probability theory

[gài lǜ lùn]

Branch of mathematics

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Probability theory is research Random phenomenon Quantitative mathematics Branch. Random phenomenon is relative to Decisive phenomenon In other words, the phenomenon that certain results must occur under certain conditions is called decisive phenomenon. For example, in Standard atmospheric pressure Next, pure water When heated to 100 ℃, water will inevitably boil. Random phenomenon refers to that under the condition that the basic conditions remain unchanged, before each test or observation, it is not sure which results will appear chance 。 For example, when a coin is tossed, the front or the back may appear. The realization and observation of random phenomena are called Random test 。 Each possible result of a random test is called a Basic Events , one or a group of basic events are collectively called random events, or simply events. Typical random tests are Roll dice , coin toss, poker draw, roulette, etc.

The probability of an event is a measure of the likelihood that the event will occur. Although the occurrence of an event in a random experiment is accidental, those random experiments that can be repeated in large numbers under the same conditions often show obvious quantitative laws.

Chinese name: probability theory
Foreign name: Probability Theory
Presenter: Giroramo Caldano
Definition: Likelihood of things happening

Common Experiments: Roll dice, coins, etc
application area: Mathematics statistics , financial mathematics, etc
Time of occurrence: sixteenth century
Type: Branch of mathematics

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origin

Probability theory is a branch of mathematics that studies the quantitative laws of random phenomena, and it is a science that studies the possibility of things happening. But the origin of the original probability theory gambling The problem is related. In the 16th century, Italian scholars Giroramo Caldano (Girolamo Cardano) began to study some simple problems in gambling such as dice.

Probability vs Statistics Some of concept And simple methods, which were mainly used for gambling and Demographic model 。 With the development of human beings social practice People need to understand the inevitable laws implied in various uncertain phenomena and use Mathematical method By studying the possibility of various results, probability theory came into being and gradually developed into a rigorous discipline. The methods of probability and statistics are increasingly used in various fields natural science 、 economics 、 Medical Science 、 Financial insurance even to the extent that humanities Medium.^[1]

development

With the development of science in the 18th and 19th centuries, people noticed that in some biological, physical and social phenomenon There is some similarity between chance games and chance games, so the probability theory originated from chance games is applied to these fields; At the same time, this has greatly promoted the development of probability theory itself. The founder of probability theory as a branch of mathematics is the Swiss mathematician Bernoulli He established the first limit theorem , i.e Bernoulli's law of large numbers The frequency of the event is stable in its probability. subsequently Desmoff and Laplace The second fundamental limit theorem is derived（ central limit theorem ）In its original form.

Based on the systematic summary of previous work, Laplace wrote the Analytical Probability Theory, which clearly gave the classical definition of probability, and introduced more powerful analysis tools into probability theory, pushing probability theory to a new development stage.

At the end of the 19th century, Russian mathematicians Chebyshev 、 Markov 、 Liapunov Et al. established the Law of large numbers and central limit theorem The general form of random variable Approximate obedience Normal distribution 。 At the beginning of the 20th century, stimulated by physics, people began to study random processes. In this regard Kolmogorov 、 Wiener , Markov, Xinqin Levy and Feiler And others have made outstanding contributions.^[1]

definition

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Traditional probability

The traditional probability is also called Laplacian probability, because it is defined by French mathematicians Laplace Proposed. If the unit events included in a random test are limited, and the probability of each unit event is equal, the random test is called Laplace test. In the Laplace test, the probability P (A) of event A in event space S is:

For example, in a random experiment in which one coin and one die are rolled at the same time, if event A is to obtain the national emblem surface and the number of points is greater than 4, then the probability of event A should have the following calculation method: S={(national emblem, 1 point), (number, 1 point), (national emblem, 2 points), (number, 2 points), (national emblem, 3 points), (number, 3 points), (national emblem, 4 points), (number, 4 points), (national emblem, 5 points) , (number, 5 points), (national emblem, 6 points), (number, 6 points)}, A={(national emblem, 5 points), (national emblem, 6 points)}, according to the definition of Laplace, The probability of A is 2/12=1/6. It is noted that there are several doubts in the Laplacian test. Whether there is such a test in reality, the probability of unit event has the exact same probability value, because people do not know whether the coin and dice are "perfect", that is, whether the dice are made evenly, and whether their center of gravity is located in the center, And whether the wheel tends to a certain number, etc.

Nevertheless, the traditional probability is widely used in practice to determine the Probability value The theoretical basis is: If there is not enough evidence to prove that the probability of one event is greater than the probability of another event, then the probability values of these two events can be considered equal 。 If you look closely at this definition, you will find that Laplace explained probability with probability. The word "equal possibility" (the original text is "equal possibility") is used in the definition, which actually means "the same probability". This definition does not say what probability is and how to use numbers to determine probability. In real life, there are also a series of problems that cannot be explained by traditional probability definitions, for example, Life Insurance Company It is impossible to determine the probability that a 50 year old person will die in the next year.^[1]

axiomatic definition

How to define probability and how to establish probability theory on a strict logical basis are the difficulties in the development of probability theory. The exploration of this problem has lasted for three centuries. Completed in the early 20th century Lebesgue measure With integral theory and the subsequent development of abstract measure and integral theory Axiom of probability The establishment of the system has laid the foundation.

In this context, Soviet mathematicians Kolmogorov In 1933, he first gave the definition of the measure theory of probability and a set of strict axioms in his book "Basis of Probability Theory". His axiomatic method It has become the basis of modern probability theory, making probability theory a rigorous mathematical branch and playing a positive role in the rapid development of probability theory.

The following is the axiomatic definition:

Set random experiment E sample space Is Ω. If, according to some method, a real number P (A) is assigned to each event A of E, and the following axioms are met:

（1） Non negative: P(A)≥0；

（2） Normative: P(Ω)=1；

（3） Countability (full) additivity: For an infinite number of events A1 that are incompatible with each other, A2，……，An，……， yes

, the real number P (A) is the probability of event A.^[1]

Statistical definition

Let the number of random event A in n repeated tests be n A If the test number n is large, the frequency n A /N oscillates stably near a certain value p, and the amplitude of its oscillation becomes smaller and smaller as the number of tests n increases, then the number p is called the probability of random event A, recorded as P (A)=p.^[1]

event

Events include unit events, event space, random events, etc.

The unique and mutually independent results that may occur in a random test are called Unit Event , denoted by e. The set of all unit events that may occur in random tests is called event space, which is represented by S. For example, in a random experiment of rolling dice, if the number of points obtained is used to represent unit events, a total of 6 unit events may occur, and the event space can be expressed as S={1, 2, 3, 4, 5, 6}. The above event space is composed of countable finite unit events. In fact, there is also an event space composed of countable infinite and uncountable unit events, such as once until national emblem In the face up random coin toss experiment, the event space is composed of countable infinite unit events, which is expressed as: S={countries, countable countries, countable countries, countable countries, countable countries, countable countries,...}. Note that in this example, "countable countries" are unit events. Throw two chopsticks on the table at random, and the Angle of intersection Assuming α, the composition of the event space of this random test can be expressed as

。

Random events are in the event space S subset , which is composed of unit elements in the event space S, and is represented by capital letters A, B, C. For example, in the random experiment of rolling two dice, if the random event A="the sum of points obtained is greater than 10", then A can be composed of the following three unit events: A={(5, 6), (6, 5), (6, 6)}. If all possible unit events in the event space occur in the random test, this event is called Inevitable event , denoted by

； Correspondingly, if the event space does not contain any unit event, it is called impossible event, which is expressed as

。

Calculation of events

Because the event is partly based on aggregate Therefore, the set computing method can be directly applied to the calculation of events, that is, in the calculation process, events can be treated as sets.

A's complement Occurrence of events not belonging to A	Union A → B Either A or B or A and B occur simultaneously	Intersection A ∨ B Events A and B occur simultaneously
Difference A B Event A not belonging to B occurs)	Empty set A ∨ B=∅ A, B events occur at different times	Subset B ⊆ A If A occurs, B must also occur

Illustration

In the roulette game, it is assumed that A represents the event "the ball falls in the red area", B represents the event "the ball falls in the black area", and C represents the event "the ball falls in the green area". Because event A and B have no common unit event, it can be expressed as probability P (AB)=0.

Note that events A and B are not complementary, because there is a unit event C in the whole event space S, which is neither red nor black, but green. Therefore, the complements of A and B should be represented as follows:

as well as

。

conditional probability

probability theory

The probability that an event A will occur after an event B is determined to occur is called the probability that B gives A conditional probability ； Its value is

(When

When). If B gives A the same conditional probability as A, then A and B are called Independent event 。 And this relationship between A and B is symmetric, which can be seen from the same valence statement: "When A and B are independent events, P (A ∨ B)=P (A) P (B)".^[1]

Relevant cases

It is generally believed that a bad feeling, or insecurity (commonly known as "dot back"), about the probability of what will happen is real. The following examples can vividly illustrate people's sometimes wrong understanding of probability:

probability theory

(1) Liuhecai: In Liuhe color (6 out of 49), there are 13983816 possibilities. It is generally believed that if you buy a different number every week, you can win the first prize at the latest after 13983816/52 (week)=268919. In fact, this understanding is wrong, because the probability of winning the prize is equal every time, and the probability of winning the prize will not increase with the passage of time.

(2) Birthday paradox: There are 23 people (2 × 11 players and 1 referee) on a football field. It is inconceivable that at least two of the 23 people are born on the same day. The probability is greater than 50%.

(3) Roulette: In the game, players generally believe that after red appears for many times in a row, the probability of black appears will be increasing. This judgment is also wrong, that is, the probability of black is equal every time, because the ball itself has no "memory", it will not realize what happened before, and its probability is always 18/37.

(4) Three door problem: In the game program of guessing the car hidden behind the door held by the TV station, there are three closed doors opposite the competitors, of which only one door is behind a car, and the other two doors are behind goats. The rule of the game is that the contestant first selects a door that he thinks has a car behind it, but this door remains closed. Then the host opens the door with a goat behind the other two doors that have not been selected by the contestant. At this time, the host asks the contestant whether to change his mind and choose another door, so as to make the chance of winning a car more likely?

The correct result is that if the contestant changes his original intention, his chance of winning the prize will become 2/3. Because the moment the Goat Gate was opened, the original selection result had changed from 1/3 chance to 1/2 chance. If the original intention was changed, it would be 1/2 chance to win the prize.

There are three possible situations, all of which have equal possibilities (1/3): the contestant chooses goat No. 1, and the host chooses goat No. 2. The conversion will win the car. The contestant picks goat number two, and the host picks goat number one. The conversion will win the car. The contestant picks the car, and the host picks either of the two goats. The conversion will fail. In the first two cases, the contestant can win the car by changing the choice. The third situation is the only one in which the contestant wins by keeping the original choice. Because two of the three cases are won through conversion, the probability of winning through conversion is 2/3.^[1]

calculation

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It should be mentioned that the following nine theorems for calculating probability have nothing to do with the calculation of the events mentioned above. All theorems on probability are derived from the three axioms of probability, and are applicable to all probability theories, including Laplacian probability and statistical probability.

Theorem 1

also called Complementarity principle.

The probability of complementary events with A is always 1-P (A).

The probability that the red color does not appear in the first rotation is 19/37, according to product rule The probability that there will be no red color for the second time is

Therefore, the complementary probability here refers to the probability that at least one of the two consecutive rotations is red, which is

Theorem 2

The probability of an unlikely event is zero.

Prove that Q and S are complementary events. According to axiom 2, there is P (S)=1, and then according to theorem 1 above, P (Q)=0 is obtained

Theorem 3

If A1... An events cannot occur at the same time (i.e mutex Event), and several events A1, A2,... An ∈ S are empty set Relationship, then all these events aggregate The probability of is equal to the sum of the probabilities of individual events.

For example, in a dice roll, you get 5 or 6 points probability Yes:

Theorem 4

If event A, B is Difference set Relationship, there is

Theorem 5

Arbitrary event addition rule:

For any two events A and B in the event space S, there is the following theorem: probability

Theorem 6

Multiplication rule:

The probability of simultaneous occurrence of events A and B is:

, provided that events A and B are related.

Theorem 7

Multiplication rule of unrelated events:

The probability of two unrelated events A and B occurring at the same time is: notice that this theorem is actually a special case of Theorem 6 (multiplication rule). If events A and B are not related, then there are P (A | B)=P (A), and P (B | A)=P (B). Observe two consecutive rotations in Roulette. P (A) represents the probability of the first occurrence of red, and P (B) represents the probability of the second occurrence of red. It can be seen that A is not related to B. Using the formula mentioned above, the probability of two consecutive occurrences of red is:

Ignoring this theorem is the root cause of many players' failures. It is generally believed that after several consecutive occurrences of red, the probability of black will increase. In fact, the probability of each occurrence of the two colors is equal. There is no connection between the previous red and the subsequent black, because the ball itself has no "memory" and it does not "know" What happened before.

Therefore, the probability of 10 consecutive occurrences of red is

。^[1]

Statistical probability

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Statistical probability Based on frequency theory John Wayne (John Venn, 1834-1923) and Austria Mathematician Richard Von Mises (1883-1953) proposed that they believed that the only way to obtain the probability value of an event was to conduct 100, 1000 or even 10000 independent random tests on the event, and record each test Absolute frequency value and Relative frequency value hn (A) With the increase of test times n, the following facts will appear, that is, the relative frequency value will tend to be stable, and it will float up and down at a specific value, that is, there is a Limit value P (A), the relative frequency value tends to this limit value.

This limit value is called statistical probability and is expressed as:

。

For example, if you want to know the probability value of getting 6 points in a random experiment of rolling dice, you can carry out 3000 independent throwing tests, record the number of 6 points after each test, and then calculate the relative Frequency value The statistical probability value tending to a certain number can be obtained.

Throws	Obtain the absolute frequency of 6 points	Obtain the relative frequency of 6 points
one	one	one
two	one	zero point five zero zero zero zero
three	one	zero point three three three three three
four	one	zero point two five zero zero zero
five	two	zero point four zero zero zero zero
ten	two	zero point two zero zero zero zero
twenty	five	zero point two five zero zero zero
one hundred	twelve	zero point one two zero zero zero
two hundred	thirty-nine	zero point one nine five zero zero
three hundred	forty-six	zero point one five three three three
four hundred	seventy-two	zero point one eight zero zero zero
five hundred	seventy-six	zero point one five two zero zero
six hundred	one hundred and two	zero point one seven zero zero zero
seven hundred	one hundred and twenty	zero point one seven one four three
one thousand	one hundred and seventy	zero point one seven zero zero zero
two thousand	three hundred and forty-three	zero point one seven one five zero
three thousand	five hundred and six	zero point one six eight six seven

The above mentioned is related to Relative frequency The experience value of is also called Law of large numbers , Yes Frequency theorist Define the basis of probability theory. However, no one can throw the dice infinitely, so the law of large numbers cannot be effectively proved in practice. Many arguments from mathematical theories have not been successful so far. Nevertheless, statistical probability is of great significance in today's practice. It is mathematical statistics The foundation of.^[1]

Complete probability

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N events H1, H2,... Hn are independent of each other and together form the whole event space S, that is

, and

。 Then the probability of A can be expressed as

For example, a random test tool consists of a die and three drawers in a cabinet. There are 14 drawers in drawer 1 White ball And 6 black balls. There are 2 white balls and 8 black balls in drawer 2, and 3 white balls and 7 black balls in drawer 3. The test rule is to roll dice first. If less than 4 points are obtained, drawer 1 is selected. If 4 or 5 points are obtained, drawer 2 is selected. In other cases, drawer 3 is selected. Then draw a ball randomly from the selected drawer, and the probability that the ball drawn last is a white ball is:

P (white)=P (white | extract 1) · P (extract 1)+P (white | extract 2) · P (extract 2)+P (white | extract 3) · P (extract 3)

=(14/20)·(3/6)+(2/10)·(2/6)+(3/10)·(1/6)

=28/60=0.4667

As can be seen from the example, complete probability is particularly suitable for analyzing Multilayer structure Of the random test.^[1]

Bayesian theorem

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Bayesian theorem By British mathematicians Bayes (Thomas Bayes, 1702-1761) development, used to describe two conditional probability The relationship between, such as P (A | B) and P (B | A). According to the multiplication rule of Theorem 6,

, you can immediately derive Bayesian theorem:

The above formula can also be transformed into, for example:

。

A villa has been stolen twice in the past 20 years. The owner of the villa has a dog that barks three times every night on average. The probability of the dog barking when thieves invade is estimated to be 0.9. The question is: what is the probability of the invasion when the dog barks?

People assume that event A is a dog barking at night and event B is a burglar invasion, then

，

According to the formula, it is easy to get the results:

。

Another example is that there are two containers, A and B, respectively. In container A, there are seven red balls and three white balls, and in container B, there are one red ball and nine white balls. Now we know that one ball is randomly drawn from these two containers, and it is a red ball. What is the probability that the red ball comes from container A?

Assuming that the red ball has been pulled out as event B and the ball has been pulled out from container A as event A, then there are:

，

According to the formula, there are:

。

Although probability theory first appeared in the 17th century, its axiom system was established and developed rapidly in the 1920s and 1930s. In the past half century, probability theory has shown its applicability and practicability in more and more emerging fields, such as: Physics , chemistry, biology, medicine, psychology, sociology, politics, education, economics and almost all engineering And other fields.

It is particularly worth mentioning that probability theory is the basis of mathematical statistics today, and its results are often used as the analysis data of questionnaire surveys, and also for forecasting economic prospects.^[1]

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