around a fixed number, showing certain stability. R.von mises defines this number as the probability of an event, which is the frequency definition of probability. Theoretically, the frequency definition of probability is not rigorous enough.
Statistical definition
Under certain conditions, the experiment was repeated n times, where nA is the number of times that Event A occurred in n times. If the frequency nA/n gradually stabilizes around a certain value p with the gradual increase of n, then the value p is called the probability of the occurrence of event A under this condition, and it is recorded as P (a) = P .. This definition becomes the statistical definition of probability.
In history, Jacob Bernoulli was the first person to give a strict meaning and mathematical proof to the assertion that "when the number of experiments n increases gradually, the frequency nA is stable at its probability p"? .
From the statistical definition of probability, it can be seen that the numerical value p is a quantitative index to describe the possibility of event A under this condition.
Due to frequency
It is always between 0 and 1. According to the statistical definition of probability, for any event A, there are 0≤P(A)≤ 1, p (ω) = 1, and p (φ) = 0. Where ω and φ represent inevitable events (events that must happen under certain conditions) and impossible events (events that must not happen under certain conditions) respectively.
Axiomatic definition
Andre Andrey Kolmogorov gave an axiomatic definition of probability in 1933, as follows:
Let E be a random experiment and S be its sample space. For each event A of E, assign a real number, which is recorded as P(A), which is called the probability of event A. Here, P(A) is a set function, and P(A) must meet the following conditions:
(1) Nonnegativity: for each event A, there is p (a) ≥ 0;
(2) Normality: for the inevitable event ω, there is p (ω) =1;
(3) Countable additivity: Let A 1, A2…… ...... become mutually incompatible events, that is, for i≠j, Ai∩Aj=φ, (I, J = 1, 2 ...), and then P (A/kloc.
Nature:
Probability has the following seven different attributes:
Attribute1:p (φ) = 0;
Property 2: (limited additivity) When n events A 1, …, An are incompatible with each other: p (a1∧ ... ∪ an) = p (a1)+...+p (an);
Property 3: For any event, a: p (a) = 1-p (not a);
Property 4: When events A and B satisfy that A is included in B: P(B-A)=P(B)-P(A), p (a) ≤ p (b);
Property 5: For any event A, p (a) ≤1;
Property 6: For any two events A and B, p (b-a) = p (b)-p (ab);
Property 7: (addition formula) For any two events A and B, P(A∪B)= P(A)+P(B)-P(A∪B).