📚 A-Level Edexcel Mathematics: Probability Key Summary | A-Level Edexcel 数学：概率 考点精讲
Welcome to this focused revision guide on probability for A‑Level Edexcel Mathematics. This article consolidates essential concepts, formulas, and techniques required for the exam, from basic probability rules and tree diagrams to binomial and normal distributions. Each section provides English‑Chinese paired explanations to help you master the content with clarity. Whether you are revising for Paper 3 or strengthening your statistics foundation, use this as your quick‑reference companion.
欢迎阅读本篇 A‑Level Edexcel 数学概率考点精讲。本文浓缩了考试必备的核心概念、公式与方法，涵盖从基础概率运算、树图到二项分布与正态分布的全部要点。每个部分均采用中英文配对讲解，帮助你清晰理解、高效复习。无论你是为 Paper 3 冲刺，还是夯实统计基础，都可以将本文作为快速回顾的得力助手。
1. Basic Concepts of Probability | 概率基本概念
An experiment or trial produces outcomes. The set of all possible outcomes is called the sample space, denoted by S. An event event A, written P(A), is a measure of how likely A is to occur, satisfying 0 ≤ P(A) ≤ 1.
实验或试验会产生结果。所有可能结果的集合称为样本空间，记作 S。事件是样本空间的一个子集。事件 A 的概率，记作 P(A)，是 A 发生可能性的度量，满足 0 ≤ P(A) ≤ 1。
If S has n equally likely outcomes and event A contains m outcomes, then P(A) = m / n.
如果样本空间有 n 个等可能结果，事件 A 包含 m 个结果，则 P(A) = m / n。
The complement of A, denoted A', is the event that A does not occur, and P(A') = 1 – P(A).
事件 A 的补集，记作 A'，表示 A 不发生的事件，且满足 P(A') = 1 – P(A)。
2. Addition Rule and Mutually Exclusive Events | 加法定则与互斥事件
For any two events A and B, the addition rule states:
对于任意两个事件 A 和 B，加法定则可表示为：
P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
Events A and B are mutually exclusive if they cannot occur at the same time, i.e. A ∩ B = ∅. In that case, the intersection is zero and the rule simplifies to P(A ∪ B) = P(A) + P(B).
若事件 A 与 B 不能同时发生，即 A ∩ B = ∅，则称它们互斥。此时交集为空，公式简化为 P(A ∪ B) = P(A) + P(B)。
Always check whether events are mutually exclusive before applying the simpler addition rule.
在应用简化的加法定则之前，务必确认事件互斥。
3. Multiplication Rule and Independent Events | 乘法定则与独立事件
Two events A and B are independent if the occurrence of one does not affect the probability of the other. For independent events,
如果事件 A 与 B 中一个的发生不影响另一个的概率，则它们是独立的。对于独立事件，有
P(A ∩ B) = P(A) × P(B)
The general multiplication rule, which holds for any events, is P(A ∩ B) = P(A) × P(B|A) = P(B) × P(A|B).
通用的乘法定则适用于任何事件：P(A ∩ B) = P(A) × P(B|A) = P(B) × P(A|B)。
To test independence, you can compare P(A|B) with P(A); if they are equal, A and B are independent.
检验独立性时，可以比较 P(A|B) 与 P(A)；若相等，则 A 和 B 独立。
4. Conditional Probability | 条件概率
The probability of A given B is defined as
在事件 B 发生的条件下 A 的概率定义为
P(A|B) = P(A ∩ B) / P(B), provided P(B) > 0
This formula is fundamental for solving problems where prior information restricts the sample space. It is also the basis for probability trees and Bayes-type reasoning.
该公式是解决先验信息缩小样本空间问题的基石，也是树图和贝叶斯类推理的基础。
Rearrange the definition to find intersections: P(A ∩ B) = P(B) × P(A|B).
将定义变形即可求交集：P(A ∩ B) = P(B) × P(A|B)。
5. Probability Tree Diagrams | 概率树图
Tree diagrams help visualise multistage experiments and conditional probabilities. Each branch is labelled with a probability; the sum of probabilities at any branching point must equal 1. Multiply along a path to find the probability of a sequence of events (AND), and add probabilities of different paths when events are mutually exclusive (OR).
树图有助于直观表现多阶段试验和条件概率。每个分支标注概率；任一分支点的概率之和必须为 1。沿路径相乘得到连续事件（且）的概率；互斥路径相加得到（或）的概率。
Always write the outcome at the end of each path and check that the total of all final probabilities equals 1. Tree diagrams are especially useful for conditional probability questions where the second set of branches shows probabilities given the outcome of the first stage.
务必在每个路径末尾写出结果，并核对所有最终概率之和为 1。当第二级分支表示给定第一阶段结果的概率时，树图对条件概率问题尤为有用。
6. Venn Diagrams and Set Notation | 维恩图与集合符号
Venn diagrams represent the sample space as a rectangle and events as circles inside it. Overlapping circles show intersections. You should be comfortable with notation: A ∩ B (intersection), A ∪ B (union), A' (complement), and n(A) for the number of elements in A.
维恩图用矩形表示样本空间，内部圆形表示事件。圆的交叠表示交集。应熟练使用符号：A ∩ B（交集），A ∪ B（并集），A'（补集），以及 n(A) 表示集合 A 中元素个数。
by considering the number of outcomes in each region divided by the total. The formula P(A ∪ B) = P(A) + P(B) – P(A ∩ B) is directly related to the overlapping regions.
根据维恩图或集合符号，可通过各区域元素个数除以总数来求概率。公式 P(A ∪ B) = P(A) + P(B) – P(A ∩ B) 直接对应交叠区域的关系。
7. Discrete Random Variables and Probability Distributions | 离散随机变量与概率分布
₁, p₂, ... . The probability distribution must satisfy Σ pᵢ = 1 and 0 ≤ pᵢ ≤ 1 for each i.
离散随机变量 X 取可数个值，如 x₁, x₂, ...，对应概率为 p₁, p₂, ...。概率分布必须满足 Σ pᵢ = 1，且对于每个 i 都有 0 ≤ pᵢ ≤ 1。
Below is an example of a simple probability distribution:
以下是一个简单概率分布的示例：
| x | 1 | 2 | 3 |
|---|---|---|---|
| P(X=x) | 0.2 | 0.5 | 0.3 |
that 0.2 + 0.5 + 0.3 = 1. If a value is missing, you can find it using the sum condition.
检验 0.2 + 0.5 + 0.3 = 1。若缺少某个值，可利用概率和条件求出。
8. Expected Value and Variance | 期望与方差
For a discrete random variable X with values xᵢ and probabilities pᵢ, the expected value (mean) is
对于取值为 xᵢ、概率为 pᵢ 的离散随机变量 X，期望（均值）为
E(X) = μ = Σ xᵢ pᵢ
The variance is
方差为
Var(X) = Σ (xᵢ – μ)² pᵢ = Σ xᵢ² pᵢ – μ²
= a² Var(X). Note that adding a constant shifts the mean but does not affect the variance.
对于线性函数，有如下法则：E(aX + b) = aE(X) + b，Var(aX + b) = a² Var(X)。注意加常数会使均值平移，但不影响方差。
The standard deviation is the square root of the variance: SD(X) = √Var(X).
标准差为方差的平方根：SD(X) = √Var(X)。
9. The Binomial Distribution | 二项分布
A binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success p. If X ~ B(n, p), then
二项分布适用于描述固定次数的独立试验中成功次数的概率模型，每次试验成功概率 p 不变。若 X ~ B(n, p)，则
P(X = k) = nCk pk (1 – p)n – k
Here nCk = n! / [k!(n – k)! the binomial coefficient. The expected value and variance are E(X) = np and Var(X) = np(1 – p).
其中 nCk = n! / [k!(n – k)!] 为二项式系数。期望和方差分别为 E(X) = np，Var(X) = np(1 – p)。
The conditions for a binomial distribution are: a fixed number of trials n; each trial is independent; there are only two outcomes (success/failure); and the probability of success p is constant. Check these conditions carefully in exam questions before applying the binomial model.
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