Probability, also known as probability, chance rate or probability and possibility, is the basic concept of mathematical probability theory, a real number between 0 and 1, and a measure of the possibility of random events. The number representing the probability of an event is called the probability of the event. It is a measure of the possibility of random events and one of the most basic concepts of probability theory. People often say how sure someone is to pass the exam and how likely something is to happen. These are examples of probability. However, if the probability of an event is 1/n, it does not mean that the event must occur once in n events, but that the frequency of this event is close to the value of1/n.

Defining the frequency definition of probability With the increase of the complexity of people's problems, 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. Andre Andrey Kolmogorov gave an axiomatic definition of probability in 1933. The strict definition of probability assumes that E is a random test and ω is its sample space. For each event A of E, assign a real number, which is called the probability of event A. Here, p () is a set function, and p () must meet the following conditions: (1) is nonnegative: for each event A, there is P(A) ≥ 0; (2) Normality: for the inevitable event S, there is p (s) =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. In fact, in life, production and economic activities, people are often concerned about the possibility of random events. For example: (1) toss a uniform coin, and the probability of the front and side is 1/2 respectively. (2) What are the chances of winning the lottery? The above-mentioned positive opportunities, as well as the probability or hit rate of winning the lottery, are used to measure the probability of random events. The probability of a random event A is called the probability of this event and is expressed by P(A). The probability is a number between 0 and 1. The greater the probability, the greater the possibility of an event; The smaller the probability, the less likely the event will happen. In particular, the probability of impossible events is 0, and the probability of inevitable events is 1, that is, p (φ) = 0 and p (ω) = 1. If an experiment meets two conditions: (1) There are only a limited number of basic results in the experiment; (2) The possibility of each basic result in the experiment is the same. Such an experiment becomes a classic experiment. For event A in the classical experiment, its probability is defined as: P(A)=m/n, 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. The statistical definition of probability is that, under certain conditions, the experiment is repeated n times, where nA is the number of times that event A occurs 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, and this definition becomes the statistical definition of probability. In history, Jacob Bernoulli (A.D. 1654 ~ 1705), the most important scholar in the history of early probability theory, was the first to give a strict meaning and mathematical proof to the assertion that "when the number of experiments N gradually increases, 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. Since the frequency nA/n is always between 0 and 1, it can be seen from the statistical definition of probability that for any event A, there are 0≤P(A)≤ 1, p (ω) = 1, and p (φ) = 0. ω and φ represent inevitable events (events that must occur under certain conditions) and impossible events (events that must not occur under certain conditions) respectively. The first person who edited this historical system to calculate probability was cardano in16th century. This is recorded in his book. The content of probability in the book was translated from Latin by Gould. Cardano's mathematical works have a lot of advice for gamblers. These suggestions are all written in the essay. For example: "Who, when should gamble?" Why did Aristotle condemn gambling? Are those who teach others to gamble good at it? "and so on. However, it was in a series of letters between Pascal and Fermat that a systematic study of probability was first put forward. These communications were originally put forward by Pascal, who wanted to ask Fermat some questions about Chevalier de Meyer. Chevalier de Meyer is a famous writer, an outstanding figure in the court of Louis XIV, and an avid gambler. There are two main problems: the problem of rolling dice and the distribution of competition bonuses. Edit two categories of classical probability in this paragraph. The object of classical probability discussion is limited to the case that all possible results of random experiments are finite and equal, that is, the basic space consists of finite elements or basic events, the number of which is recorded as n, and the possibility of each basic event is the same. If event A contains m basic events, the probability of event A is defined as p(A)=m/n, that is, the probability of event A is equal to the number of basic events contained in event A divided by the total number of basic events in the basic space. This is the classical definition of probability by P.-S. Laplace, or the classical definition of probability. Historically, classical probability was produced by studying the problems in gambling games such as dice. To calculate the classical probability, all the basic events can be listed by exhaustive method, and then the number of basic events contained in an event can be counted and divided, that is, the calculation process can be simplified by combined calculation. Geometric Probability Related Geometric Probability If there are infinitely many basic events in the random test, and the probability of each basic event is equal, then the classical probability can't be used, so the geometric probability is generated. The basic idea of geometric probability is to correspond an event to a geometric region and calculate the probability of an event by using the measurement of the geometric region. Buffon's needle throwing problem is a typical example of applying geometric probability. In the early stage of the development of probability theory, people noticed that it is not enough to consider only a limited number of test results in classical probability, but also an infinite number of test results. For this reason, an infinite number of test results can be represented by a certain region S in Euclidean space, and the test results have the so-called "uniform distribution" property. The precise definition of "uniform distribution" is similar to the concept of "equal possibility" in classical probability theory. It is assumed that the area S and any small area A that may appear in it are measurable, and the measured size is expressed by μ(S) and μ(A) respectively. For example, the length of one-dimensional space, the area of two-dimensional space and the volume of three-dimensional space. Suppose this metric has various attributes, such as length, such as nonnegativity and additivity. The strict definition of geometric probability is that an event A (also a region in S) contains A, and its metric size is μ(A). If P(A) is used to represent the probability of the occurrence of event A, considering the "uniform distribution", the probability of the occurrence of event A is taken as: P(A)=μ(A)/μ(S), so the calculated probability. ◆ If φ is an impossible event, that is, φ is an empty area in ω, and its metric size is 0, then its probability p (φ) = 0. If a series of tests have the following three items, edit the independent test sequence in this paragraph: (1) Each test has only two results, one marked as "success" and the other marked as "failure", with P{ success }=p, P{ failure }= 1-p=q (2) Success in each test. Then this series of tests is called independent test sequence, also known as Bernoulli probability. Edit the inevitable and impossible events in this paragraph. In a specific random experiment, every possible result is called a basic event, and the set of all basic events is called a basic space. Random events (events for short) are composed of some basic events. For example, in a random experiment of rolling dice twice in a row, z and y are used to represent the first and second appearance points respectively. Z and y can take the values of 1, 2, 3, 4, 5, 6, and each point (z, y) represents a basic event, so the basic space contains 36 elements. " The sum of points is 2 "an event, which consists of a basic event (1, 1), and can be represented by the set {( 1, 1)}". "The sum of points is 4" is also an event, which consists of (1, 3. If "the sum of points is 1" is also regarded as an event, then it is an event that does not contain any basic events and is called an impossible event. This event can't happen in the experiment. If "the sum of points is less than 40" is regarded as an event, it contains all the basic events, and this event must occur in the experiment, so it is called an inevitable event. If A is an event, then "Event A didn't happen" is also an event, which is called the opposite event of Event A. In real life, it is necessary to study various events and their interrelationships, various subsets of elements in the basic space and their interrelationships. For example, Xiao Ming will put five balls in four drawers, and one drawer will have two balls, which is an inevitable event. For another example, Xiao Ming will put three balls in five drawers. If there are balls in every drawer, then this is an impossible event, a random event, a basic event and other possible events. Every experiment with possible results is called a basic event. Usually, an event in an experiment consists of basic events. If an experiment has n possible results, that is, the experiment consists of n basic events, and all the results are equally likely to appear, then such events are called allelic events. Two events that cannot happen at the same time are called mutually exclusive events. There must be a mutually exclusive event called an antagonistic event. That is, p (inevitable event) = 1 P (possible event) =(0- 1) (score can be used) p (impossible event) =0 to edit the properties of this paragraph. Attribute 1.p (φ) = 0. Characteristic 2 (limited additivity). When n events A65438+,. ∪ an) = p (a 1)+...+p (an)。 Real estate 3. For any event, a: p (a) = 1-p (not a). Property 4. When events A and B satisfy that A is contained in B, P (b-a). P (B-A) = P (B)-P (AB)。 Attribute 7 (addition formula). For any two events a and b, P(A∪B)= P(A)+P(B)-P(A∪B). (Note: The number after A is 655. For example, there is a coin toss test done by predecessors (table below P.44), saying that the stable value P of frequency μn is the probability of event A, and P(A)=p [statistical definition of probability] P(A) is objective, while Fn(A) is empirical. In statistics, the value of Fn(A) is sometimes used as an approximation of probability when n is large. Edit the three basic attributes of this paragraph 1. [Nonnegativity]: Any event A, p (a) ≥ 0 2. [completeness]: p (ω) = 1 3. [law of addition] if event a and event b are incompatible, that is, AB=φ, then P (A+B) = p (. Repeat the experiment n times independently. For example, remember that the frequency of event A is μA and Fn(A), the frequency of event B is μB and Fn(B), and the frequency of event A+B is μA+B and Fn(A+B). It is easy to know that μA+B =μA +μB, ∴. Their stable values should also be: P(A+B)=P(A)+P(B) [addition rule] If events A and B are incompatible, that is, AB=φ, then P (a+b) = P (a)+P. Please think about it: if A and B are not incompatible, they are compatible. Further research shows that P(A+B)=P(A)+P(B)-P(AB), which is the so-called "retreat more and make up less"! Edit the fuzzy sum probability 1 in this paragraph. Is uncertainty random? Do likelihood ratio and probability represent all uncertainties? Bayesian camp: Probability is a kind of subjective prior knowledge, not a measure of frequency and objectivity. Lindley: Probability is the only effective and sufficient description of uncertainty, and all other methods are not similar enough: uncertainty is expressed by numbers between unit intervals [0, 1], and they have propositional differences in set, correlation, connection and distribution: treatment. Classical set theory represents events that are impossible in probability. And is fuzziness based on (1)? Consider whether it can logically or partially violate the "non-contradiction theorem" (one of Aristotle's three "thinking theorems", the middle identity theorem). These are all classical theorems in black and white. The emergence of ambiguity (contradiction) is the end of western logic. (2) Can conditional probability operators be derived? In classical set theory: fuzzy theory: it is a unique problem of fuzzy sets to consider the extent to which a superset is a subset of its subsets. 2. Fuzziness and probability: whether the degree of the event is fuzzy and how fuzzy it is. Randomness is the uncertainty of whether an event occurs or not. Example: There is a 20% chance of light rain tomorrow (including compound uncertainty). Parking space problem The probability that there is an apple in the refrigerator is reversed from the event that there is half an apple in the refrigerator. The fuzziness of the origin of the earth's evolution and recovery is a kind of definite uncertainty, which is the characteristic of physical phenomena. The result of expressing uncertainty with fuzziness will be shocking, and people need to re-examine the realistic model. The economic concept of editing probability in this paragraph [1] Probability refers to the possibility of producing a certain result. Introduction is a difficult concept to formalize, because its formation depends on the nature of uncertain events and people's subjective judgment. The objective measure introduced comes from the frequency of similar events in the past. In the case that it is impossible to judge according to past experience, the formation of probability depends on subjective judgment based on intuition. At this time, different people will form different judgments and make different choices.