基于图论的高中数学全概率公式教学探讨

Discussion on the Teaching of the Total Probability Formula in High School Mathematics Based on Graph Theory

摘要: 在高中数学中，概率不仅是学生需要掌握的基础知识，更是培养其随机思维的关键载体。全概率公式作为概率的核心公式，为复杂事件概率的计算提供了严谨的逻辑框架，已成为近年高考的热门考点。本文首先系统梳理全概率公式“样本空间建模→分割构造→公式推导”的逻辑链，帮助学生理解公式的理论基础；其次，提出有向图模型，并结合高考真题与教学案例，从图论视角探索公式教学设计的新思路。

Abstract: In high school mathematics, probability is not only a fundamental knowledge that students need to master, but also a key carrier for cultivating their random thinking. As a core formula in probability, the total probability formula provides a rigorous logical framework for calculating the probability of complex events. It has become a popular examination point in recent years’ college entrance examinations. This paper first systematically sorts out the logical chain of the total probability formula, namely “modeling of sample space → construction of partition → derivation of formula”, to help students understand the theoretical basis of the formula. Secondly, we propose a directed graph model and, in combination with true questions from college entrance examinations and teaching cases, explore new ideas for formula teaching design from the perspective of graph theory.