📚 AS Mathematics: Probability Revision Guide | AS 数学：概率考点精讲
Welcome to our comprehensive revision guide for probability in AS Mathematics. This article covers essential topics including basic probability, mutually exclusive and independent events, conditional probability, tree diagrams, Venn diagrams, discrete random variables, and expected values. Whether you are following the CAIE or Edexcel syllabus, these concepts form the foundation of your statistics knowledge. Work through each section carefully and test yourself with the examples provided.
欢迎阅读我们的AS数学概率综合考点精讲。本文涵盖基本概率、互斥事件与独立事件、条件概率、树图、维恩图、离散随机变量以及期望值等核心内容。无论你学习的是CAIE还是Edexcel课程，这些概念都是概率与统计的基础。请仔细研读每个部分，并利用给出的示例进行自我检测。
1. Basic Probability Concepts | 基本概念
Probability is a measure of the likelihood that an event occurs. It is always a number between 0 and 1 inclusive. The sample space S is the set of all possible outcomes of a random experiment. For an event A, if all outcomes are equally likely, the probability of A is given by P(A) = number of favourable outcomes for A / total number of outcomes in S, often written as P(A) = n(A) / n(S).
概率是对事件发生可能性的度量，其值总是在0到1之间（含0和1）。样本空间 S 是随机试验所有可能结果的集合。对于事件 A，若所有结果等可能，则 A 的概率为 P(A) = A 的有利结果数 / S 中的总结果数，常写作 P(A) = n(A) / n(S)。
The probability of an impossible event is 0, written P(∅) = 0. The probability of the whole sample space is 1, P(S) = 1. For any event A, 0 ≤ P(A) ≤ 1.
不可能事件的概率为0，记作 P(∅) = 0。整个样本空间的概率为1，P(S) = 1。对于任意事件 A，有 0 ≤ P(A) ≤ 1。
The complement of A, denoted by A’ or Ac, consists of all outcomes not in A. Its probability is P(A’) = 1 – P(A).
A的补集，记作 A’ 或 Aᶜ，包含所有不在A中的结果，其概率为 P(A’) = 1 – P(A)。
2. Mutually Exclusive Events | 互斥事件
Two events A and B are mutually exclusive (disjoint) if they cannot happen at the same time. This means their intersection is empty: A ∩ B = ∅, so P(A ∩ B) = 0.
若两个事件 A 与 B 不能同时发生，则称它们互斥（不相容）。这意味着它们的交集为空：A ∩ B = ∅，因此 P(A ∩ B) = 0。
For mutually exclusive events, the probability that either A or B occurs is simply the sum of their individual probabilities: P(A ∪ B) = P(A) + P(B).
对于互斥事件，A或B发生的概率就是它们各自概率的和：P(A ∪ B) = P(A) + P(B)。
This addition rule can be extended to more than two mutually exclusive events. If A₁, A₂, …, Aₙ are pairwise mutually exclusive, then P(A₁ ∪ A₂ ∪ … ∪ Aₙ) = P(A₁) + P(A₂) + … + P(Aₙ).
这一加法法则可以推广至多个互斥事件。若 A₁, A₂, …, Aₙ 两两互斥，则 P(A₁ ∪ A₂ ∪ … ∪ Aₙ) = P(A₁) + P(A₂) + … + P(Aₙ)。
3. Independent Events | 独立事件
Two events A and B are independent if the occurrence or non‑occurrence of one does not affect the probability of the other. The formal definition is: P(A ∩ B) = P(A) × P(B).
若一个事件的发生或不发生不影响另一个事件的概率，则称事件 A 与 B 独立。其正式定义为：P(A ∩ B) = P(A) × P(B)。
Do not confuse independence with mutual exclusivity. Mutually exclusive events with non‑zero probabilities can never be independent because if one occurs the other cannot, so P(A|B) = 0 while P(A) ≠ 0.
不要混淆独立与互斥。概率非零的互斥事件绝不可能独立，因为如果一件发生则另一件必不发生，于是 P(A|B) = 0 而 P(A) ≠ 0。
An equivalent condition for independence is that P(A|B) = P(A) and P(B|A) = P(B), whenever the conditional probabilities are defined.
独立性的等价条件是：只要条件概率有定义，便有 P(A|B) = P(A) 且 P(B|A) = P(B)。
4. Conditional Probability | 条件概率
Conditional probability quantifies the likelihood of event A given that event B has already occurred. It is defined as P(A|B) = P(A ∩ B) / P(B), provided P(B) > 0.
条件概率量化了在事件 B 已经发生的条件下事件 A 发生的可能性。其定义为 P(A|B) = P(A ∩ B) / P(B)，其中 P(B) > 0。
Similarly, P(B|A) = P(A ∩ B) / P(A). Rearranging gives the general multiplication rule: P(A ∩ B) = P(A) × P(B|A) = P(B) × P(A|B).
类似地，P(B|A) = P(A ∩ B) / P(A)。移项可得一般乘法法则：P(A ∩ B) = P(A) × P(B|A) = P(B) × P(A|B)。
When solving problems, identify the condition and the event of interest, then use the formula. Drawing a tree diagram often helps visualize the sequential conditioning.
解题时，先明确条件和目标事件，再代入公式。绘制树图通常有助于将顺序条件关系可视化。
5. Probability Tree Diagrams | 概率树图
A probability tree diagram is a visual tool for multi‑stage experiments. Each branch represents a possible outcome at a given stage, labelled with its probability. Probabilities on the branches from a single node must sum to 1.
概率树图是处理多阶段试验的可视化工具。每个分支代表某一阶段的一个可能结果并标出其概率。从同一节点出发的各分支概率之和必须为1。
To find the probability of a particular path, multiply the probabilities along the branches. To find the probability of an event that consists of several paths, add the probabilities of those paths.
求某一路径的概率，就将路径上各分支的概率相乘。求由多个路径组成的事件的概率，则将这些路径的概率相加。
Tree diagrams are particularly useful when probabilities change after the first stage, e.g. when items are drawn without replacement. The second‑stage probabilities are then conditional on the first outcome.
当概率在第一阶段后发生改变时，树图尤为有用，例如不放回抽取。此时第二阶段的概率是以第一阶段结果为条件的条件概率。
Example: A bag contains 3 red and 2 blue balls. Two balls are drawn without replacement. The tree shows four paths: RR, RB, BR, BB, with calculated probabilities. The probability of drawing at least one red is P(RR) + P(RB) + P(BR) = 1 – P(BB).
示例：袋中有3个红球和2个蓝球，不放回地抽取两个球。树图显示四条路径：RR、RB、BR、BB，并计算出相应的概率。至少抽到一个红球的概率为 P(RR)+P(RB)+P(BR) = 1 – P(BB)。
6. Venn Diagrams and Set Notation | 维恩图与集合符号
Venn diagrams represent events as regions inside a rectangle that denotes the sample space S. Overlapping regions show intersections, while combined regions show unions. They help visualise probabilities and apply set operations.
维恩图将事件表示为矩形（代表样本空间 S）内的区域。重叠区域表示交集，合并的区域表示并集。它们有助于将概率问题可视化并运用集合运算。
Key set notation: A ∪ B (union, either A or B or both), A ∩ B (intersection, both A and B), A’ (complement, not A), and A \ B (difference, A but not B). The number of outcomes in a region is often written as n(A) and probability as P(A) = n(A) / n(S).
关键集合符号：A ∪ B（并集，A 或 B 或两者同时发生），A ∩ B（交集，A 与 B 同时发生），A’（补集，非 A），以及 A \ B（差集，属于 A 但不属于 B）。某区域的元素个数常写作 n(A)，概率写作 P(A) = n(A) / n(S)。
From a Venn diagram, you can read off intersections and unions to apply the addition rule: P(A ∪ B) = P(A) + P(B) – P(A ∩ B), which corrects for double‑counting the overlap.
通过维恩图可以读出交集与并集的信息，从而应用加法法则：P(A ∪ B) = P(A) + P(B) – P(A ∩ B)，该式纠正了重叠部分被重复计算的问题。
7. The Addition Rule | 加法法则
The general addition rule for any two events A and B is: P(A ∪ B) = P(A) + P(B) – P(A ∩ B). This formula works whether or not the events are mutually exclusive.
对于任意两个事件 A 和 B，一般的加法法则为：P(A ∪ B) = P(A) + P(B) – P(A ∩ B)。无论事件是否互斥，该公式都成立。
If A and B are mutually exclusive, then P(A ∩ B) = 0 and the rule reduces to P(A ∪ B) = P(A) + P(B). Always check whether the events can occur together before adding probabilities blindly.
若 A 与 B 互斥，则 P(A ∩ B) = 0，公式简化为 P(A ∪ B) = P(A) + P(B)。在盲目相加概率之前，务必先检查事件是否能同时发生。
When three events are involved, the addition rule extends to P(A ∪ B ∪ C) = P(A) + P(B) + P(C) – P(A ∩ B) – P(A ∩ C) – P(B ∩ C) + P(A ∩ B ∩ C). At AS level you usually only need the two‑event version.
当涉及三个事件时，加法法则扩展为 P(A ∪ B ∪ C) = P(A) + P(B) + P(C) – P(A ∩ B) – P(A ∩ C) – P(B ∩ C) + P(A ∩ B ∩ C)。在AS阶段通常只需掌握两个事件的情况。
8. The Multiplication Rule | 乘法法则
The general multiplication rule links conditional probabilities: P(A ∩ B) = P(A) × P(B|A) = P(B) × P(A|B). It is used when combining probabilities from dependent stages.
一般乘法法则将条件概率联系起来：P(A ∩ B) = P(A) × P(B|A) = P(B) × P(A|B)。该式用于组合来自相依阶段的概率。
If A and B are independent, the rule simplifies to P(A ∩ B) = P(A) × P(B). This is both the definition of independence and a quick way to compute joint probabilities for independent events.
若 A 和 B 独立，该法则简化为 P(A ∩ B) = P(A) × P(B)。这既是独立性的定义，也是计算独立事件联合概率的快捷方法。
A typical application: in a sequence of trials, such as rolling a die twice, the probability of getting a six on the first roll and an odd number on the second is P(6) × P(odd) = (1/6) × (1/2) = 1/12, because the rolls are independent.
典型应用：在一系列试验中，例如掷两次骰子，第一次得6且第二次得奇数的概率为 P(6) × P(odd) = (1/6) × (1/2) = 1/12，因为两次投掷独立。
When events are not independent, you must use the appropriate conditional probability. For example, drawing two cards without replacement requires adjusting the probability for the second draw based on the first outcome.
当事件不独立时，必须使用相应的条件概率。例如，不放回地抽两张牌，需要根据第一张的结果调整第二张的概率。
9. Discrete Random Variables | 离散随机变量
A discrete random variable X takes a countable number of possible values, each with an assigned probability. The probability distribution of X lists all possible values xᵢ and their probabilities pᵢ = P(X = xᵢ), satisfying Σ pᵢ = 1.
离散随机变量 X 可取可数个值，每个值都有对应的概率。X 的概率分布列出所有可能取值 xᵢ 及其概率 pᵢ = P(X = xᵢ)，且满足 Σ pᵢ = 1。
The distribution can be presented in a table:
| x | 0 | 1 | 2 |
|---|---|---|---|
| P(X=x) | 0.25 | 0.5 | 0.25 |
分布可用表格表示，如上所示。
To find the probability that X falls in a certain range, sum the probabilities for the values in that range. For instance, P(X ≥ 1) = 1 – P(X=0). Always check that the sum of all probabilities is exactly 1.
求 X 落在某个范围内的概率，只需将相应取值的概率相加。例如，P(X ≥ 1) = 1 – P(X=0)。务必检查所有概率之和恰好为1。
10. Expected Value and Variance of a Random Variable | 随机变量的期望与方差
The expected value (mean) of a discrete random variable X is denoted E(X) or μ, and is calculated as E(X) = Σ xᵢ pᵢ, where xᵢ are the values and pᵢ are their probabilities. It represents the long‑run average outcome.
离散随机变量 X 的期望值（均值）记作 E(X) 或 μ，计算公式为 E(X) = Σ xᵢ pᵢ，其中 xᵢ 为取值，pᵢ 为相应的概率。它代表了长期重复试验中的平均结果。
It is not necessary for the expected value to be a possible value of X. For a fair six‑sided die, E(X) = 1×(1/6)+2×(1/6)+…+6×(1/6) = 3.5, even though you can never roll a 3.5.
期望值不一定是 X 的一个可能取值。对于一枚公平的六面骰子，E(X) = 1×(1/6)+2×(1/6)+…+6×(1/6) = 3.5，尽管你永远掷不出3.5。
The variance measures the spread of the distribution. It is given by Var(X) = E(X²) – [E(X)]², where E(X²) = Σ xᵢ² pᵢ. The standard deviation is the square root of the variance.
方差衡量分布的离散程度，计算公式为 Var(X) = E(X²) – [E(X)]²，其中 E(X²) = Σ xᵢ² pᵢ。标准差是方差的平方根。
For example, if X has the distribution: x = 1,2,3 with probabilities 0.2, 0.3, 0.5, then E(X)=1×0.2+2×0.3+3×0.5=2.3, E(X²)=1×0.2+4×0.3+9×0.5=5.9, so Var(X)=5.9 – 2.3² = 0.61.
例如，若 X 的分布为：x=1,2,3，概率分别为0.2, 0.3, 0.5，则 E(X)=1×0.2+2×0.3+3×0.5=2.3，E(X²)=1×0.2+4×0.3+9×0.5=5.9，故 Var(X)=5.9 – 2.3² = 0.61。
11. Combining Concepts: Solving Probability Problems | 综合应用：解概率问题
In AS examinations, questions often require you to combine several concepts. For instance, you may be given a real‑life scenario, need to construct a tree diagram, apply conditional probability to find a path probability, and then use the addition rule to find ‘at least one’ probabilities.
在AS考试中，题目常需你综合运用多个概念。例如，给出一个实际情景，需要构造树图，利用条件概率求出路径概率，再通过加法法则计算“至少一个”的概率。
Another common type is to interpret data from a two‑way table or Venn diagram and calculate probabilities like P(A|B) or P(A∪B). Practice recognising which rule to apply: look for key words such as ‘given that’ (conditional), ‘and’ (intersection), ‘or’ (union), ‘replace’/’independent’ vs ‘without replacement’/’dependent’.
另一类常见题型是解读双向表或维恩图数据，并计算如 P(A|B) 或 P(A∪B) 的概率。练习识别该用哪条法则：留意关键词，如“已知”（条件概率）、“和”（交集）、“或”（并集）、“放回”/“独立”与“不放回”/“相依”。
Always set out your working clearly, define events, and write down the formula you are using. This helps avoid mistakes and earns method marks even if the final answer is slightly off.
解题时务必清晰地展示过程，定义事件，并写出所使用的公式。这有助于避免错误，且即使最终答案略有偏差也能获得方法分。
12. Common Pitfalls and Tips | 常见易错点与提示
Pitfall 1: Using the addition rule without subtracting the intersection when events are not mutually exclusive. Always check if events can occur together.
易错点1：当事件不互斥时，使用加法法则却没有减去交集。务必检查事件是否能同时发生。
Pitfall 2: Confusing P(A|B) and P(B|A). They are generally not equal. P(A|B) = P(A∩B)/P(B), while P(B|A) = P(A∩B)/P(A).
易错点2：混淆 P(A|B) 与 P(B|A)。它们通常不相等。P(A|B) = P(A∩B)/P(B)，而 P(B|A) = P(A∩B)/P(A)。
Pitfall 3: Assuming events are independent without justification. Unless the question explicitly states independence or the process involves replacement, probabilities usually change.
易错点3：未经证实就假设事件独立。除非题目明确指出独立或过程涉及放回，否则概率通常会发生改变。
Tip: Draw a tree or Venn diagram whenever you can. Visual representation reduces errors in complicated chains of probability.
提示：尽可能画树图或维恩图。可视化表示能减少复杂概率链中的错误。
Tip: When calculating expected value, make a table listing xᵢ and pᵢ, then compute xᵢ pᵢ step‑wise to minimise arithmetic mistakes. The same applies to variance calculations with xᵢ² pᵢ.
提示：计算期望值时，可列出 xᵢ 和 pᵢ 的表格，然后逐步计算 xᵢ pᵢ 以减少算术错误。对于方差计算中的 xᵢ² pᵢ 也应如此。
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