IB Mathematics: Core Concepts of Probability Theory | IB数学：概率论核心概念全梳理

📚 IB Mathematics: Core Concepts of Probability Theory | IB数学：概率论核心概念全梳理

Probability is the mathematical language of uncertainty. This article reviews the core ideas that IB Mathematics students must master, from sample spaces and conditional probability to binomial and normal distributions.

概率论是描述不确定性的数学语言。本文系统梳理IB数学学生必须掌握的核心概念，从样本空间、条件概率到二项分布和正态分布。

1. Sample Space and Events | 样本空间与事件

A random experiment is a process whose outcome cannot be predicted with certainty. The sample space, often denoted Ω, is the set of all possible outcomes. Each individual outcome is called a sample point. An event is any subset of Ω, usually written with capital letters such as A or B.

随机试验是指结果无法预先确定的过程。样本空间通常记为 Ω，是试验所有可能结果组成的集合。每一个单独结果称为样本点。事件是 Ω 的任意子集，通常用 A、B 等大写字母表示。

Sample space Ω | 样本空间 Ω: the set of all possible outcomes. For a fair six-sided die, Ω = {1, 2, 3, 4, 5, 6}.

样本空间 Ω: 所有可能结果的集合。例如均匀六面骰子，Ω = {1, 2, 3, 4, 5, 6}。
Event | 事件: a subset of the sample space. The event “rolling an even number” is A = {2, 4, 6}.

事件: 样本空间的子集。例如“掷出偶数”这一事件为 A = {2, 4, 6}。
Empty event ∅ | 空事件 ∅: the event with no outcomes; it is impossible.

空事件 ∅: 不含任何结果的事件，即不可能事件。
Certain event Ω | 必然事件 Ω: the whole sample space; it is guaranteed to occur.

必然事件 Ω: 整个样本空间，代表必然发生的事件。

When two dice are rolled, the sample space contains 36 ordered pairs. The event “the sum is 7” is {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)}.

掷两枚骰子时，样本空间包含 36 个有序数对。事件“点数之和为 7”为 {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)}。

2. Probability Basics and Axioms | 概率基础与公理

A probability is a number assigned to an event that describes how likely it is to occur. The Kolmogorov axioms are the foundation of probability theory.

概率是赋予事件的一个数，用来刻画该事件发生的可能性大小。柯尔莫哥洛夫公理是概率论的基础。

Axiom 1 | 公理1: For any event A, 0 ≤ P(A) ≤ 1.

公理1: 对任意事件 A，有 0 ≤ P(A) ≤ 1。
Axiom 2 | 公理2: P(Ω) = 1. The probability of the entire sample space is 1.

公理2: P(Ω) = 1，即整个样本空间的概率为 1。
Axiom 3 | 公理3: If A₁, A₂, … are mutually exclusive, then P(∪ Aᵢ) = Σ P(Aᵢ).

公理3: 若 A₁, A₂, … 两两互斥，则 P(∪ Aᵢ) = Σ P(Aᵢ)。

From these axioms, the addition rule follows: P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

由这些公理可推导出加法公式：P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。

If all outcomes in a finite sample space are equally likely, then P(A) = n(A) / n(Ω), where n(A) means the number of outcomes in event A.

若有限样本空间中每个结果等可能，则 P(A) = n(A) / n(Ω)，其中 n(A) 表示事件 A 中结果的个数。

3. Complementary and Mutually Exclusive Events | 对立事件与互斥事件

The complement of event A, written A’, contains all outcomes not in A. Because A and A’ partition the sample space, P(A’) = 1 − P(A).

事件 A 的对立事件记为 A’，包含所有不在 A 中的结果。由于 A 与 A’ 构成样本空间的一个划分，因此 P(A’) = 1 − P(A)。

Two events are mutually exclusive if they cannot happen at the same time. In set notation, A ∩ B = ∅, so P(A ∩ B) = 0.

两个事件互斥是指它们不可能同时发生。用集合语言表示即 A ∩ B = ∅，因此 P(A ∩ B) = 0。

For mutually exclusive events, the addition rule simplifies to P(A ∪ B) = P(A) + P(B).

对于互斥事件，加法公式简化为 P(A ∪ B) = P(A) + P(B)。

It is important not to confuse mutually exclusive events with independent events. If two events are mutually exclusive, then knowing one occurs tells you the other does not.

注意不要将互斥事件与独立事件混淆。若两事件互斥，则知道其中一个发生就说明另一个不会发生。

4. Independent Events and Conditional Probability | 独立事件与条件概率

The conditional probability of A given B, written P(A|B), is the probability that A occurs knowing that B has already occurred.

在 B 已发生的前提下 A 发生的条件概率记为 P(A|B)，表示已知 B 发生后 A 发生的概率。

P(A|B) = P(A ∩ B) / P(B), where P(B) > 0

P(A|B) = P(A ∩ B) / P(B)，其中 P(B) > 0

Rearranging gives the multiplication rule: P(A ∩ B) = P(A|B) × P(B).

变形可得乘法公式：P(A ∩ B) = P(A|B) × P(B)。

Events A and B are independent if the occurrence of one does not affect the probability of the other. That is, P(A|B) = P(A) or P(B|A) = P(B), which leads to:

事件 A 与 B 独立是指一个事件的发生不影响另一个事件发生的概率，即 P(A|B) = P(A) 或 P(B|A) = P(B)，因此有：

A and B are independent ⇔ P(A ∩ B) = P(A) × P(B)

A 与 B 独立 ⇔ P(A ∩ B) = P(A) × P(B)

Example: draw a card from a standard deck, then replace it. Let A be “draw a heart” and B be “draw an Ace”. These are independent if the card is replaced, so P(A ∩ B) = P(A)P(B) = 1/4 × 1/13 = 1/52.

例如：从一副标准扑克牌中有放回地抽一张。设 A 为“抽到红心”，B 为“抽到A”。因为有放回，A 与 B 独立，所以 P(A ∩ B) = P(A)P(B) = 1/4 × 1/13 = 1/52。

5. Probability Trees and Venn Diagrams | 概率树与韦恩图

Probability trees are useful for multistage experiments. Each branch is labelled with a conditional probability, and probabilities along a path are multiplied together.

概率树适用于多阶段试验。每条分支标注条件概率，沿路径的概率相乘。

Rule 1 | 规则1: The probabilities on branches from the same node sum to 1.

规则1: 从同一节点出发的所有分支概率之和为 1。
Rule 2 | 规则2: The probability of an outcome is the product of probabilities along the path.

规则2: 某一结果的概率等于沿路径各概率之积。
Rule 3 | 规则3: If a final event can be reached through several paths, add the path probabilities.

规则3: 若同一最终事件可通过多条路径到达，则将各路径概率相加。

Example: a box has 3 green and 2 yellow marbles. Two marbles are drawn without replacement. The probability that both are green is P(first green) × P(second green | first green) = 3/5 × 2/4 = 3/10.

例：盒中有 3 个绿球和 2 个黄球，不放回地取两次。两次都取到绿球的概率为 P(第一次绿) × P(第二次绿 | 第一次绿) = 3/5 × 2/4 = 3/10。

Venn diagrams display events as regions inside the sample space. They help visualise P(A ∩ B), P(A ∪ B), P(A’) and P(A|B).

韦恩图将事件表示为样本空间内的区域，有助于直观理解 P(A ∩ B)、P(A ∪ B)、P(A’) 与 P(A|B)。

6. Bayes’ Theorem | 贝叶斯定理

Bayes’ theorem links P(A|B) to P(B|A). It is especially powerful in diagnostic testing and decision making.

贝叶斯定理将 P(A|B) 与 P(B|A) 联系起来，在医学检测和决策分析中尤其重要。

P(A|B) = P(B|A) × P(A) / P(B)

If the sample space is split into A and A’, then P(B) = P(B|A)P(A) + P(B|A’)P(A’), giving the extended form:

若样本空间被分割为 A 与 A’，则 P(B) = P(B|A)P(A) + P(B|A’)P(A’)，于是得到扩展形式：

P(A|B) = [P(B|A)P(A)] / [P(B|A)P(A) + P(B|A’)P(A’)]

Example: a disease affects 1% of a population. A test is 99% sensitive, so P(positive | disease) = 0.99, and 95% specific, so P(negative | healthy) = 0.95, which means P(positive | healthy) = 0.05. If a patient tests positive, the probability they actually have the disease is:

例：某种疾病在人群中的患病率为 1%。某检测的灵敏度为 99%，即 P(阳性 | 患病) = 0.99；特异度为 95%，即 P(阴性 | 健康) = 0.95，因此 P(阳性 | 健康) = 0.05。若某人检测结果为阳性，其真正患病的概率为：

P(Disease | Positive) = (0.99 × 0.01) / (0.99 × 0.01 + 0.05 × 0.99) ≈ 0.167

P(患病 | 阳性) = (0.99 × 0.01) / (0.99 × 0.01 + 0.05 × 0.99) ≈ 0.167

The surprisingly low result occurs because false positives from healthy people dominate the denominator.

这个看似较低的结果是因为健康人群中的假阳性在分母中占据主导。

7. Discrete Probability Distributions | 离散概率分布

A discrete random variable X takes values that can be listed, such as whole numbers. The probability distribution of X lists each possible value x with its probability P(X = x).

离散随机变量 X 的取值可以一一列出，例如整数。X 的概率分布列出每个可能取值 x 及其概率 P(X = x)。

For a valid discrete probability distribution:

一个有效的离散概率分布必须满足：

Condition 1 | 条件1: P(X = x) ≥ 0 for every value x.

条件1: 对每个取值 x，有 P(X = x) ≥ 0。
Condition 2 | 条件2: Σ P(X = x) = 1, where the sum is over all possible values.

条件2: 对所有可能取值求和，Σ P(X = x) = 1。

Example: for a fair six-sided die, the probability distribution of X is P(X = x) = 1/6 for x = 1, 2, 3, 4, 5, 6.

例如：掷均匀六面骰子，X 的概率分布为当 x = 1, 2, 3, 4, 5, 6 时，P(X = x) = 1/6。

8. Expectation and Variance | 期望与方差

The expectation μ = E(X) measures the centre of a probability distribution. It is the long-run average value of the random variable.

期望 μ = E(X) 度量概率分布的中心位置，是随机变量在长期试验中的平均值。

E(X) = Σ x P(X = x)

The variance measures the spread about the mean. A useful formula is:

方差度量随机变量围绕均值的离散程度。常用公式为：

Var(X) = E(X²) − [E(X)]²

The standard deviation is σ = √Var(X).

标准差为 σ = √Var(X)。

Key linear properties of expectation and variance are:

期望与方差的重要线性性质如下：

E(aX + b) = aE(X) + b for constants a and b.

E(aX + b) = aE(X) + b，其中 a、b 为常数。
Var(aX + b) = a² Var(X). Adding a constant does not change variance.

Var(aX + b) = a² Var(X)。加上常数不会改变方差。
If X and Y are independent: Var(X + Y) = Var(X) + Var(Y).

若 X 与 Y 独立: Var(X + Y) = Var(X) + Var(Y)。

9. Binomial Distribution | 二项分布

A binomial distribution models the number of successes in a fixed number n of independent trials, each with the same probability of success p.

二项分布用于描述固定次数 n 次独立试验中的成功次数，每次试验的成功概率 p 保持不变。

A situation must satisfy these conditions:

该情形必须满足以下条件：

Fixed number of trials: there are n identical trials.

试验次数固定: 共有 n 次相同试验。
Two outcomes: each trial is either a success or a failure.

两种结果: 每次试验只有成功或失败两种结果。

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IB Mathematics: Core Concepts of Probability Theory | IB数学：概率论核心概念全梳理