A measure of the probability of random events. 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.

■ Frequency definition of probability

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. Andre Andrey Kolmogorov gave an axiomatic definition of probability in 1933.

■ Strict definition of probability

Let E be a random experiment and S be its sample space. For each event A of E, assign a real number, which is denoted as P(A), which is called the probability of event A. Here, P () is a set function, and P () must meet the following conditions:

(1) Nonnegativity: 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.

■ Classical definition of probability

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 becomes a classic experiment.

For event A in the classic 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.

■ Statistical definition of probability

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, Joko Bernoulli (1654 ~ 1705 AD), 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 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.

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.

[Edit this paragraph] Examples in life

It is generally believed that people always have a bad premonition or insecurity about the probability that will happen, commonly known as "point back". The following examples can vividly describe people's misunderstanding of probability sometimes:

■ 1. Mark Six Lottery: In Mark Six Lottery (6 out of 49), one * * has 139838 16 possibilities (see Combinatorial Mathematics). Generally speaking, if you buy different numbers every week, you can buy them at 1398 3865438 at the latest. In fact, this understanding is wrong, because the winning probability of each prize is equal, and the winning probability will not increase with the passage of time.

■2. Birthday Paradox: There are 23 people (2× 1 1 athlete and 1 referee) on a football field. Incredibly, the probability that at least two of these 23 people have birthdays on the same day is greater than 50%.

■3. Roulette game: In the game, players generally think that the probability of black will increase after red appears several times in a row. This judgment is also wrong, that is, the probability of appearing black every time is equal, because the ball itself has no "memory" and will not realize what happened before, and its probability is always 18/37.

■4. Three questions: In the game program of guessing the car hidden behind the door held by the TV station, there are three closed doors opposite the contestants, only one of which has a car behind it, and the other two doors are goats. The rule of the game is that the contestant first chooses a door that he thinks there is a car behind, but this door remains closed, and then the host opens the other two doors that are not selected by the contestant and have goats behind them. At this time, the host asked the contestant if he wanted to change his mind and choose another door to make the car have a better chance of winning. The correct result is that if the player changes his mind and chooses another closed door at this time, his chances of winning the car will double.

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King William:

I have a little humble opinion about M4. Three questions:

Participants have a 50% chance of winning the car.

Because no matter whether the participants choose one of the three doors for the first time, the host will open a goat's back. And after the opening, players can also choose. In this way, the contestants actually only need to choose one of the two doors. The probability is 1/2. This winning probability does not need to consider the probability of three doors.

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N43e 120 revision: one of the three probability games, 2009-0 1- 12.

The same logical example:

A prison guard randomly selected one of the three criminals to be released and the other two will be executed. The guards know who will be released, but they are not allowed to reveal any information about their identity to the criminals. We call the criminals X, Y, Z, Y, Z. The criminal X privately asked the prison guard Y or Z which one would be executed, because he already knew that at least one of them would die, and the prison guard could not disclose any information about his status. The guards told X and Y that they would be executed. X is happy because he thinks that he or Z will be released, which means that the probability of his release is 1/2. Is he right? Or his chances or 1/3?

Solution:

The probability formula of the key items of the parties is: 2/3 * 1/2 = 1/3.

Description:

2/3 is at the beginning, and the probability of making any mistakes is 2/3; Then the probability of choosing the right one is1/3;

Next, remove an item;

1/2 At this time, the party entered the sub-event group, and his arbitrary choice was right or wrong.

It's easy to be misunderstood here

Next, remove any items;

-And-

Next, consciously remove one item; (For example, for items without flowers, remove the second number in the middle)

different

Next, consciously remove one item;

-And-

Next, get rid of a wrong item;

different

These are independent events,

similar

The time point of stopping childbearing has nothing to do with the probability of birth sex.

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TANKTANK98 correction: What is the probability here?

I think this question confuses many people. In fact, there are two possibilities:

1. For the whole door-opening event, including from the beginning, the probability of the contestants rose from 1/3 to 2/3, because there were three doors selected by the contestants (there were 1/3).

The other two doors (each 1/3), and later the host decided that one of them had no car, which increased the total probability of the remaining two doors having cars to 100%, while the original total probability of these two doors was 66%. Who is the extra 33% allocated to?

When a contestant chooses one of the remaining two doors, his chance of getting the car is 50%.

The objects of probability must be clearly distinguished! The probability of choosing 1 Shihou Zhang or the probability from beginning to end will really confuse people.

Yi u is full of flavor:

"If the competitor changes his mind and chooses another closed door, his chances of winning this car will be doubled." This statement. The odds are always 50%.

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Harvey: As the protagonist of The Final is 2 1 said, "I will definitely change, because that's the probability given to me by the host." The fact is that the player's choice is random (the car has a 33% chance and the sheep has a 66% chance), but when the owner chooses the sheep, he must choose the remaining sheep (66%)! Of course, in this case, the result can only be a "car." Then players always choose to change, and only choose sheep (33%) when choosing a car. At this time, your chances of getting a car in the game have doubled (33% to 66%), so if you are smart, will you choose to change or not? I think you already have the answer in your heart now.

Backward thinkers, about three questions: this is a problem with preconditions, and everyone is confused by serious thinking.

1. Results: The probability of winning the car by changing doors is 2/3. The concept of winning a car without changing doors is 1/3 (established).

Premise: The same person plays the same game for more than 3 times, so the winning probability of the car is 2/3 every time you choose to change doors.

2. Results: The winning probability of cars with and without changing doors is 1/2 (established).

Premise: The same person has only one chance to play the same game, so after the host determines a door, the probability that he will change or not is 1/2.

The problem with the results of 2/3 and 1/2 is that they are not a category at all, but two probability categories. 2/3 probability is a relative space. In 100 chances, you will have a 2/3 chance to win. The probability of 1/2 is the probability of occurrence under limited circumstances, so it is different.

[Edit this paragraph] Two kinds of probabilities

■ Classical probability correlation

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 correlation

Set Probability If there are infinitely many basic events in the random experiment, and each basic event has the same possibility, 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.

◆ Strict definition of geometric probability

Let the event A (also a region in S) contain A, and its metric size is μ(A). If P(A) represents the probability of event A, considering the "uniform distribution", the probability of event A is taken as: P(A)=μ(A)/μ(S), and the calculated probability is called geometric 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.

[Edit this paragraph] Independent test sequence

If a series of experiments have the following three items:

(1) There are only two results in each experiment, one is marked as "success" and the other as "failure", P{ success }=p, P{ failure} =1-p = q;

(2) The probability of success, p, remains constant in each experiment;

(3) Experiments are independent of each other.

Then this series of tests is called independent test sequence, also known as Bernoulli probability.

[Edit this paragraph] Necessary events and impossible events

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 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" is an event, which consists of a basic event (1, 1). The set {( 1, 1)} can be used to indicate that "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 of Event A. In real life, it is necessary to study various events and their relationships, and various subsets of elements and their relationships in the basic space.

Random events, basic events, possible events, mutually exclusive events, opposing events.

Events that may or may not occur under certain conditions are called random 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.

[Edit this paragraph] The essence of probability

Attribute 1.p (φ) = 0.

Characteristic 2 (limited additivity). When n events A 1, …, An are incompatible with each other: p (a1∧ ... ∪ an) = p (a1)+...+p (an).

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Real estate 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.

Real estate 6. For any two events a and b, 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 (AB).

(Note: the numbers 1, 2, ... a are all subscripts. )