The content of classical probability has been in high school mathematics textbooks for many years. In the previous class, the emphasis was on calculating classical probability by permutation and combination. After the implementation of senior high school curriculum standard textbook, the concept of classical probability is introduced, which weakens the calculation of classical probability and strengthens the understanding of probability itself. This kind of change forces the classroom teaching to make a major change. At the fifth seminar on "Research on the Core Concepts, Thinking Methods, Structure System and Instructional Design of Middle School Mathematics", there are two research classes on classical probability, and the textbooks used are Mathematics 3 (Compulsory) and 3.2. 1 Classical Probability published by People's Education Press. After class, the teachers all think that these two classes have failed to achieve the new teaching objectives well, and one of the important reasons is that the concept of basic events has not been clearly explained. So the teachers discussed how to grasp the basic concept of events involved in this lesson and formed two different opinions. In view of everyone's views, let's talk about my personal thoughts on this content teaching.

First, the cause of the dispute

The teaching goal of this course is to understand classical probability and its probability calculation formula through examples, and to calculate the basic number of events and the probability of events through enumeration. The focus of teaching should be to let students understand classical probability through examples, and initially learn to turn some practical problems into classical probability, rather than focusing on how to count. However, due to the influence of traditional teaching, teachers still spend too much teaching time in class to calculate the probability of events, which fails to make students really understand the classical probability. Some students still won't turn their practical problems into classical probability, and as a result, they know nothing about the calculated probability. Another key reason for this phenomenon is that the teacher has not clearly explained an important concept of this course-basic events. So, after class, everyone discussed the concept of how to deal with basic events in this class.

One view holds that the key to determining whether an event is a basic event lies in its inseparability; Another view is that to judge whether an event is a basic event, we should proceed from specific problems, and every possible outcome can be regarded as a basic event, not an irreducible standard.

In fact, the above two views are reasonable, and the reason for the difference lies in the different starting points. The former simply looks at the basic event concept itself, while the latter looks at the basic event concept with some specific problems. The key to solving the dispute is to find out what the classical probability class should teach students. Only by grasping the concept of basic events from the teaching tasks of this lesson can we have a correct positioning of the concept of basic events.

Second, let's go back to the concept first.

Since the argument is caused by the concept, we might as well go back to the concept first, and then find out what this course teaches students. In this lesson, basic event and classical probability are two closely related core concepts, and understanding either concept requires understanding the other concept at the same time.

(1) basic event

1. The meaning of basic events

Since the concept of basic events is the basis of the concept of classical probability, only by understanding the concept of basic events can we understand classical probability. However, the textbook did not give the concept of basic events before introducing classical probability, but only pointed out that basic events have the following characteristics:

(1) Any two basic events are mutually exclusive;

(2) Any event (except impossible events) can be expressed as the sum of basic events.

But it is difficult for students to judge whether an event is a basic event according to the above characteristics. For example, in the random experiment of dice rolling, we can think that there will be two results: one is that the number of points on the top is odd, and the other is that the number of points on the top is even. For these two events, they are mutually exclusive, but their sum cannot be used to represent such an event as "the number of points above is not less than 3". So, can we judge that these two events are not basic events?

Events have different degrees of complexity. In probability theory, complex events are often "decomposed" into simpler events under the same random phenomenon. Among them, some events can no longer be "decomposed" into simpler events. Events that can no longer be "decomposed" within a certain research scope like this are called basic events. According to this definition, the basic event should be the simplest event within the research scope.

2. How to identify basic events

The definition of the above-mentioned basic events has two conditions, one is "within a certain research scope" and the other is "cannot be decomposed". If only "can't be decomposed" is the standard, in the random test of dice, the top points are 1, 2, 3, 4, 5 and 6 respectively, and only these six events are basic events. They also have two obvious characteristics in textbooks, and their sum can represent any event except impossible events, including events such as "the number of points above is not less than 3". However, if we still consider "within a certain research scope", then when we study the case of odd and even numbers of upper points, the two events of "odd number of upper points" and "even number of upper points" can also be regarded as basic events. Because these two events can be regarded as the simplest events in the interval with odd and even upper points. However, these two projects cannot be regarded as basic projects if the number of points on the upper side is not less than 3. The points above are 1, 2, 3, 4, 5 and 6 respectively, but these six events are the basic events in various research fields in the random test of dice. In this regard, it is difficult for students to understand when they first start learning. The key to teaching is that teachers should guide students to grasp the basic events step by step, so that students can grasp the basic events with the standard of "no longer decomposing", and then gradually understand "within a certain research scope" and gradually achieve a correct grasp of the basic events.

In addition, when talking about the characteristics of the basic events of the two classes, the teacher guides the students to focus on the mutual exclusion of events. Although this chapter gives the concept of mutual exclusion, it mainly considers the needs of related content. In essence, mutual exclusion is not a concept of probability theory, and its definition has nothing to do with probability. Therefore, the teaching of basic event concepts should not focus on the understanding of mutual exclusion, as long as students can distinguish whether events are mutually exclusive according to practical problems.

(2) Classical probability

1. The meaning of classical probability

The textbook calls the probability model with the following characteristics as classical probability:

(1) The number of possible basic events in the experiment is limited;

(2) The possibility of each basic event is equal.

In both classes, the teacher introduced the classical probability from the front with the example of throwing coins and dice. This can't help students understand classical probability well. Teachers should also list some counterexamples that do not meet the above characteristics for students to judge, so as to help students better understand this concept. For example, in a random experiment to study the situation of wine bottles falling to the ground, there are three results after the wine bottles thrown upward fall to the ground, that is, the bottle body is down, the bottle mouth is down, and the bottle bottom is down, but the possibilities of these three results are not equal, so this probability model is not a classical probability model. For another example, in the random experiment to study the position where the bullet hits the target when shooting, there are infinitely many possible results, so this probability model is not a classical probability model.

2. Classical probability is a mathematical model.

In the textbook, the probability model with the above characteristics is called the classical probability model, but in the communication with students after class, it is found that they are not clear about what a probability model is. This also affects their understanding of classical probability. The reason is that the concept of mathematical model was not well understood in the past. Prior to this, the textbook only introduced the function model, so students will naturally measure other mathematical models through some characteristics of the function model. As a result, it is difficult to understand the probability model as well as the mathematical model. Therefore, it is necessary for teachers to briefly introduce the concept of mathematical model in class. Generally speaking, a mathematical model refers to a structure expressed in a general and approximate way in a formal mathematical language according to the research purpose.

When students understand the concept of mathematical model, teachers should guide students to understand classical probability through typical examples. For example, when you flip a coin, you can think that there are only two results, namely "heads up" and "tails up", and each result has the same possibility, so it conforms to the classical probability. It is worth noting that we should focus on the understanding of concepts and don't waste time on some details. For example, some people think that in the coin throwing experiment, the actual situation may be that the coin is upright, and whether the texture of the coin is uniform can only be approximate. This should also make students understand that classic concepts are not an accurate description of reality. In our view, this is unnecessary.

(3) Teaching should deal with the relationship between basic events and classical probability.

Although basic events and classical probability are the two core concepts of this lesson, from the perspective of teaching objectives, the focus of teaching is to understand classical probability, and understanding the concept of basic events is to better understand classical probability. Therefore, in classroom teaching, teachers should not let students know these two concepts in isolation, but should link them and highlight their understanding of classical probability. For a probability model, students should first set out from practical problems, determine basic events according to the research scope, and dialectically understand the concept of basic events in the process; Then we will see if these basic events are finite and equally possible, so as to determine whether they are classical probabilities. In this way, students' attention focuses on practical problems, and their understanding of the two concepts is closely combined with specific problems at the same time, rather than isolated and abstract. The key to judge whether students know basic events and classical probability lies in whether they can turn practical problems into classical probability.

Third, what should the classical probability course teach students?

The purpose of understanding the concepts of basic events and classical probability is to better carry out the teaching of this course. So, what should the classical probability course teach students?

(A) will turn some practical problems into classical probability.

In the classical probability problem, the key is to give the correct model. Teachers should list more specific problems, so that students have more opportunities to try to turn practical problems into classical probabilities instead of focusing on calculating probabilities. But the teaching of both courses is biased towards this. For example, a teacher gave the concept of classical probability with the following three questions:

Question 1 What are the probabilities of "heads up" and "tails up" in the coin toss experiment?

Question 2: What are the probabilities of the six basic events "1 point", "2 points", "3 points", "4 points", "5 points" and "6 points" in the dice test?

Question 3: What is the probability of "even score" in the dice-throwing experiment?

As can be seen from the above questions, the teacher focused on the calculation of probability. From the actual teaching point of view, the whole teaching link is basically discussing the calculation of probability, but it is not enough to help students understand concepts and guide students to sum up specific problems. Therefore, affected by this, in the follow-up teaching, students also focus on the calculation of probability when facing specific problems, and do not form the conscious consciousness of transforming specific problems into classical probabilities, which leads to insufficient training in transforming practical problems into classical probabilities. In addition, because too much time is spent discussing the calculation of probability, one class has no time to learn some examples in the textbook, and the other class has not enough time for students to analyze and discuss how to transform into classical probability, which leads to the focus of this class can not be better highlighted.

(2) It will turn some practical problems into different classical models.

The same problem can also be solved with different classical probabilities. Therefore, the teaching of this course should not only let students learn to turn some practical problems into classical probabilities, but also learn to turn some practical problems into different classical probabilities. For example, both classes discussed the following questions:

Throw a dice with uniform texture and find the probability of even points.