The year 1987 was the centenary of the birth of the legendary Indian mathematician Srinivasa Ramanujan (1887-1920). A series of events were held in his honour. A prominent contemporary statistician, C. Radhakrishna Rao, born in India, was also invited to give three lectures. The Indian Statistical Institute, based on the lectures of Rao, published a book on Statistics and Truth in 1989 in his honour.
As a student, I majored in mathematics, a logic of inferring results from given premises. Later, I studied statistics, a rational method of learning from experience, and a logic of verifying premises from given results. I have recognized the importance of mathematics and statistics in all human endeavors to advance natural knowledge and to manage daily affairs effectively.
I believe:
In the final analysis, all knowledge is history.
In the abstract sense, all science is mathematics.
In a rational world, all judgments are statistics.
This passage summarizes the importance of mathematics and statistics and their respective implications.
For a long time, high school mathematics has covered topics of probability, in which classical probability (i.e. the probability of explaining probability by the same probability) is also a significant proportion. Therefore, probability is often associated with array combinations, while array combinations are more complex mathematical puzzles. Although students are sometimes distracted by those complex problems, they are sometimes dazed. But that is only a skillful aspect, in terms of cognition, usually not too confused. In recent years, given the importance of statistics, in high school mathematics, statistics has gradually been added.
Another prominent statistician, Jerzy Neyman (1894-1981), who was born in Poland and emigrated to the United States in 1938, was the first to make the statement in a 1934 speech. After his speech, the president of the conference, Arthur Lyon Bowley (1869-1957), said, "I'm not sure this confidence is a confidence trick". The concept of the Neyman confidence interval was first proposed by most statisticians, including the British statistician Sir Ronald Aylmer Fisher (1890-1962), who is considered the founder of modern statistical methodology.
Over the years, more than seventy years have passed, and today's statisticians, of course, have fully understood the meaning of the trust interval. However, in universities, whether in probability and statistics, statistics, and mathematical statistics, the trust interval is usually in the second half of the textbook. That is, college students in the relevant courses, when they begin to come into contact with the trust interval, there is a fairly adequate basis for probability statistics.
Why is it that a somewhat profound topic is so well placed in high school math curriculum? The main reason for this is probably its importance. This can be understood by looking at the confidence intervals and confidence levels that are often published in the media.
In some statistical textbooks, confidence intervals account for a large portion of a chapter. Different parameters, different distributions, can have different confidence intervals; even if the same parameters have the same distribution, different methods can be used to obtain different confidence intervals. Sometimes, due to insufficient conditions, or for reasons of computational complexity, it is enough to retreat and seek a second, to obtain an approximate confidence interval. Of course, this requires some conditions, and some theorems.
In his interpretation of the normal distribution, the confidence interval and the confidence level, Yang said:
High school statistical inference only estimates the expected value of random variables, and the theory behind it is central limit theory. To introduce central limit theory, you need to introduce a normal distribution. This part is only a general introduction to actively build students' intuition about central limit theory. For a fixed confidence level, give a confidence interval formula, then have students run a random number table simulation or experiment to cast a copper plate with a positive probability of p, enter the confidence interval formula, explain the meaning of the confidence interval; and with this interpretation, explain why most students will cover p with the confidence intervals obtained.
This is not a statistical view. Because the interpretation of the curriculum is unclear, high school mathematics teachers who teach seriously and want to teach students to understand, just have to delve into the principles, their own interpretation. Some also propose articles that claim to be able to clarify these concepts.
Why is the concept of the confidence interval often reduced to a kind of "spoiler" scenario?
The probability of getting an even number is that there are six faces in a cluster, under a cluster. The probability of getting an even number is assumed to be the same on each side, i.e. 1/6, whereas the probability of getting an even number on each side is 1/6, whereas the probability of getting an even number on each side is 1/6, 2, 4, 6 etc. Therefore, the probability is 3/6. This is the so-called classical probability, and the basic assumption is that the same probability exists. There are several possible phenomena to be observed in advance, and a few more are of interest to us.
In late July and early August 2009, World Golf Champion Tiger Woods played in the Buick Open in Michigan, United States. He finished the first round eight strokes behind the leader and ranked 95th. This triggered a two-game losing streak in his career.
这时大家看法丕变，一致认为这座冠军盃，几乎可说是他的囊中物了。因过去的纪录显示，伍兹如能带着54洞领先进入决赛圈，战绩是35胜1败。你要不要猜后来他赢了没有？运动比赛，往往有过去资料可参考，此时相同的可能性便不宜用了。36次中成功35次，“相对频率”为35/36(约0.972)。这种以相对频率来解释概率，是常有的作法。适用能重复观测的现象。会不会有爆出冷门的时候？当然有。只是对一特定事件，用过去多次同样情况下，该事件发生的相对频率，来估计下一次事件发生的概率，乃是在没有更多资讯下，常被认为一属于客观的办法。
A man looks at a girl, surprised, and thinks that this is the bride of his life. After evaluating, he is full of confidence, and the chance of chasing after himself is 80%. The others do not look good, and ask him how he came up with this figure of 8%. The man shows a calendar, one sign after another, showing that the girl is very nice to him.
Subjective probabilities, of course, can also be based on some objective facts. But even when faced with the same information, different people may have different judgments, and therefore give different subjective probabilities. Have you ever seen He's Just Not That Into You?
For example, if you're chasing a girl, you're going to have a few girls, and you're going to have them experiment, chase them repeatedly, and then you're going to count a few of them, and then you're going to determine the probability that she's going to be chased by you. For this kind of phenomenon that can't be repeatedly observed, subjective probability is often used when talking about probability.
Although it is subjective, it is still reasonable. For example, if the probability of passing an exam is 0.9, it is fine, one should always be a little confident, but if at the same time one is afraid that the probability of failing is 0.8, it is not possible. The probability of various possibilities is 1. Even if subjective, it can be discussed separately, but it must be solved.
The above three are common explanations of the probability, usually a number of ways of thinking about the size and likelihood of an event. Although they are for different situations, they can often be used interchangeably. Everybody has heard the example of a murderer with the same name as his grandfather. A good-hearted person told his grandmother that he had killed his grandfather.
Of course, you can be incredulous, no matter what the result of the casting, everyone thinks that it is only a temporary situation, willfully firmly believe that this is a fair copper plate. This is not impossible, as there will be a mother, even more witnesses, she does not believe her son will kill, as long as she does not see it with her own eyes. To know that random phenomena, events as long as the probability is positive, no matter how small the probability value, everything can happen. After all, the probability of the appearance of a positive copper plate, only God knows.
Although the above three explanations of probability cover many real-life situations, mathematicians of course do not stop there. They like abstraction and generalization. Like solving equations, they seek formulas to express the solution of a certain kind of equation, rather than satisfy themselves with finding a specific solution. And once you have a complete understanding of the real number system, you define the real number system in an axiomatic way.
What is called the axiomatic way to introduce probability? First, there is a set, called the sample space, which is the set of all possible outcomes of an observation. There can be this observation, or it can be only virtual. Some sub-sets of the sample space, which we are interested in, are events. All events are also a set.
This does not require too much sample space, but it cannot be an empty set. The set of events must meet certain conditions. Simply put, you cannot have too few events of interest. For example, you cannot only be interested in an event A and not be interested in A. Therefore, the set of events must be large enough, at least everything should be included.
Under the structure of probability spaces, no matter how one interprets probability, one can express it and find the meaning of probability. But since abstraction is no longer limited to copper boards, dice, and poker cards, one can discuss more general problems and have enough theory to dig into.
The development of probability theory was later than in other areas of mathematics; however, after its formalization, probability theory quickly developed far and wide and became an important area of mathematics. This is thanks to the important probability theorist of the twentieth century, Russian chemist Andrei Nikolaevich Kolmogorov (1903-1987), who laid the foundations of probability in his booklet Foundationsof the Theory of Probability, published in 1933, which is less than 100 pages long. In this book, he says:
The theory of probability as a mathematical discipline can and should be developed from axioms in exactly the same way as Geometry and Algebra.
The French Newton, Pierre-Simon, Marquis de Laplace (1749-1827), said:
This science, which originated in the consideration of games of chance, should have become the most important object of human knowledge. Themost important questions of life are, for the most part, really only problems of probability).
The probability is for random phenomena. But not everything in the world is random, we have said inevitability. Suppose you throw one or both sides of a human head on a copper plate, and observe that you get that side. You know this is an inevitable phenomenon, but you can still say that the probability of a human head is 1 and the probability of other situations is 0. That is, consider this as a degenerate random phenomenon.
Some physicists, for example, believe that a copper plate can be calculated by the speed, angle, elasticity of the ground, shape and weight of the copper plate, and so on. When the copper plate lands, it will face up, so it is not random. As for the lottery prize, as long as the initial conditions are determined, the ball will be opened, so it is also not random.
Some theologians may think that everything is in fact going according to God's will, but we don't know. We can't say for sure. Have you ever seen Jason and the Argonauts? This is a movie based on Greek mythology, about Aries in the twelve zodiac signs, released in 1963.
With the advancement of technology, people are gradually understanding the pulse of many phenomena. For example, we know that once a woman becomes pregnant, the sex of the baby is determined. But for a woman with a large stomach, the good thing is that due to the unknown, she can still guess the probability of giving birth to a boy. On the eve of the exam, the students, although they are seriously preparing, still speculate, each of them thinks that the probability is high.
But for the teacher of a well-posed problem, it makes no sense to judge the probability that the problem will be solved. Because for him, the probability of each problem is only 1 or 0, no other value. Similarly, for the person who sees the fruit behind the fruit, the probability of the fruit being a monkey or an apple, will only say 1 or 0. Randomness is different from randomness.
In section 2, we introduced probabilities in a probability space. Since the sample space can be virtual, the event is also virtual. But suppose there really is an observation, such as projecting a four-sided object, the four-sided marker points 1, 2, 3, 4, and observing the resulting number of points. The sample space is the set of 1, 2, 3, 4. The set of events can be taken to be the largest, that is, the set that contains all subsets of the sample space.
Even if you've accepted the concept of a probability space, which is a mathematician's favorite definition, you might be wondering, what does the so-called probability of a point number 1 being 0.1 mean?
假设投掷n次，点数1出现a次，则相对频率a/n与0.1之差的绝对值，会大于一给定的正数(不管它多小)之概率，将随着n的趋近至无限大，而趋近至0。
You, a pragmatic person, probably don't think that such an explanation is very practical. First, you have to ask yourself what is approaching infinity? You just keep throwing, don't stop, the sun rises and sets, spring and autumn come, keep throwing, even if your grandfather succeeds, infinity is still not reached, you still have to throw. That math graduate, when he heard you ask about infinity, like fish get water, this is one of the few tricks he learned in his four-year window in mathematics.
Trying to explain the meaning of the probability value will be a layer after layer of rotation in probability and infinity. This is like trying to define what to call a point, the result will be like falling into the online group, learning step by step. Finally, it is difficult to say that the point is an undefined noun. But anyway, you should understand that for the above four facets, only one throw, is not able to show the probability of the number 1 occurring 0.1, the meaning of the minor 0.1.
This is a simplified version of one of the laws of large numbers. Mathematically, it means that the relative frequency of occurrence of an event, the probability of meeting converges to the probability of occurrence of the event. In a random world, there are still some laws to follow, and the law of large numbers is one of the most important ones.
Events can occur as long as the probability is positive. So, no matter how large the number of observations, it is not possible to exclude very biased events (such as 1,000,000 observations where the number of occurrences of 1 is 0, or 1,000,000 times). However, when the statistician jumps out, it is possible to make a determination to determine whether the probability of the occurrence of 1 is really 0.1, which is in the category of statistical assumptions to determine the "testing hypothesis".
If it is unusual, then the initial assumption is not acceptable. Note that if a copper plate is assumed to be fair, 100 rolls will result in at least 80 positives, compared to 10 rolls, resulting in at least 8 positives. The former is more unusual because the probability of its occurrence is much lower than the latter.
In a random world, it is often unknown which is true. It is often impossible to prove that what is true is true. It is a hypothesis, and it is up to you to accept that hypothesis. The probability of the occurrence of the number 1 of the 4 faces is 0.1, and even if you throw it several times, it cannot be proven true.
In addition, the probability of the occurrence of the number 1 can also be estimated for four facets, and there are some different estimation methods, which can be obtained different estimates. In mathematics, using different methods, must lead to the same result. The so-called homogeneity of probabilities. But in statistics, there is often no one-size-fits-all method, unless some limitation is made.
We often make estimates of an unknown quantity. The unknown quantity can be the probability of an event, a parameter of a distribution (such as the expected value and the number of variables, etc.) or the lifetime of an object. These unknown quantities are known as parameters. Sometimes parameters are estimated in a range and given a range that covers the probability of the parameter.
Data is the main basis for statistical decision-making. If data is lacking, they tend to overlook it. Let's look at a simple and common situation. Suppose you want to estimate the probability of a copper plate appearing positive p. Naturally, you throw a few times, say n times, and observe the result n times. This process is called sampling. In this case, the results of each throw are not important.
Since this involves two distributions, the calculation is more complicated, and if n is large enough (n is not too small), we can often approximate by means of a normal distribution. This is another important rule in probability theory, the central limit theorem. It must be mentioned that the central limit theorem is only used when approximating with a normal distribution, and not for all confidence intervals.
For estimating the probability of the copper plate appearing positive p, before sampling, the confidence interval is a random interval, if the confidence level is set to 95%, then there is a probability of (or more precisely a probability of 0.95, if the confidence interval is only approximate) 0.95, the confidence interval will contain p; after sampling, a fixed interval is obtained. Then p will belong to the probability range, and will not be 1 and 0 and no longer p. Why is this?
Let's start with the following example. Suppose a department store celebrates its anniversary and customers buy a certain amount, they can draw 1 lottery ticket from 1 to 10. If they draw 5, they can get 30% of the company's spending today. Before the draw, you know that there is a 0.1 chance of getting a mortgage ticket, the chance is not small.
There are many examples of this. Before hitting the stick, you can say that the probability of hitting the jackpot is 0.341, after hitting it is not an odd, and 0.341 has not been sent to the field. Give another example. Suppose a lottery ticket issued by a bank, each issue from 1 to 42, opens 6 yards as the main prize number. You bet 6 yards, before the jackpot, you know that it is easy to hit at least 1 yard, because the probability is about 0.629.
Also, as stated in the syllabus, it is possible for random number tables to simulate the appearance of a positive p (the syllabus has less than two letters in the positive bracket, which is incomprehensible) the probability of the copper plate n times, in order to obtain the confidence interval. You see, p is basically predetermined, one of the results of the simulation is a fixed range, p does not fall within it, a look at it, how can you say that the range covers the probability of p is 0.95?
What is the use of that 95%? 0.95 is a probability value, and a probability value is never the result of a one-time experiment. Roughly speaking, if you repeat the experiment and get a lot of confidence intervals, it will contain a number of confidence intervals p, about 95% of the total range. So, 0.95 means the same as we explained the probability in the previous section.
Since probability is related to our life habits, using it well will help us make more accurate decisions in a random world. However, it is often difficult to apply probability and the resulting probability values are often considered wrong.
In the past, in mathematics classes, we would encounter so-called application problems. The problem is to understand that after writing a mathematical formula, you solve the mathematics. This is when you can throw away the original long narrative.
In the movie Final 21 (English title is 21), the mathematics professor asks a question in the classroom. There are three doors, one behind the car, the other two behind the goat. After you choose the first door, the presenter opens the second door and sees the goat.
Yes, because my chance of getting the carwill increase from 33.33% to66.67% by switching from door 1 to door 3.
The professor said, "Very good!" and agreed with his opinion, that is, it should be changed.
A more accurate way of saying this would be that if the presenter knew in advance that the car was behind that door, he would open one door and then the goat's door (which is a more reasonable approach, otherwise the game would not be possible). If the third door is selected, then, as the student in the film says, the probability of getting the car will increase from 1/3 to 2/3; but if the presenter does not know in advance that the car is behind that first door (which is of course a rare case), but only randomly chooses one door from the second and third doors, and then the goat is right behind the door, then it is not necessary to change, because of the change or not, the probability of getting the car is 1/2.
However, the reader may have noticed that in the case where the presenter knows in advance that the car is behind that door, we are actually implying a hypothesis. That is, if the 2nd and 3rd doors are all goats, the presenter opens the 2nd or 3rd door at random (i.e. with a probability of 1/2 each). In fact, there can be a more general hypothesis. When the 2nd and 3rd doors are all goats, assuming the presenter opens the 2nd or 3rd door with a probability of q1 and?q respectively, and 0≤q≤1 switches the 3rd door, the probability of getting the car is 1/1+q (see footnote 2) ; this probability is influenced by how the presenter opens the 2nd door!
Consider another example. A couple has just moved into a neighborhood, and everyone knows they have two children, but they don't know the gender. One day, a community administrator sees a housewife playing with a child in the house. If the child is a girl, ask for the probability that both children are girls. Many people think this problem is not difficult, thinking that the probability is 1 / 3.
Finally, look at another example that often appears in probability textbooks. There is a unit circle on the plane, randomly drawing a string, the probability of which is greater than the length of the triangle's edge. Using geometry, the innermost edge of the unit circle, the length of the triangle's edge can be found. But how to draw a string at random?
The above examples tell us that when dealing with probability problems, the situation must be clearly defined. In terms of terminology, the probability space must be clearly given, otherwise it will lead to all sorts of things. Sometimes, although the probability space is not given, the situation is simpler, everyone has a common opinion, why the probability space is not particularly emphasized at this time, there is no problem.
In addition to contextual interpretation, some unique concepts in probability, such as conditional probability, independence, and random sampling, should also be carefully considered when applying probability.