Abstract.In this paper, the authors carry out an analysis on the problems that exist in the teaching of the course Probability and Statistics now, and take advantage of the teaching cases of the course Probability and Statistics in the real word to guide the students to implement thinking, association, comparison and conclusion with a higher frequency. The purpose of this paper is promoting the students to obtain a deeper understanding of some concepts, making an enhancement to the students" awareness in mathematics, and training the students to possess the ability to make use of the mathematical knowledge to resolve the practical problems with a gradual step.
Key words： Probability and Statistics; Educational Reform; Teaching Cases
1. Introduction
At the present time, the class hours, which are arranged by the schools for the course Probability and Statistics, continue to be compressed. However, in this process, the requirements on the teaching of Probability and Statistics often go to rise, but not to fall.
Therefore, it is highly necessary for the modern scholars to make full use of the modern scientific points of view to re-examine, select and organize the teaching contents of the course Probability and Statistics, and introduce the cutting-edge science and technology as well as the advanced methods.
All these can make the boring teaching of the course Probability and Statistics lively and actively, and can provide the helps for the students to carry out the scientific research in the future at the same time.
2.Using Introduction as a Key to Stimulate the Interests of Students in Probability and Statistics
The lesson for the Introduction is mainly aiming at providing the students with the general concepts and characteristics of the course Probability and Statistics as well as the overview and application of the discipline development and the problems that the teachers and students are necessary to pay attention to.
For this reason, in the process of giving an explanation, it is highly necessary for the teachers of the course Probability and Statistics to give full attention to this aspect.
Probability theory was originated from the games. From the 15th century to the 16th century, Italian mathematicians Pacioli, Tartaglia and Cardano had made a discussion on these probability questions in their works.
For example, if the game between the two parties came to an end ahead of time, how did people allocate the stake? How is the probability for two students at least in a class (n≤365 students) to be born in the same day? In the process of implementing the teaching of the Introduction, it is highly necessary to pay attention to the introduction on the background knowledge of the course Probability and Statistics. On the one hand, this can make the students know the development history of this course. On the other hand, this can stimulate the students to generate the curiosity in learning.
3.Attaching Higher Importance to the Interests of Students in the Teaching of Probability and Statistics
For example, in the process of giving an introduction to the course Probability and Statistics, the teachers can provide the students with an introduction for the following questions.
Question 1: There is a competition (three out of five sets) between two football teams (A and B) of the same level, and the winner can receive a prize of 500,000 RMB. By now, three sets have been completed, in which the team A get two wins and one loss. At the present time, however, the team B quits the sets in the latter time for some reasons, and asks to allocate the bonus in accordance with the times of the win. In other words, 2/3 is allocated to the team A, and 1/3 is allocated to the team B. From the above, the teacher can ask whether it is reasonable to allocate the prize of 500,000 RMB.
Question 2: At the present time, the lotteries, which can be seen at any place, have attracted a larger number of people to buy. At the same time, it is reported that somebody wins a prize from time to time. Therefore, the teacher can ask the question about how the possibility for the students to win a large prize.
For the above two questions, the students seem to be highly interested in the answers.
Besides, in the process of introducing the mathematical expectation, the questions can be introduced in the following.
Question 1: At some place, a mass screening on a disease is implemented, and therefore it is necessary to verify the blood of each individual. If there is an N population at the local place, it is necessary to test the blood of the N persons. At the present time, two ways can be used for implementing the test: (1) the blood of each individual receives a test respectively, and is necessary to be tested for the N times in total; (2) The groups can be established according to the k persons, and the blood that is taken from the k persons is mixed together to have a test. If the mixed blood tends to be a negative reaction, it indicates the blood of the k persons to be negative. In such a way, the blood of the k persons is necessary to be tested for only once. However, if the mixed blood tends to be positive, it is necessary to carry out a test on the blood of the k persons again in order to know clearly those whose blood is positive in the k persons. In such a way, the total times for the k persons to receive the test is k +1. Therefore, it can be assumed that the probability for the persons to be negative in blood is P(0a) for the insurance participant if there is no on-suicide death within five years. Therefore, the teacher can raise a question about how to determine b to make the insurance company gain the expected return; how about the expected income of the insurance company if there are m persons to buy the insurance?
By resolving the above two problems, the students can easily get an understanding that the mathematical expectation is a theoretical average actually.
4. Actual Teaching Cases
4.1 Combing visual sense, illusion and theory together and mobilizing the enthusiasm of students
For example, (question about a ballot) there is a movie ticket, and there are seven persons wanting to get it; the seven persons decide to take advantage of the drawing-lot way to win it. Therefore, the teacher can raise a question about the probability for the ith person to get the movie ticket (i=1, 2… 7).
Concerning about this kind of question, the students may think that the possibility for the persons who draw the lots first to get the movie ticket is larger than that for the persons who draw the lots latter. In fact, the reorganization on the possibility does not rely on the intuition; the probability for the persons who draw the lots first or latter is the same actually.
Solving: Assume = "the ith person gets the movie ticket" (i=1, 2… 7)
Obviously, there is ,
If the second person did not get the movie ticket, it is necessary for the first person not to get it.
This means that there is . Therefore, the equation can be established.
From the above, the multiplication formula of the probability can be used. This is because that the second person has a greater probability to get the movie ticket in the remaining six lots if the first person did not get the movie ticket.
Thus, ,
,
Similarly to get
,
…
.
For example, (question about birthday) there are n persons (n A and B are incompatible with each other
Event A and eventB are mutually exclusive
(S is the sample space)
(2) A and B are not necessarily incompatible with each other if they are mutually independent, and also are not necessarily independent with each other if they are mutually incompatible.
For example, decipher
(i) Decoding password: the probability for A is 1/2; the probability for B is 1/3.
A: A can be decoded; B: B can be decoded.
Thus, A and B are independent with each other, but are not exclusive with each other. This is not because they are the events that can’t occur simultaneously. However, this can only suggest that A & B are not necessarily incompatible with each other if they are mutually independent.
(ii) Tossing coin: A means that the obverse side is upward, and B means that the reverse side is upward, and therefore it can be known that A and B are incompatible with each other. A and B, however, are not independent with each other, why? This is because the occurrence of an event will give rise to the non-occurrence of another event. Therefore, how can they be independent with each other?
Therefore, in the practical applications, it can be seen that the definitions are not often used to make a judgment on whether the events are independent with each other, but the practical significance of the events are used to make the judgment.
(3) If A & B are two events whose probability is not zero (i.e. P (A)>0 and P (B)>0), it is impossible for these two events to be independent and also exclusive with each other. Why is this?
First: If A & B are independent, there is P (AB)＝P(A)P(B)>0, and thus it is impossible for them not to be incompatible with each other, and otherwise there is P (AB)＝0.
Second: If A & B are incompatible with each other, there is P(AB)＝0, there are is no P(A)P(B).
4.2.2 Necessary event and impossible event
Necessary event: The sample space S is a subset of it, and always occurs in each experiment, and therefore is called as necessary event.
Impossible event: The empty set φ does not contain any sample point, and does not occur in each experiment, and therefore is called as impossible event.
Thus, it can be easily known that there is
However, consequently, the students may misunderstand this.
Therefore, it can be learn that the event whose probability is equal to 1 must be necessary; the event whose probability is equal to 0 must be impossible.
Contrary case: Assume the continuous random variable X has a probability density:
Therefore, the probability for the event is equal to 1, but is not a necessary event.
The probability of is equal to 0, but is not an impossible event.
In addition, there are a number of some easily-confused concepts such as the "frequency and probability" and the "probability and conditional probability". Therefore, in the process of teaching, it is necessary for the teacher to provide the students with a thorough explanation, so that the students can correctly get an understanding of the relationship among these concepts.
5. Conclusion
With the purpose of mobilizing the enthusiasm of the students in the learning of the course Probability and Statistics, it is necessary to integrate the heuristic teaching method into the teaching process, to inspire the students to think and discover things. In addition, in the process of teaching, it is necessary to pay attention to the combination of the contrast and associative methods, to guide the students to make a comparison or association on the concepts that the students have learnt.
Also, in the process of teaching, it is necessary for the teachers to creatively introduce the multiple teaching methods such as multimedia, slide projectors and teaching models to coordinate the explanation in accordance with the needs of the teaching contents. With the multimedia technology, some random phenomena can be simulated, so that the e ability of the students in the observation of problems can be trained, and the students can pay attention to the application of the mathematical software applications such as Mathematic, Matlab and SAS. In this way, it can not only help the students further understand the concepts and get rid of the complicated calculations and demonstrations, and also can make an increase to the amount of the teaching information and save a lot of time.
6. References
[1]Zhongxin Ni, Min Chen. Modern Teaching Method of Probability and Statistics of Technology Focusing on Statistical Idea [J]. College Mathematics: 2004, (2).
[2]Xiaolong Chen, Qingsheng Shi, Xiaowei Deng. Teaching Reform and Practice of Probability and Mathematical Statistics Courses [J]. Journal of Nanjing University of Technology (Social Science Edition), 2004 (3).
[3]Qunying Wu. A Probe of Teaching in Interest and Heuristic Method in Probability Statistics Course [J]. Guangxi Higher Education Research: 2001, (3).
[4]Xiaolong Chen, Qingsheng Shi, Xiaowei Deng. Probability and Mathematical Statistics [M]. Nanjing: Southeast University Press, 2003.