Probability, also called "probability", is a reflection ofRandom eventThe probability of occurrence.Random events refer to events that may or may not occur under the same conditions.For example, randomly select one product from a batch of products with genuine and defective products. "The selected product is genuine" is a random event.Suppose a random phenomenon has been tested and observed n times, in which event A occurs m times, that is, its frequency is m/n.After a lot of repeated tests, m/n is often closer to a certain constant (see Bernoulli's law of large numbers for the proof of this assertion).This constant is the probability of occurrence of event A, which is usually expressed by P (A).
The first person to systematically calculate the probability issixteenth centuryOfCaldano。It is recorded in his book Liber de Ludo Aleae.The content about probability in the book is translated from Latin by Gould.
Many of Caldano's mathematical works givegamblerRecommendations.These suggestions are written in short passages.However, the system research probability was first proposed inPascalandFermatIn a series of letters.These communications were originally proposed by Pascal, who wanted to ask Fermat for some questions about Chevvalier de Mere.Chevvalier de Mere is a famous writer,Louis XIVAn important figure of the court is also a fanatical gambler.There are mainly two problems: the dice problem and the bonus allocation problem.
Probability is a numerical value that measures the possibility of an accidental event.If the test is repeated many times (represented by X), the accidental event (represented by A) occurs several times (represented by Y).The numerical value (represented by P) is formed by taking X as the denominator and Y as the numerator.In many tests, when P is relatively stable at a certain value, P is called the probability of occurrence of A.If the probability of an accidental event is determined by long-term observation or a large number of repeated tests, it is statistical probability or empirical probability.
The discipline that studies the internal laws governing accidental events is called probability theory.Belonging to a branch of mathematics.Probability theory reveals the manifestation of internal laws contained in accidental phenomena.Therefore, probability plays an important role in people's understanding of natural and social phenomena.For example, social products should be deducted before being allocated to individual consumption. How much should be deducted and how much should be accumulated in national income should be usedprobability theoryTo determine.
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The word "probability" comes from the Latin word "probability", which can also be interpreted as "probability"Probity means "integrity and honesty". In Europe, probity is used to express the authority of witness testimony in court cases, and is usually related to the reputation of witnesses.In a word, the meaning of probability is different from that of probability in modern sense.
Classical definition
If one test meets two conditions:
(1) The test has only limited basic results;
(2) The possibility of each basic result of the test is the same.
Such experiments are classical experiments.
For event A in classical experiment, its probability is defined as: P (A)=
Where n represents the total number of all possible basic results in the test.M represents event AcontainThe number of basic results of the test.This method of defining probability is called the classical definition of probability.[1]
Frequency definition
With the increase of the complexity of the problems people encounter, 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 tests, with the increase of the number of tests, the frequency of an event always swings around a fixed number, showing a certain stability.R.vonMizezThis fixed number is defined as the probability of the event, which is the frequency definition of probability.Theoretically, the frequency definition of probability is not rigorous enough.
Statistical definition
Under certain conditions, repeat n tests, nAIs the number of times that event A occurs in n tests. If the frequency increases with nA/When n gradually stabilizes near a certain value p, the value p is called the probability of event A under this condition and recorded as P (A)=p.This definition is called the statistical definition of probability.
In history, the first statement that "when the number of tests n gradually increases, the frequency nA is stable on its probability p" is strictly meaningful and mathematically provedJacob Bernoulli （Jacob Bernoulli）[2]。
From the statistical definition of probability, we can see that the value p is one that depicts the probability of occurrence of event A under this conditionQuantitative indicators。
Due to frequency
It is always between 0 and 1. From the statistical definition of probability, for any event A, there is 0 ≤ P (A) ≤ 1, P (Ω)=1, P (Φ)=0.Wherein, Ω and Φ respectively representInevitable event(events that must occur under certain conditions) andImpossible event(An event that must not occur under certain conditions).
axiomatic definition
KolmogorovThe axiomatic definition of probability was given in 1933, as follows:
Let E be a random test and S be itssample space 。Each event A of E is assigned to a real number, recorded as P (A), and called the probability of event A.Here P (A) is a set function, and P (A) must meet the following conditions:
(3) Countability and additivity: set Aone，Atwo... is an incompatible event, that is, for i ≠ j, Ai∩Aj=φ. (i, j=1, 2...), then P (Aone∪Atwo∪……)=P(Aone)+P(Atwo)+……
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Probability has the following 7 different properties:
Nature 1:
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Nature 2:（Finite additivity）When n events Aone，…，AnWhen two are incompatible:
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Property 3: For any event A:
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Nature 4: When events A and B satisfy that A is included in B:
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Property 5: For any event A,
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Property 6: For any two events A and B,
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Property 7: (addition formula) For any two events A and B,
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event
In a specific randomized trial, each possible result is called aBasic EventsThe collection of all basic events is called the basic space.Random events (events for short) are composed of some basic events. For example, in a random experiment in which dice are rolled twice in a row, Z and Y are used to represent the number of points that appear for the first time and the second time, respectively. Z and Y can take the values of 1, 2, 3, 4, 5, and 6. Each point (Z, Y) represents a basic event, so the basic space contains 36 elements."The sum of points is 2" is an event, which is composed 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 is composed of three basic events (1, 3), (2, 2), (3, 1) and can be represented by the set {(1, 3), (3, 1), (2, 2)}.If "the sum of points is 1" is also regarded as an event, it is an event that does not contain any basic events, calledImpossible event。P (impossible event)=0.This event is unlikely to occur during the test.If "the sum of points is less than 40" is regarded as an event, it contains all basic events, and this event must occur in the test, which is calledInevitable event。P (inevitable event)=1.In real life, various events and their relationships, as well as various elements in the basic spacesubsetAnd their interrelationships[3]。
Events that may or may not occur under certain conditions are calledRandom event。
Usually an event in an experiment consists of basic events.If there are n possible results in an experiment, that is, the experiment consists of n basic events, and all the results are equally likely to occur, then such events are calledAnd other possible events。
The object of classical probability type discussion is limited toRandom testAll possible results are finite and so on, that is, the basic space is composed of finite elements orBasic EventsThe number of components is recorded as n. 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)=
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, which is P-S. Laplace's classical probability definition, or the classical definition of probability.Historically, classical probability was caused by studying problems in gambling games such as dice.The classical probability can be calculated byExhaustion methodList all basic events, and then count the number of basic events contained in an event to divide, that is, the calculation process can be simplified with the help of combination calculation.
Geometric probability
If there are infinitely many basic events in the random test, and eachBasic EventsThe occurrence is so possible that the classical probability can not be used, so the geometric probability is generated.The basic idea of geometric probability is to correspond events to geometric regions, and use the measurement of geometric regions to calculate the probability of events,Bufeng needleQuestion is a typical example of applying geometric probability[3]。
Suppose that an event A (also a region in S), S contains A, and its measurement size is μ (A). If P (A) represents the probability of event A's occurrence, considering the "uniform distribution", the probability of event A's occurrence is taken as: P (A)=μ (A)/μ (S), so the calculated probability is called geometric probability.If Φ is an impossible event, that is, Φ is an empty area in Ω, its measurement size is 0, so its probability P (Φ)=0.
stayprobability theoryAt the early stage of development, people noticed that it was not enough for the classical probability model to consider only the limited number of test results, but also the infinite number of test results.Therefore, infinite test results can be represented by a region S of European space, whose test results have the property of so-called "uniform distribution". The precise definition of "uniform distribution" is similar to the concept of "equal possibility" in the classical probability model.Assume that area S and any possible small area A in it can be measured, and the size of the measurement is represented by μ (S) and μ (A) respectively.asOne-dimensional spaceLength of,2D spaceArea of,three-dimensional spaceVolume, etc.It is assumed that this metric has the same properties as length, such as nonnegativity and additivity.
Distinguishing frequency
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"Probability" is introduced into the quantification of the probability of event occurrence.Independent repeated testTotal times n, frequency of event A μ, frequency of event A Fn(A) =μ/n, frequency F of An(A) Is there a stable value?If there is, the stable value p of frequency μ/n is the probability of occurrence of event A, recorded as P (A)=p (statistical definition of probability).
P (A) is objective, while Fn(A) It depends on experience.In statistics, sometimes F is used when n is largen(A) Value is an approximation of probability.
