【People's Education Press】High School Mathematics Elective Compulsory Volume 3 (A Edition)

📚 Content Summary

This textbook primarily covers advanced high school mathematics content, including counting principles (classification addition, step-by-step multiplication, permutations and combinations, and the binomial theorem), random variables and their distributions (conditional probability, discrete distributions, binomial distribution, and normal distribution), and statistical analysis of paired data (simple linear regression, independence testing).

Explore the rules of counting, grasp the laws of randomness, and master the core of data analysis.

Author: Zhang Jianyue, Li Zenghu

Acknowledgments: This book has been reviewed and approved by the Expert Committee of the National Textbook Committee (2019)

🎯 Learning Objectives

1. Accurately distinguish and apply the classification addition counting principle and the step-by-step multiplication counting principle to solve real-world problems.
2. Understand the properties of binomial coefficients (symmetry, monotonicity, sum), and use Pascal’s Triangle to solve problems involving sums of combinatorial numbers.
3. Construct mathematical models to analyze the number of execution paths in computer programs and the coding capacity of vehicle license plates, and prove generalized forms of the binomial theorem.
4. Skillfully apply the total probability formula and Bayes’ formula to solve probability problems in complex contexts.
5. Grasp the concept of discrete random variables, understand the properties of probability distributions, and independently compute expected values (means) and variances.
6. Accurately identify n-trial Bernoulli experiments, and distinguish between scenarios applicable to binomial and hypergeometric distributions (with replacement vs. without replacement sampling).
7. Differentiate between correlation and functional relationships; use scatter plots to determine positive or negative correlation, and calculate the sample correlation coefficient r to measure the strength of linear association.
8. Master the least squares method for estimating parameters in simple linear regression, establish empirical regression equations, and perform reasonable predictions and residual analysis.
9. Understand the fundamental principles of independence testing, formulate null hypotheses, and use the \chi^2 statistic to assess independence between categorical variables.

🔹 Lesson 1: Counting Principles

Overview: This lesson covers the foundational logic of counting, focusing on the definitions, distinctions, and applications of the classification addition counting principle and the step-by-step multiplication counting principle. It also delves into the structure of the binomial theorem, including numerical patterns in Pascal’s Triangle, symmetry and monotonicity of binomial coefficients, and practical applications of these mathematical models in computer program testing and civilian license plate regulations.

Learning Outcomes:

• Accurately distinguish and apply the classification addition counting principle and the step-by-step multiplication counting principle to solve real-world problems.
• Understand the properties of binomial coefficients (symmetry, monotonicity, sum), and use Pascal’s Triangle to solve problems involving sums of combinatorial numbers.
• Construct mathematical models to analyze the number of execution paths in computer programs and the coding capacity of vehicle license plates, and prove generalized forms of the binomial theorem.

🔹 Lesson 2: Random Variables and Their Distributions

Overview: This module presents a comprehensive knowledge system from conditional probability to discrete random variables and their distributions. Core topics include the three fundamental formulas of conditional probability (multiplication, total probability, Bayes’), their applications in artificial intelligence and game theory, and a comparative analysis of binomial and hypergeometric distributions.

Learning Outcomes:

• Skillfully apply the total probability formula and Bayes’ formula to solve probability problems in complex contexts.
• Grasp the concept of discrete random variables, understand the properties of probability distributions, and independently compute expected values (means) and variances.
• Accurately identify n-trial Bernoulli experiments, and distinguish between scenarios suitable for binomial and hypergeometric distributions.

🔹 Lesson 3: Statistical Analysis of Paired Data

Overview: This course aims to guide students in understanding non-deterministic relationships between variables through statistical analysis of paired data. Students will learn to move from intuitive observations in scatter plots to quantitative characterization via the sample correlation coefficient, mastering the construction, evaluation, and application of simple linear regression models.

Learning Outcomes:

• Differentiate between correlation and functional relationships; use scatter plots to determine positive or negative correlation, and calculate the sample correlation coefficient r to measure the strength of linear association.
• Master the least squares method for estimating parameters in simple linear regression, establish empirical regression equations, and perform reasonable predictions and residual analysis.
• Understand the fundamental principles of independence testing, formulate null hypotheses, and use the \chi^2 statistic to assess independence between categorical variables.