📚 Mastering Probability for Edexcel A-Level Mathematics | 掌握爱德思A-Level数学概率
Probability is a cornerstone of the Edexcel A-Level Mathematics syllabus, underpinning key topics in Statistics and applied modules. This bilingual revision companion covers set notation, mutually exclusive and independent events, conditional probability, discrete random variables, expectation, variance, linear transformations, and the binomial distribution. Each section pairs a clear English explanation with a matching Chinese translation so that learners can revise both the mathematics and the specialist vocabulary.
概率是爱德思 A-Level 数学大纲的基石，支撑着统计与应用模块中的关键主题。本文提供全面的双语复习指南，涵盖集合符号、互斥与独立事件、条件概率、离散随机变量、期望、方差、线性变换和二项分布。每一小节均配有清晰的英文解释与对应的中文译文，帮助学习者同时复习数学内容与专业词汇。
1. Basic Concepts and Set Notation | 基本概念与集合符号
Probability measures how likely an event is to occur. The sample space S is the set of all possible outcomes of an experiment. An event A is a subset of S. When all outcomes are equally likely, the probability of A is given by the number of favourable outcomes divided by the total number of outcomes.
概率衡量事件发生的可能性。样本空间 S 是实验所有可能结果的集合。事件 A 是 S 的子集。当所有结果等可能时，A 的概率等于有利结果数除以总结果数。
For any event A, its complement A' means "not A". The probability of the complement is 1 minus the probability of A. The empty set ∅ represents an impossible event with probability 0, while the whole sample space S is a certain event with probability 1.
对于任何事件 A，其补集 A' 表示 "非 A"。补集的概率等于 1 减去 A 的概率。空集 ∅ 表示不可能事件，概率为 0；整个样本空间 S 是必然事件，概率为 1。
P(A) = n(A) / n(S) P(A') = 1 − P(A) P(∅) = 0 P(S) = 1
- Sample space S: the set of all possible outcomes.
样本空间 S：所有可能结果的集合。 - Event A: a subset of the sample space.
事件 A：样本空间的一个子集。 - Complement A': the event that A does not occur.
补集 A'：A 不发生的事件。
2. Mutually Exclusive Events and the Addition Rule | 互斥事件与加法法则
re mutually exclusive if they cannot happen at the same time. In set notation this means A ∩ B = ∅ probability that either A or B occurs is simply the sum of their individual probabilities.
如果两个事件 A 和 B 不能同时发生，则称它们互斥。用集合符号表示即 A ∩ B = ∅。对于互斥事件，A 或 B 发生的概率就是它们各自概率之和。
When events are not mutually exclusive, we must subtract the probability of the intersection to avoid double counting. This is called the general addition rule. It appears regularly in exam questions involving Venn diagrams and two-way tables.
当事件不互斥时，我们必须减去交集的概率以避免重复计算。这称为一般加法法则。它经常出现在涉及维恩图和双向表的考题中。
If A ∩ B = ∅, then P(A ∪ B) = P(A) + P(B)
In general, P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
- Mutually exclusive: A ∩ B = ∅, so P(A ∩ B) = 0.
互斥：A ∩ B = ∅，因此 P(A ∩ B) = 0。 - General addition rule: always subtract the overlap.
一般加法法则：始终减去重叠部分。
3. Independent Events and the Multiplication Rule | 独立事件与乘法法则
Events A and B are independent if the occurrence of one does not affect the probability of the other. For independent events, the probability that both occur is the product of their individual probabilities. Equivalently, the conditional probability of A given B equals the unconditional probability of A.
如果事件 A 和 B 中一个的发生不影响另一个的概率，则称它们独立。对于独立事件，两者同时发生的概率等于各自概率的乘积。等价地，已知 B 发生时 A 的条件概率等于 A 的无条件概率。
Students often confuse mutually exclusive events with independent events. They are different ideas. If two events are mutually exclusive and both have positive probability, they cannot be independent, because if one occurs, the other cannot occur.
学生常常混淆互斥事件与独立事件。它们是不同的概念。如果两个事件互斥且概率都为正，它们不可能是独立的，因为若一个发生，另一个就不可能发生。
P(A ∩ B) = P(A) × P(B) P(A | B) = P(A)
- Independent: P(A ∩ B) = P(A) × P(B).
独立：P(A ∩ B) = P(A) × P(B)。 - Mutually exclusive: P(A ∩ B) = 0.
互斥：P(A ∩ B) = 0。
4. Conditional Probability and Tree Diagrams | 条件概率与树状图
of an event occurring given that another event has already occurred. It is defined as P(A | B) = P(A ∩ nts, where we multiply probabilities along branches and add probabilities across branches.
条件概率是在另一个事件已经发生的条件下，某事件发生的概率。其定义为 P(A | B) = P(A ∩ B) / P(B)，其中 P(B) > 0。树状图对多阶段实验特别有用：沿分支相乘，跨分支相加。
On a tree diagram, the first set of branches represents the first stage, and the second set represents the second stage. Conditional probabilities appear on the second set of branches. Always check whether the question says "given that" or asks for an unconditional probability.
在树状图中，第一组分支代表第一阶段，第二组代表第二阶段。条件概率出现在第二组分支上。务必检查题目是否说 "已知" 或要求的是无条件概率。
P(A | B) = P(A ∩ B) / P(B) P(A ∩ B) = P(A | B) × P(B)
- Multiply along: P(A then B) = P(A) × P(B | A).
沿分支相乘：P(A 后 B) = P(A) × P(B | A)。 - Add across: combine all branches giving the required event.
跨分支相加：合并所有满足目标事件的分支。
5. Venn Diagrams and Two-Way Tables | 维恩图与双向表
Venn diagrams display the union, intersection and complement of events. Two-way tables display frequencies for two categorical variables. Both tools help convert word problems into clear probability statements. From a two-way table, conditional probabilities can be calculated directly using row or column totals.
维恩图展示事件的并集、交集和补集。双向表展示两个分类变量的频数。这两种工具都能帮助把文字题转化为清晰的概率表述。根据双向表，可以直接利用行总计或列总计计算条件概率。
For example, if a table shows the numbers of students who study Physics and Mathematics, then P(Physics | Mathematics) equals the number in both divided by the total number studying Mathematics. Always divide by the total of the condition, not by the grand total.
例如，若表格显示学习物理和数学的学生人数，则 P(物理 | 数学) 等于两者都学的人数除以学数学的总人数。始终除以条件所在组的总数，而不是总计。
P(A | B) = n(A ∩ B) / n(B)
- Venn diagram: useful for showing overlaps and complements.
维恩图：适合展示重叠与补集。 - Two-way table: useful for frequency data and conditional probability.
双向表：适合频数数据和条件概率。
6. Discrete Random Variables and Probability Distributions | 离散随机变量与概率分布
bility P(X = x). The sum of all probabilities must equal 1. A cumulative distribution function gives P(X ≤ x).
离散随机变量 X 取可数个值。其概率分布列出每个可能值 x 及其对应概率 P(X = x)。所有概率之和必须等于 1。累积分布函数给出 P(X ≤ x)。
Probability distributions can be presented in a table, as a formula, or as a graph. When a question gives a probability distribution with an unknown constant, use the fact that the probabilities sum to 1 to find it. Always check that no probability is negative.
概率分布可以用表格、公式或图形表示。当题目给出带未知常数的概率分布时，利用概率之和为 1 来求该常数。始终检查概率不为负。
Σ P(X = x) = 1 F(x₀) = P(X ≤ x₀)
| x | 0 | 1 | 2 |
| P(X = x) | 0.2 | 0.5 | 0.3 |
7. Expectation and Variance | 期望与方差
e results. The variance measures the spread of the distribution around the mean. A useful computational formula is Var(X) = E(X²) − [E(X)]².
期望 E(X) 是随机变量的长期平均值。计算方法是将每个值乘以对应概率后求和。方差衡量分布围绕均值的离散程度。一个常用的计算式是 Var(X) = E(X²) − [E(X)]²。
X). In exam questions, show the working for E(X²) clearly, because a small arithmetic error can affect both the variance and the standard deviation.
标准差是方差的平方根，即 SD(X) = √Var(X)。在考试中，要清楚写出 E(X²) 的计算过程，因为小的算术错误会影响方差和标准差。
E(X) = Σ x P(X = x)
Var(X) = Σ x² P(X = x) − [E(X)]² SD(X) = √Var(X)
- E(X): weighted mean of the distribution.
E(X)：分布的加权平均值。 - Var(X): average squared deviation from the mean.
Var(X)：偏离均值的平方的平均值。
8. Linear Transformations of Random Variables | 随机变量的线性变换
tion and variance change in predictable ways. Adding a constant b shifts the mean but does not change the spread. Multiplying by a constant a scales the mean and changes the variance by a factor of a².
如果随机变量 X 经过线性变换变为 Y = aX + b，则期望和方差会以可预测的方式变化。加上常数 b 会移动均值，但不改变离散程度。乘以常数 a 会缩放均值，并使方差变为原来的 a² 倍。
The standard deviation is affected by the absolute value of a, because standard deviation is a measure of spread and must be non-negative. These rules are frequently tested in questions about combining or scaling random variables.
标准差受 a 的绝对值影响，因为标准差是离散程度的度量，必须非负。这些规则在有关随机变量组合或缩放的题目中经常考查。
E(aX + b) = a E(X) + b Var(aX + b) = a² Var(X)
SD(aX + b) = |a| SD(X)
9. The Binomial Distribution | 二项分布
The binomial distribution models the number of successes in n independent trials, where each trial has exactly two outcomes and the probability of success p is constant. We write X ~ B(n, p). The probability of exactly r successes is given by the binomial formula.
二项分布描述 n 次独立试验中成功的次数，每次试验只有两种结果，且成功概率 p 恒定。记作 X ~ B(n, p)。恰好有 r 次成功的概率由二项公式给出。
p(1 − p). Before using the binomial model, always check the four conditions: fixed number of trials, independent trials, two possible outcomes, and constant probability of success.
二项分布的均值和方差分别为 E(X) = np 和 Var(X) = np(1 − p)。使用二项模型前，务必检查四个条件：试验次数固定、试验独立、只有两种结果、成功概率恒定。
X ~ B(n, p) P(X = r) = C(n, r) × pʳ × (1 − p)ⁿ⁻ʳ
E(X) = np Var(X) = np(1 − p)
- n: fixed number of trials.
n：固定的试验次数。 - p: constant probability of success.
p：恒定的成功概率。 - r: number of successes of interest.
r：关注的成功次数。
10. Exam Technique and Common Pitfalls | 考试技巧与常见陷阱
Read each probability question carefully and identify whether the events are mutually exclusive, independent, or neither. Write down the relevant formula before substituting numbers. Use exact fractions where possible, and only round at the final stage unless the question states otherwise.
仔细阅读每道概率题，判断事件是互斥、独立还是两者都不是。代入数字前先写出相关公式。尽可能使用精确分数，除非题目另有说明，否则只在最后一步取近似值。
Common pitfalls include dividing by the wrong total in conditional probability, forgetting to subtract the intersection in the general addition rule, and confusing Var(aX + b) with aVar(X) + b. Always check that probabilities sum to 1 and that no probability lies outside the interval from 0 to 1.
常见陷阱包括：条件概率中除以错误的总数；一般加法法则中忘记减去交集；把 Var(aX + b) 与 aVar(X) + b 混淆。始终检查概率之和为 1，且没有概率落在 0 到 1 之外。
- Check the condition: P(A | B) uses the total of B.
检查条件：P(A | B) 使用 B 的总数。 - Check the range: 0 ≤ P(A) ≤ 1 for every event A.
检查范围：每个事件 A 都满足 0 ≤ P(A) ≤ 1。 - Check the sum: all probabilities in a distribution add to 1.
检查求和：分布中所有概率之和为 1。
11. Formula Summary and Quick Revision | 公式汇总与快速复习
Use this summary table for rapid revision. It brings together the main formulas from the previous sections. Memorising these formulas and understanding when to apply them is essential for success in Edexcel A-Level Mathematics probability questions.
使用下面的汇总表进行快速复习。它汇集了前几节的主要公式。记住这些公式并理解何时使用它们，是成功解答爱德思 A-Level 数学概率题的关键。
| Topic | Formula | Notes |
| Complement | P(A') = 1 − P(A) | "Not A" |
| Addition rule | P(A ∪ B) = P(A) + P(B) − P(A ∩ B) | General case |
| Multiplication rule | P(A ∩ B) = P(A) × P(B | A) | General case |
| Independent events | P(A ∩ B) = P(A) × P(B) | If independent |
| Expectation | E(X) = Σ x P(X = x) | Discrete X |
| Variance | Var(X) = E(X²) − [E(X)]² | Discrete X |
| Binomial | P(X = r) = C(n, r) pʳ (1 − p)ⁿ⁻ʳ | X ~ B(n, p) |
Finally, practise past-paper questions under timed conditions. Probability questions reward careful reading, clear notation and accurate arithmetic. Keep a personal list of mistakes and review it before the exam.
最后，在限时条件下练习历年真题。概率题奖励仔细阅读、清晰符号和准确计算。整理个人错题清单，并在考试前复习。
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