IB Maths Core: Probability | IB 数学核心：概率

📚 IB Maths Core: Probability | IB 数学核心：概率

Probability is a fundamental branch of mathematics that quantifies uncertainty and randomness. In IB Mathematics, the core probability topics provide students with essential tools for analysing chance events, from simple experiments like rolling dice to real-world applications in science, economics, and decision-making. This article revisits the key concepts, rules, and distributions required for the IB core syllabus, with clear examples and explanations.

概率是数学中量化不确定性和随机性的基础分支。在 IB 数学中，核心概率主题为学生提供了分析随机事件的基本工具，从简单的掷骰子实验到科学、经济学和决策等现实世界应用。本文回顾了 IB 核心教学大纲所需的关键概念、法则和分布，并配有清晰的例子和解释。

---

1. Introduction to Probability | 概率导论

Probability is a measure of the likelihood that an event will occur, expressed as a number between 0 and 1. A probability of 0 means the event is impossible, while 1 indicates certainty. In most practical contexts, probabilities are given as fractions, decimals, or percentages.

概率是度量事件发生可能性的数值，在 0 到 1 之间。概率为 0 表示事件不可能，1 表示必然发生。在多数实际情境中，概率以分数、小数或百分比给出。

e equally likely, P(Event) = (Number of favourable outcomes) / (Total number of outcomes).

概率论的基础在于等可能结果的概念，例如抛掷公平硬币或掷公平骰子。当所有结果等可能时，P(事件) = (有利结果的数量) / (总结果数量)。

---

2. Sample Spaces and Events | 样本空间与事件

An event is any subset of the sample space, such as rolling an even number: E = {2, 4, 6}.

样本空间，记作 S 或 Ω，是实验所有可能结果的集合。对于单次掷骰子，S = {1, 2, 3, 4, 5, 6}。事件是样本空间的任意子集，例如掷出偶数点：E = {2, 4, 6}。

(single outcome) or compound (multiple outcomes). The complement of an event A, written A' or Aᶜ, is the set of outcomes not in A. P(A') = 1 – P(A).

事件可以是简单的（单个结果）或复合的（多个结果）。事件 A 的补集，记作 A' 或 Aᶜ，是不在 A 中的结果集合。P(A') = 1 – P(A)。

---

3. Probability Rules | 概率法则

P(A) ≤ 1. (2) P(S) = 1. (3) For any sequence of mutually exclusive events A1, A2, ..., P(∪ A_i) = Σ P(A_i).

基本概率公理：(1) 对于任何事件 A，0 ≤ P(A) ≤ 1。(2) P(S) = 1。(3) 对于任意互斥事件序列 A₁, A₂, ...，P(∪ A_i) = Σ P(A_i)。

Addition Rule: For any two events A and B, P(A ∪ B) = P(A) + P(B) – P(A ∩ B). If A and B are mutually exclusive, P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B).

加法法则：对于任意两个事件 A 和 B，P(A ∪ B) = P(A) + P(B) – P(A ∩ B)。若 A 和 B 互斥，则 P(A ∩ B) = 0，因此 P(A ∪ B) = P(A) + P(B)。

---

4. Mutually Exclusive Events | 互斥事件

they cannot occur simultaneously. In set notation, their intersection is the empty set: A ∩ B = Ø. Example: Drawing a heart and a spade from a deck of cards in one draw.

若两个事件不能同时发生，则它们互斥。用集合符号表示，它们的交集为空集：A ∩ B = Ø。示例：从一副纸牌中抽一张，抽出红心和黑桃。

When using Venn diagrams, mutually exclusive events are shown as non-overlapping circles. The probability of the union is simply the sum of individual probabilities.

使用韦恩图时，互斥事件表示为不重叠的圆。并集的概率就是各事件概率之和。

---

5. Independent Events | 独立事件

the occurrence of one does not affect the probability of the other. Formal condition: P(A ∩ B) = P(A) × P(B), or P(A|B) = P(A).

若事件 A 和 B 的发生互不影响，则它们独立。形式条件：P(A ∩ B) = P(A) × P(B)，或 P(A|B) = P(A)。

ls, such as repeated coin tosses. Be careful: independence differs from mutual exclusivity—independent events can occur together (unless P(A)=0 or P(B)=0).

独立性是许多概率模型（如重复抛硬币）的关键假设。注意：独立与互斥不同——独立事件可以同时发生（除非 P(A)=0 或 P(B)=0）。

---

6. Conditional Probability | 条件概率

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

条件概率 P(A|B) 是在 B 已发生的条件下 A 发生的概率。定义为 P(A|B) = P(A ∩ B) / P(B)，前提是 P(B) > 0。

Conditional probability enables us to update probabilities based on new information. For example, if we know a card drawn is red, the probability it is a heart becomes 1/2 (since half the red cards are hearts).

条件概率使我们能根据新信息更新概率。例如，若已知抽到的牌是红色，则它是红心的概率变为 1/2（因为红牌中有一半是红心）。

---

广告 / Sponsored

7. Probability Tree Diagrams | 概率树图

Tree diagrams are useful for representing sequential events with probabilities on branches. Each branch probability is conditional on the previous event. To find the probability of a complete path, multiply along the branches.

树图有助于表示带有分支概率的序列事件。每条分支概率是以前一事件为条件的。要计算完整路径的概率，沿分支相乘。

The sum of probabilities for all final outcomes equals 1. Trees can include with/without replacement scenarios, which affect whether probabilities remain constant or change.

所有最终结果概率之和等于 1。树图可包含放回/不放回情形，这会影响概率是否保持不变。

---

8. Venn Diagrams and Set Notation | 韦恩图与集合符号

Venn diagrams visually display sets and their relationships: union (∪), intersection (∩), complement (A'). They are particularly helpful for solving probability problems involving overlapping events.

韦恩图直观地展示集合及其关系：并集（∪）、交集（∩）、补集（A'）。它们在解决涉及重叠事件的概率问题时尤为有用。

In IB exams, you may be asked to fill probabilities into a Venn diagram or to find probabilities of combined events. Always start by placing the intersection probability, then work outward.

在 IB 考试中，你可能需要将概率填入韦恩图，或求复合事件的概率。始终先从交集概率入手，然后向外推导。

---

9. Bayes' Theorem | 贝叶斯定理

Bayes' Theorem connects P(A|B) and P(B|A): P(A|B) = [P(B| eversing conditional probabilities, often applied in medical testing and machine learning.

贝叶斯定理将 P(A|B) 与 P(B|A) 联系起来：P(A|B) = [P(B|A) × P(A)] / P(B)。它在反转条件概率时极为有用，常应用于医学检测和机器学习。

A typical problem: Given the sensitivity and specificity of a test and the disease prevalence, find the probability that a person actually has the disease given a positive test result.

典型问题：给定检测的灵敏度和特异度以及疾病流行率，求检测呈阳性者实际患病的概率。

---

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

lists each possible value x with its P(X = x). The sum of all probabilities must equal 1.

离散随机变量 X 为随机实验的结果赋予数值。其概率分布列出每个可能的 x 值及 P(X = x)。所有概率之和必须等于 1。

ents the long-run average. The variance Var(X) = E((X - μ)²) = Σ (x - μ)²·P(X = x) = E(X²) – [E(X)]².

期望值（均值）E(X) = Σ x·P(X = x)，代表长期平均值。方差 Var(X) = E((X - μ)²) = Σ (x - μ)²·P(X = x) = E(X²) – [E(X)]²。

---

11. Binomial Distribution (Core) | 二项分布（核心）

rials with only two outcomes (success/failure) and constant probability p of success. If X ~ B(n, p), then P(X = r) = C(n, r) p^r (1 – p)^{n – r}, where C(n, r) = n!/(r!(n–r)!).

二项实验由 n 次独立试验组成，每次只有两种结果（成功/失败），且成功概率 p 恒定。若 X ~ B(n, p)，则 P(X = r) = C(n, r) p^r (1 – p)^{n – r}，其中 C(n, r) = n!/(r!(n–r)!)。

For a binomial random variable, E(X) = np, Var(X) = np(1 – The calculator function binompdf/binomcdf or spreadsheets can compute these probabilities.

对于二项随机变量，E(X) = np，Var(X) = np(1 – p)。可使用计算器函数 binompdf 或电子表格计算这些概率。

---

12. Normal Distribution Introduction | 正态分布简介

The normal distribution is a continuous probability distribution with a bell-shaped curve, defined by its mean μ and standard deviation σ. The total area under the curve is 1. It models many natural phenomena.

正态分布是一种连续概率分布，曲线呈钟形，由其均值 μ 和标准差 σ 定义。曲线下的总面积为 1。它可模拟许多自然现象。

In IB Core, you may need to use the standard normal distribution (Z ~ N(0,1)) and convert values using z = (x – μ)/σ. Probabilities are found using tables or calculator invNorm functions.

在 IB 核心课程中，你可能需要使用标准正态分布（Z ~ N(0,1)）并通过 z = (x – μ)/σ 进行转换。概率可使用表格或计算器 invNorm 功能求得。

---

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）