Mastering IB Probability and Statistics | 精通IB概率与统计

📚 Mastering IB Probability and Statistics | 精通IB概率与统计

Probability and statistics form a crucial part of the IB Mathematics curriculum, whether you are taking Analysis & Approaches (AA) or Applications & Interpretation (AI). This guide breaks down the core concepts—from basic probability rules to hypothesis testing—helping you build a solid understanding and prepare effectively for your exams.

概率与统计是IB数学课程的关键组成部分，无论你学习的是分析与方法(AA)还是应用与解释(AI)。本指南分解了从基本概率法则到假设检验等核心概念，帮助你建立扎实的理解，并高效备考。

1. Sample Space, Events and Probability | 样本空间、事件与概率

In any probability experiment, the sample space U is the set of all possible outcomes. An event A is a subset of the sample space. The probability of an event, P(A), is a number between 0 and 1 that measures the likelihood of A occurring. For equally likely outcomes, P(A) = n(A)/n(U).

在任何概率实验中，样本空间 U 是所有可能结果的集合。事件 A 是样本空间的一个子集。事件 A 的概率 P(A) 是介于 0 和 1 之间的一个数，衡量 A 发生的可能性。对于等可能结果，P(A) = n(A)/n(U)。

We often represent sample spaces using lists, tables, or tree diagrams. The complement of A, denoted by A’ or Aᶜ, satisfies P(A’) = 1 – P(A). Two events are mutually exclusive if they cannot occur simultaneously, so P(A ∩ B) = 0.

我们常用列表、表格或树形图表示样本空间。A 的补集，记作 A’ 或 Aᶜ，满足 P(A’) = 1 – P(A)。如果两个事件不能同时发生，则它们是互斥的，因此 P(A ∩ B) = 0。

2. Probability Rules and Set Operations | 概率法则与集合运算

The addition rule states that for any two events A and B, P(A ∪ B) = P(A) + P(B) – P(A ∩ B). This prevents double-counting outcomes that belong to both events. When events are mutually exclusive, the intersection probability is zero, simplifying the rule.

加法法则指出，对于任意两个事件 A 和 B，P(A ∪ B) = P(A) + P(B) – P(A ∩ B)。这样可以避免重复计算同属于两个事件的结果。当事件互斥时，交集的概率为零，从而简化该法则。

Venn diagrams are powerful visual tools for understanding union, intersection, and complement. In IB problems, you may need to fill Venn diagrams with given probabilities or frequencies. Always label each region carefully and translate worded conditions into set notation.

韦恩图是理解并集、交集和补集的有力视觉工具。在IB题目中，你可能需要根据给定的概率或频数填充韦恩图。务必仔细标注每个区域，并将文字条件转换为集合符号。

3. Conditional Probability and Tree Diagrams | 条件概率与树形图

Conditional probability, P(A|B), represents the probability of event A occurring given that B has already occurred. It is defined by P(A|B) = P(A ∩ B) / P(B), provided P(B) > 0.

条件概率 P(A|B) 表示在事件 B 已经发生的情况下事件 A 发生的概率。其定义为 P(A|B) = P(A ∩ B) / P(B)，前提是 P(B) > 0。

Tree diagrams are especially useful for multi-stage experiments where probabilities depend on earlier outcomes. Multiply probabilities along the branches and add probabilities of different paths leading to the same final outcome. Always check that the probabilities from any node sum to 1.

树形图特别适用于多阶段实验，其中概率依赖于先前的结果。沿分支相乘概率，并将不同路径通向同一最终结果的概率相加。务必检查从任一节点出发的概率之和为 1。

4. Discrete Random Variables and Expectation | 离散随机变量与期望

A discrete random variable X takes a countable number of values, each with an associated probability. The probability distribution can be displayed as a table. The sum of all probabilities must equal 1.

离散随机变量 X 取可数个值，每个值对应一个概率。其概率分布可表示为表格。所有概率之和必须等于 1。

The expected value E(X) = μ = Σ xᵢ p(xᵢ) gives the long-run average. The variance Var(X) = σ² = Σ (xᵢ – μ)² p(xᵢ) or equivalently E(X²) – [E(X)]². These measures help describe the centre and spread of a distribution.

期望值 E(X) = μ = Σ xᵢ p(xᵢ) 给出长期平均值。

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