📚 A-Level CIE Maths: Probability Key Concepts Revision | A-Level CIE 数学：概率考点精讲
In CIE A-Level Mathematics, probability plays a central role in the statistics components S1 and S2. A solid grasp of probability rules, conditional reasoning, random variables, and the binomial distribution is essential for success. This revision guide breaks down every key topic with clear explanations, worked-style notation, and exam-focused strategies.
在 CIE A-Level 数学中，概率是统计部分 S1 和 S2 的核心内容。熟练掌握概率法则、条件推理、随机变量和二项分布是取得高分的关键。本考点精讲将每个重要主题拆解为清晰的讲解、标准化符号以及考试策略，助你高效备考。
1. Basic Probability Definitions and Notation | 基本概率定义与符号
Probability measures how likely it is for an event to occur, on a scale from 0 (impossible) to 1 (certain). The set of all possible outcomes is called the sample space S. For an event A with equally likely outcomes, P(A) = n(A) / n(S), where n(A) is the number of outcomes in A and n(S) is the total number of outcomes.
概率衡量事件发生的可能性，在 0（不可能）到 1（确定）的尺度上取值。所有可能结果的集合称为样本空间 S。对于等可能结果的事件 A，有 P(A) = n(A) / n(S)，其中 n(A) 是 A 中包含的结果数，n(S) 是总结果数。
The complementary rule states that P(A’) = 1 − P(A), where A’ is the event that A does not happen. We write intersection as A ∩ B (‘A and B’) and union as A ∪ B (‘A or B’). These building blocks appear in every probability problem.
互补事件法则为 P(A’) = 1 − P(A)，其中 A’ 表示 A 不发生的事件。我们将交集写作 A ∩ B（“A 且 B”），并集写作 A ∪ B（“A 或 B”）。这些基本构件会出现在每一道概率题中。
P(A) = n(A) / n(S) | P(A’) = 1 − P(A)
2. Mutually Exclusive and Independent Events | 互斥事件与独立事件
Mutually exclusive events cannot happen at the same time. This means P(A ∩ B) = 0, and the addition rule simplifies: P(A ∪ B) = P(A) + P(B). Two events are independent if the occurrence of one does not affect the probability of the other. The test for independence is P(A ∩ B) = P(A) × P(B).
互斥事件不能同时发生，因此 P(A ∩ B) = 0，加法法则简化为 P(A ∪ B) = P(A) + P(B)。两个事件如果彼此发生不影响对方的概率，则称为独立事件。独立性的检验条件是 P(A ∩ B) = P(A) × P(B)。
Students often confuse ‘mutually exclusive’ with ‘independent’; they are different concepts. Mutually exclusive events are never independent unless one event has zero probability. Always check the context and do not apply the simplified multiplication rule unless independence is given or proven.
学生常把“互斥”和“独立”混淆，但它们是不同的概念。除非某个事件概率为零，否则互斥事件绝不独立。除非题目明示或已证明独立性，否则切勿随意使用简化乘法法则。
3. Addition and Multiplication Rules | 加法法则与乘法法则
The general addition rule for any two events is P(A ∪ B) = P(A) + P(B) − P(A ∩ B). The subtraction avoids double‑counting the intersection. The general multiplication rule uses conditional probability: P(A ∩ B) = P(A) × P(B|A), which also connects independence and conditionality.
对任意两个事件，一般加法法则为 P(A ∪ B) = P(A) + P(B) − P(A ∩ B)，减去交集是为了避免重复计数。一般乘法法则引入条件概率：P(A ∩ B) = P(A) × P(B|A)，这将独立性与条件性质联系起来。
When solving multi‑step problems, identify whether events are dependent or independent. Use the addition rule when at least one event occurs, and the multiplication rule when both events must occur. Drawing a Venn diagram or a tree diagram often helps visualise the structure.
在解决多步骤问题时，先判断事件是相依还是独立。当至少一个事件发生时用加法法则，当两个事件都必须发生时用乘法法则。绘制韦恩图或树形图往往有助于理解结构。
4. Conditional Probability | 条件概率
Conditional probability P(B|A) is the probability of event B occurring given that event A has already occurred. The definition is P(B|A) = P(A ∩ B) / P(A), provided P(A) > 0. This formula allows us to update probabilities when new information becomes available.
条件概率 P(B|A) 是指在事件 A 已经发生的条件下事件 B 发生的概率。定义为 P(B|A) = P(A ∩ B) / P(A)，前提是 P(A) > 0。该公式使我们能根据新增信息更新概率。
A typical exam question might provide P(A), P(B) and P(A ∩ B), then ask for P(A|B) or P(B|A). Rearranging the definition gives P(A ∩ B) = P(B|A) × P(A), which is the foundation of tree‑diagram probabilities. Always write the conditional event after the vertical bar and keep the condition in the denominator.
典型的考题可能会给出 P(A)、P(B) 和 P(A ∩ B)，然后求 P(A|B) 或 P(B|A)。重新整理定义式可得 P(A ∩ B) = P(B|A) × P(A)，这正是树形图概率的基础。竖线后边的是条件事件，分母一定要对应条件事件的概率。
P(B|A) = P(A ∩ B) / P(A)
5. Tree Diagrams and Conditional Probability | 树形图与条件概率
Tree diagrams are powerful tools for handling conditional probabilities in multi‑stage experiments without replacement, or when successive events are affected by earlier outcomes. Each branch shows a conditional probability; multiply along a path to get the probability of that branch sequence. Add the probabilities of all paths that lead to the same final event to find its total probability.
树形图是在多阶段试验中处理条件概率的有力工具，尤其适用于不放回情境或后续事件受先前结果影响的情形。每条分支代表一个条件概率；沿一条路径相乘即可得到该路径序列的概率。将所有通向同一最终事件的路径概率相加，便得到该事件的总概率。
For example, in a two‑stage tree with branches A and A’, the probability of B can be found by the total probability formula: P(B) = P(B|A)·P(A) + P(B|A’)·P(A’). This is an application of the law of total probability and is frequently required in CIE S1 questions.
例如，在有两级分支 A 和 A’ 的树形图中，事件 B 的概率可由全概率公式求得：P(B) = P(B|A)·P(A) + P(B|A’)·P(A’)。这正是全概率公式的应用，在 CIE S1 考题中频繁出现。
6. Probability with Venn Diagrams and Two-way Tables | 韦恩图与双向表的概率应用
Venn diagrams visually represent intersections, unions, and complements. They make it easy to check the consistency of given probabilities and to find unknown regions. A two‑way table (contingency table) organises data into categories and allows direct reading of counts or probabilities; it is particularly useful when dealing with two categorical variables.
韦恩图能直观地表示交集、并集和补集，方便检查所给概率是否一致，并求出未知区域。双向表（列联表）将数据按类别组织，可以直接读取频次或概率；在处理两个分类变量时尤为有用。
In an exam, if you are given partial information, complete the Venn diagram or two‑way table systematically. Use P(A ∪ B) = P(A) + P(B) − P(A ∩ B) to fill missing overlaps. Remember that the total probability in the sample space must equal 1.
考试中遇到部分信息时，应有条理地完成韦恩图或双向表。填充缺失的重叠部分可使用 P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。始终记得样本空间的总概率必须等于 1。
7. Counting Principles: Permutations and Combinations | 计数原理：排列与组合
Many probability questions require counting the number of ways events can happen. The factorial n! is the product of all positive integers up to n. A permutation counts arrangements when order matters: ⁿPᵣ = n! / (n−r)!. A combination counts selections when order does not matter: ⁿCᵣ = n! / (r!(n−r)!).
许多概率题需要计算事件发生的不同方式数。阶乘 n! 是从 1 到 n 的所有正整数之积。排列是对顺序重要的安排计数：ⁿPᵣ = n! / (n−r)!。组合是对顺序不重要的选择计数：ⁿCᵣ = n! / (r!(n−r)!)。
| Permutations (order matters) | ⁿPᵣ = n!/(n−r)! |
| Combinations (order does not matter) | ⁿCᵣ = n!/(r!(n−r)!) |
Use combinations when selecting a committee, choosing cards, or forming a subset. Use permutations for arranging books, lining up people, or generating sequences where order is important. CIE frequently embeds combinatorial counts into probability calculations, so fluency with ⁿCᵣ and ⁿPᵣ is essential.
选择委员会、抽牌或构成子集时用组合。排列书籍、排队或在顺序重要的序列生成中用排列。CIE 经常将组合计数嵌入概率计算中，因此熟练运用 ⁿCᵣ 和 ⁿPᵣ 至关重要。
8. Discrete Random Variables and Probability Distributions | 离散随机变量与概率分布
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 or a function. Two conditions must always hold: 0 ≤ P(X = x) ≤ 1 for every value, and the sum of all probabilities Σ P(X = x) = 1.
离散随机变量 X 取有限个或可数个值，每个值对应一个概率。概率分布可用表格或函数表示。必须始终满足两个条件：对每个取值 0 ≤ P(X = x) ≤ 1，且所有概率之和 Σ P(X = x) = 1。
Building a probability distribution in an exam often involves using given data and previously computed probabilities. Once the distribution is complete, you can find probabilities of events like P(X > 2) by summing the relevant terms. Keep expressions exact when possible, or give decimal answers to the required accuracy.
考试中建立概率分布时，常常要用到给定数据和之前计算出的概率。分布完成后，可以通过累加相应项求出类似 P(X > 2) 的事件概率。尽量保留精确表达式，或按题目要求的精度给出小数答案。
9. Expectation and Variance of a Discrete Random Variable | 离散随机变量的期望与方差
The expectation E(X) (or mean μ) gives the long‑term average value. It is calculated as E(X) = Σ x · P(X = x). The variance Var(X) measures spread and is defined as Var(X) = E[(X − μ)²] = Σ (x − μ)² P(X = x). The shortcut formula Var(X) = E(X²) − [E(X)]² is often easier to compute.
期望 E(X)（或均值 μ）给出长期平均值，计算公式
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