Definition of psychological probability

what is the definition of probability?

■ The frequency definition of probability

With the increase of the complexity of people's problems, 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 defined this fixed number as the probability of the event, which is the frequency definition of probability. Theoretically, the frequency definition of probability is not rigorous enough. A.H. Andrey Kolmogorov gave an axiomatic definition of probability in 1933.

■ Strict definition of probability

Let E be a random test and S be its sample space. For each event A of E, a real number is assigned, which is recorded as P(A), which is called the probability of event A.. Here, p () is a * * * function, and p () must meet the following conditions:

(1) Nonnegativity: for each event A, there is P(A)≥;

(2) Normality: for the inevitable event S, there is P(S)=1;

(3) countable additivity: let a1, a2 ... be mutually incompatible events, that is, for i≠j,Ai∩Aj=φ, (I, j = 1, 2 ...), then there is p (a1 ∪ a2 ∪ ...

(2) The possibility of each basic result of the test is the same.

such an experiment becomes a classical 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 indicates 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 is repeated for 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, the value P is called the probability that event A occurs under this condition, and it is recorded as p (a) = p.. This definition becomes the statistical definition of probability.

in history, Jocob Bernoulli (1654-175), the most important scholar in the history of early probability theory, was the first to give strict significance 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.

because the frequency nA/n is always between and 1, from the statistical definition of probability, we can know that for any event a, there is ≤ p (a) ≤ 1, p (ω) = 1, and p (φ) = .

ω and φ respectively represent inevitable events (events that must happen under certain conditions) and impossible events (events that must not happen under certain conditions).

how to understand the definition of probability?

first of all, it should be clear that probability is defined in an axiomatic form in mathematics.

The' statistical definition of probability',' classical definition of probability' and' geometric definition of probability' appearing in various textbooks are all descriptive statements. Teachers should not try too hard to figure out and explore the language there, but should understand its essence.

it is not difficult to say the concept of probability in general, but if we discuss it in theory or philosophy, there will be a lot of problems, which are not needed to be discussed in middle school (or even university) mathematics curriculum. Here, I would like to talk about some views on' definition' in mathematics.

We don't want to talk about the necessity, function and significance of the definition in mathematics. Every math teacher knows this.

What we want to talk about is the opposite side, which is also the place where we think there are some problems, that is, we pursue definitions excessively, explore the words in the book excessively, and ignore the grasp of the overall spirit. To define any concept, you need to use some words.

Strictly speaking, these words still need to be defined. Other words are needed to define these words.

therefore, this is an infinitely upward task that cannot be completed unless it stops somewhere. In other words, there must be some undefined words to discuss the problem as a starting point.

This point is put forward in the hope that people will not be superstitious about definitions. Some people think that everything that is not defined is not strict, and it is only strict when a definition is given.

this view is not comprehensive. Secondly, some definitions, if any, are unnecessary for many people.

most scientists don't need to know the theory of real numbers (the strict definition of real numbers), and most mathematicians don't need to master the definition of natural numbers given by Piano's axiom. Although strict expression is important, the most important vitality in mathematics comes from its problems and ideas, from people's exploration, conjecture and analysis.

the statistical definition of probability can usually be described as follows: under the same conditions, a large number of repeated experiments are carried out, and the ratio of the number of times k of an event to the total number of trials n is called the frequency of this event in these n trials. When the number of tests n is large, the frequency will be' stable' near a constant.

the greater the n, the less likely the frequency deviates from this constant. This constant is called the probability of the event.

we should be clear that the above definitions are only descriptive. In fact, it is suspected of circular definition.

because' possibility' appears in the definition. This refers to probability. (Similarly,' equal probability' usually appears in the classical definition of probability).

you can try to avoid this kind of words, but its essential meaning is unavoidable. Some people explore the definition of words such as' experiment'.

actually,' do an experiment' is not difficult to understand. For example, throw a coin, touch three red balls, take ten products, and so on.

Some complicated experiments are not difficult to explain to students. Defining' doing an experiment once' as' realizing the conditions once' is even more difficult to understand.

what is' condition'? What is' realization'? This is obviously inappropriate. Besides,' experiment' is not a term in mathematics at all.

the definition of probability

the phenomena observed in nature and society can be divided into deterministic phenomena and random phenomena. Probability is a branch of mathematics, which studies the quantitative laws of random phenomena. On the one hand, it has its own unique concepts and methods. On the other hand, it is closely related to other branches of mathematics, and it is an important part of modern mathematics. Probability is widely used in almost all scientific and technological fields, such as weather forecast, earthquake forecast and product sampling survey. Various departments of industrial and agricultural production and national economy can be used to improve the anti-interference and resolution of signals in communication engineering.

Probability formula: P(A)=m/n

What is the conceptual difference between probability and probability?

probability is probability, and there is no difference between them.

probability, also known as "probability", reflects the likelihood of random events. Random events refer to events that may or may not occur under the same conditions. For example, it is a random event to randomly select one from a batch of goods with genuine products and defective products, and "what is drawn is genuine".

suppose that a random phenomenon has been tested and observed n times, in which event A has appeared m times, that is, its frequency is m/n. After a lot of repeated experiments, m/n is often closer to a certain constant (see Bernoulli's law of large numbers for details). This constant is the probability of the occurrence of event A, which is usually expressed by P (A).

Extended data:

Probabilistic events:

In a specific random experiment, every possible result is called a basic event, and the * * * of all basic events is called a basic space. Random events (events for short) are composed of some basic events.

for example, in the random experiment of rolling dice twice in succession, z and y are used to represent the points that appear for the first time and the second time respectively, and z and y can take the values of 1, 2, 3, 4, 5 and 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) and can be represented by *** {(1,1)}.

"the sum of points is 4" is also an event, which consists of three basic events (1, 3), (2, 2) and (3, 1), and can be represented by * * * {(1, 3), (3, 1) and (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 and is called an impossible event.

P (impossible event) =. This event can't happen in the experiment. If "the sum of points is less than 4" is regarded as an event, it contains all basic events, and this event must occur in the experiment, which is called an inevitable event. P (inevitable event) =1. In real life, it is necessary to study various events and their relationships, various subsets of elements in the basic space and their relationships.