When you master the tree diagram, the next key point is the probability of the function. This is mainly to be clear in concept. On the surface, it is difficult. But one thing is clear. That is, there is an inevitable connection between function and independent variable. As long as there are independent variables, there must be functions. So the probability of both is the same.
This is just my experience. Because when I went to college, neither of them could pass. Fail the exam in probability theory. Later, only by conquering these two levels can we pass the customs.
Question 2: In fact, what is the use of probability theory permeating all aspects of modern life? As Laplace, a famous French mathematician in the19th century, said: "For most of life, the most important problem is actually only the probability problem. You can say that almost all the knowledge we have is uncertain, and only a small part is what we can know for sure. Even the mathematical science itself, the main means of induction, analogy and discovery of truth are based on probability theory. Therefore, the whole human knowledge system is associated with this theory ... "
The following are some historical cases.
Ratio of male to female at birth
Most people may think that the possibility of having boys and girls is equal, so it is inferred that the ratio of boys and girls should be 1: 1, but this is not the case.
In A.D. 18 14, the French mathematician Laplace (La Place 1794- 1827) recorded some interesting statistics in his new book "Philosophical Discussion on Probability". According to the statistics of London, Petersburg, Berlin and France, he got almost the same ratio of boys to girls. That is to say, among all the babies born, 5 1.2% are boys and 48.8% are girls. Strangely, when he counted the birth rate of boys in Paris for 40 years from 1745- 1784, he got a ratio of 25: 24, with boys accounting for 5 1.02%. Laplace is confused. He is convinced of the laws of nature, and he feels that there must be profound factors behind this 1.4%. So he made an in-depth investigation and finally found that Parisians at that time "preferred boys to girls" and abandoned boys, which distorted the truth of the birth rate. After the revision, the birth rate of boys and girls in Paris is still 22: 2 1.
An excellent mathematician = 10 teachers.
During World War II, the United States once announced that an excellent mathematician had a division of more than 10. This sentence has an unusual origin.
Before 1943, British and American transport fleets were often attacked by German submarines in the Atlantic Ocean. At that time,
Limited by their strength, Britain and the United States cannot send more escort ships. For a time, the German submarine war made the Allies at a loss.
To this end, an American navy general specially consulted several mathematicians. Mathematicians analyzed by probability theory that the encounter between the fleet and the enemy submarine was a random event. From a mathematical point of view, it has certain regularity. The smaller the fleet size of a certain number of ships (100 ships), the more times it is compiled (20 ships at a time, there must be 5 times). The greater the probability of meeting the enemy. The US Navy accepted the mathematician's advice and ordered the fleet to collectively pass through the dangerous waters, and then separately sailed to the scheduled ports. As a result, a miracle appeared: the probability of the allied fleet being attacked and sunk dropped from 25% to 1%, which greatly reduced the losses and ensured the timely supply of materials.
What is a probabilistic weather forecast?
Probabilistic weather forecast is to express the possibility of forecasting quantity with probability value. What it provides is not the "yes" or "no" of a weather phenomenon or the "big" or "small" of a meteorological element value, but the possibility of weather phenomena. For example, for the prediction of precipitation, the traditional weather forecast generally predicts whether it will rain or not, while the probability forecast gives the percentage of possible precipitation. The greater the percentage, the greater the possibility of precipitation. Generally speaking, if the probability value is less than or equal to 30%, it can be considered that there is basically no precipitation; The probability value is 30%-60%, and precipitation may occur, but it is less likely; The probability is 60%-70%, and the possibility of precipitation is very high; The probability value is greater than 70%, and precipitation occurs. Probabilistic weather forecast not only reflects the certainty of weather change, but also reflects the uncertainty and uncertainty degree of weather change. In many cases, this forecasting form can better meet the decision-making needs in economic activities and military activities.
What is the probability of being infected with AIDS?
What is the probability of being infected with AIDS? According to Xu Keyi, director of ditan hospital STD Prevention and Control Center, AIDS is transmitted to others through three modes of transmission, namely blood transmission, sexual transmission and mother-to-child transmission. If a normal person loses the blood of an HIV-positive person or an AIDS patient, its >>
Question 3: How to learn probability theory? Learning Probability Theory should pay attention to the following points:
1. In the process of learning Probability Theory, we should grasp the introduction of concepts and the understanding of the background;
2. In the process of studying Probability Theory, we should carefully scrutinize the connotations of the introduced concepts, as well as their connections and differences;
3. Understand the concepts in probability theory;
4. Focus on understanding the concepts and problem-solving ideas involved in different types of questions. This can often "get twice the result with half the effort";
5. Understand the statistical significance of interval estimation and hypothesis testing, and use these eight formulas flexibly on the basis of understanding. There is absolutely no need to memorize it.
Question 4: How to learn probability theory well? Why don't you cut the scabbard? First of all, let's analyze the results of previous postgraduate entrance examinations and observe the difference between higher mathematics and probability statistics. First, the average score rate of probability statistics is often lower than that of advanced mathematics. Second, the score distribution of higher mathematics is small at both ends, that is, the ratio of low score to high score is small, the ratio of middle score is large, and the score rate of probability statistics is low and many. Advanced mathematics mainly solves the problems about the properties and images of (one-dimensional or multi-dimensional) functions by learning limits, derivatives and integrals. It is closely related to middle school mathematics and has the same thinking method and problem-solving thinking. So it is easier to understand the concept (of course, there are more abstract contents such as the mean value theorem. ). On the other hand, because many specific elementary functions are involved, there are many calculation skills in finding derivatives and integrals. Mastering these skills requires a lot of practice, so some students can answer quite a few questions correctly even if the concept is not very clear, and make certain achievements in the postgraduate entrance examination. In the study of probability theory and mathematical statistics, more attention is paid to the understanding of concepts, which is ignored by most students. When reviewing for the postgraduate entrance examination, almost half of the students still don't know what random variables are and why they should be introduced. There is no way to start with the concepts of independence and irrelevance involving random variables. On the one hand, advanced mathematics deals with "some" events. For example, the function y=f(x), when x is determined, y has a certain value corresponding to it. In probability theory, the random variable x is uncertain before sampling. We can only determine the probability of it falling in a certain area through random experiments. It is often difficult to establish a way of thinking with uncertainty. If we apply a deterministic way of thinking, we will make mistakes. Because you don't understand the basic concepts, even simple topics are difficult to score, which leads to low scores. On the other hand, because there are not many calculation skills involved in probability theory, except classical probability, geometric probability and how to determine the upper and lower limits of integral when calculating the function distribution of two-dimensional random variables, the others are only numerical values or the calculation of integral and derivative. Therefore, if the concept is clear, the problem is often solved smoothly and the correct answer is easy to get, which is the reason for the high score. According to the above analysis, it is revealed that the learning method of advanced mathematics cannot be copied to the learning of probability and statistics. Learning methods should be put forward according to the characteristics of probability and statistics, so as to get twice the result with half the effort. Here we put forward some suggestions on the learning methods of probability theory and mathematical statistics respectively. First, learning probability theory should pay attention to the following points, grasp the introduction of concepts and the understanding of the background, such as why the concept of random variables should be introduced. This is actually an abstract process. Just like when primary school students learn mathematics, one apple plus two apples always equals three apples, and then it is abstracted as 1+2=3. For a specific random event in a specific random experiment, the probability can be calculated, but it is local and isolated after all. Can we unify the different sample spaces of different random experiments and describe the whole random experiment? The introduction of random variable X (that is, a single-valued real function from sample space to real axis) can transform the probability of random events in different random experiments into the probability that random variables fall on a real number *** B, and different random experiments can be described by different random variables. In addition, if all real numbers *** B, P(X∈B) is known, then the probability of any random event in the random experiment is completely determined. So we just need to find the distribution P(X∈B) of the random variable X, and then describe the random experiment comprehensively. Its research has become the central topic of probability theory. Therefore, the introduction of random variables is an important milestone in the history of probability theory. Similarly, the axiomatic definition of probability is introduced. The introduction of concepts such as mathematical characteristics of random variables has a clear background and needs to be deeply understood in learning. 2. In the process of studying Probability Theory, we should carefully scrutinize the connotation of the introduced concepts and their relationships and differences. For example, what is the significance of the connotation of the concept of random variables? It is a single-valued real function X(w) from the sample space to the real axis, but it is different from the general function. First, its domain is sample space. Different random tests have different sample spaces, and their values are uncertain. Different test results can take different values, but the probability of taking a certain interval can be determined according to random tests, I > >.
Question 5: How to learn probability theory? Probability theory is usually studied together with mathematical statistics, which is difficult. Probability theory gives me the feeling that it theorizes the simple probability in our middle school, which is more in-depth and rigorous, so some. . . Random variable, continuous discrete type, expectation, variance. It is necessary to combine the existing probability knowledge and understand while learning.
Question 6: How to teach yourself probability? To understand the meaning and background of probability theory, you can't learn by rote, or you won't know how to do another problem. Like what is probability? What is the concept of probability? What are the applications? Waiting ... in fact, probability is the possibility of an event. To put it bluntly, it is to describe the possibility of some economic and social phenomena in probabilistic language. To know the possibility of this phenomenon, we need to know the distribution function, because the probability of continuous type is actually integral, and the integrand function is the density function of random variable (discrete type can be regarded as continuous discretization, which is the same) ... The probability is not difficult, believe in yourself!
Question 7: How to learn probability theory and mathematical statistics well is more difficult than advanced mathematics. I think those high numbers are quite simple, but probability theory can be used. . . Fortunately, it's over.
Question 8: How to learn probability theory and mathematical statistics >