📚 IGCSE Edexcel Maths: Probability Key Points Revision | IGCSE Edexcel 数学：概率 考点精讲
Probability is a core topic in the IGCSE Edexcel Mathematics syllabus, appearing in both calculator and non‑calculator papers. It tests your ability to quantify chance, analyse events, and use structured methods such as tree diagrams and Venn diagrams to handle combined events. This guide breaks down every key concept you need, with worked examples and exam tips that match the style of Edexcel questions.
概率是 IGCSE Edexcel 数学大纲中的核心主题，在计算器和非计算器试卷中都会出现。它考查你量化可能性、分析事件以及使用树形图和韦恩图等结构化方法处理组合事件的能力。本指南分解了每个你需要的关键概念，配有符合 Edexcel 题型风格的示例和考试技巧。
1. Basic Probability Concepts | 基本概率概念
Probability is a measure of how likely an event is to happen. It always lies between 0 (impossible) and 1 (certain), and can be written as a fraction, decimal or percentage. The probability of an event A is found by P(A) = number of favourable outcomes / total number of possible outcomes, provided all outcomes are equally likely.
概率是衡量事件发生可能性大小的量度。它的值总是在 0（不可能）到 1（必然）之间，可以写成分数、小数或百分数。如果所有结果等可能发生，事件 A 的概率由 P(A) = 有利结果数 / 可能结果总数 求得。
The sum of probabilities of all possible outcomes of a trial is 1. For example, when rolling a fair six‑sided die, P(1) + P(2) + P(3) + P(4) + P(5) + P(6) = 1. If the probability of an event happening is p, then the probability of it not happening (the complement) is 1 − p. We write this as P(not A) = 1 − P(A). This simple rule is extremely useful for ‘at least one’ types of questions.
一次试验所有可能结果的概率之和为 1。例如，抛一枚均匀的六面骰子，P(1) + P(2) + P(3) + P(4) + P(5) + P(6) = 1。如果事件发生的概率是 p，那么它不发生的概率（补集）为 1 − p。我们写作 P(非 A) = 1 − P(A)。这个简单的规则在“至少一个”类型的问题中非常有用。
2. Experimental vs Theoretical Probability | 实验概率与理论概率
Theoretical probability is what we expect to happen based on equally likely outcomes, while experimental probability (or relative frequency) comes from actually carrying out an experiment or survey. The formula for experimental probability is estimated P(A) = number of times A occurs / total number of trials. As the number of trials increases, the experimental probability tends to get closer to the theoretical probability – this is known as the Law of Large Numbers.
理论概率是基于等可能结果我们预期会发生的情况，而实验概率（或相对频率）来自实际进行的实验或调查。实验概率的公式是 估计的 P(A) = A 发生的次数 / 总试验次数。随着试验次数的增加，实验概率会越来越接近理论概率——这就是大数定律。
In Edexcel IGCSE exams, you may be given a table of frequencies and asked to estimate a probability. Always check whether the question expects a fraction, decimal or percentage. Be careful not to confuse the number of trials with the number of outcomes.
在 Edexcel IGCSE 考试中，你可能会遇到一个频率表并被要求估计概率。一定要检查题目要求用分数、小数还是百分数作答。注意不要把试验次数与结果数混淆。
3. Sample Space and Listing Outcomes | 样本空间与列出结果
A sample space is the set of all possible outcomes of an experiment. For a single coin toss, the sample space is {Heads, Tails}. For rolling a die and tossing a coin, the sample space can be listed systematically as (1,H), (1,T), (2,H), … or shown in a two‑way table. Systematic listing ensures you do not miss any outcomes.
样本空间是一次实验所有可能结果的集合。对于一次抛硬币，样本空间是 {正面，反面}。对于掷骰子和抛硬币的组合，样本空间可以用列表系统列出，如 (1, H), (1, T), (2, H), … 也可以用双向表表示。系统的列举可以确保你不会遗漏任何结果。
A product rule can help: if one event has m outcomes and another independent event has n outcomes, the total number of combined outcomes is m × n. For example, rolling two fair dice gives 6 × 6 = 36 equally likely outcomes. You can then count favourable outcomes to find probabilities.
乘积法则可提供帮助：如果一个事件有 m 种结果，另一个独立事件有 n 种结果，则组合结果总数为 m × n。例如，掷两枚均匀骰子产生 6 × 6 = 36 种等可能结果。然后你可以数出有利结果的数目来求概率。
4. Probability of Single Events | 单一事件的概率
For a single event, simply apply P(A) = number of favourable outcomes / total outcomes. Edexcel often embeds this in real‑world contexts: picking a red ball from a bag, a student chosen at random, or a spinner landing on an even number. You must always check whether the items are fair and equally likely.
对于单一事件，只需应用 P(A) = 有利结果数 / 结果总数。Edexcel 常将这嵌入现实情境：从袋中取出红球、随机选取一名学生，或转盘停在偶数上。你必须始终检查物品是否均匀且结果等可能。
When probabilities are given as fractions, simplify them unless the question asks for an unsimplified form. The complement rule is tested frequently: the probability of not getting a 6 on a fair die is 1 − 1/6 = 5/6. Additionally, be comfortable working with decimals and percentages – a probability of 0.2 is the same as 20%.
当概率以分数给出时，要进行约分，除非题目要求不约分的形式。补集规则经常被考查：没掷出 6 点的概率是 1 − 1/6 = 5/6。此外，要熟练使用小数和百分数——概率 0.2 相等于 20%。
5. Mutually Exclusive Events | 互斥事件
Two events are mutually exclusive if they cannot happen at the same time. For example, when rolling a die, getting a 2 and getting a 5 are mutually exclusive. For mutually exclusive events A and B, the probability of A or B happening is the sum of their individual probabilities: P(A or B) = P(A) + P(B). This OR rule is fundamental.
如果两个事件不能同时发生，则它们是互斥的。例如，掷骰子时，出现 2 和出现 5 是互斥的。对于互斥事件 A 和 B，A 或 B 发生的概率是它们各自概率之和：P(A 或 B) = P(A) + P(B)。这个“或”法则是基础。
If events are not mutually exclusive, simply adding the probabilities would count the overlap twice. In such cases, you need the general addition formula: P(A or B) = P(A) + P(B) − P(A and B). This is often best visualised with a Venn diagram, which we will cover later.
如果事件不是互斥的，简单相加会重复计算重叠部分。在这种情况下，你需要一般的加法公式：P(A 或 B) = P(A) + P(B) − P(A 和 B)。这通常用韦恩图可以最好地可视化，我们稍后会讲到。
6. Independent Events and the AND Rule | 独立事件与乘法法则
Two events are independent if the outcome of one does not affect the outcome of the other. For instance, tossing a coin and rolling a die are independent. For independent events A and B, the probability that both A and B happen is: P(A and B) = P(A) × P(B). This is the multiplication rule for independent events.
如果两个事件的结果互不影响，则它们是独立的。例如，抛硬币和掷骰子是独立的。对于独立事件 A 和 B，两者都发生的概率是：P(A 和 B) = P(A) × P(B)。这就是独立事件的乘法法则。
Beware: problems involving picking items without replacement lead to non‑independent events, where the probability of the second event changes depending on the first outcome. These must be handled with tree diagrams (see next section). In an exam, always ask yourself whether the situation is ‘with replacement’ (independent) or ‘without replacement’ (dependent).
注意：涉及不放回抽取的问题会导致事件不独立，此时第二个事件的概率取决于第一次的结果。这些情况必须用树形图处理（见下一节）。考试时，始终问自己情况是“放回”（独立）还是“不放回”（不独立）。
7. Tree Diagrams | 树形图
Tree diagrams are essential for displaying sequences of events, especially when events are dependent (without replacement). Each branch represents a possible outcome, and probabilities are written on the branches. The probabilities on each set of branches must sum to 1. To find the probability of a combined path, you multiply along the branches.
树形图对于展示事件序列至关重要，特别是当事件不独立（不放回）时。每个分支代表一种可能的结果，分支上标有概率。每组分支上的概率之和必须等于 1。要计算组合路径的概率，沿分支相乘。
For example, a bag contains 3 red and 2 blue balls. Two balls are drawn without replacement. The tree has first stage probabilities 3/5 and 2/5. For the second stage, if a red was removed, the probabilities become 2/4 for red and 2/4 for blue. The four combined probabilities are (3/5)×(2/4)=6/20, (3/5)×(2/4)=6/20, (2/5)×(3/4)=6/20, (2/5)×(1/4)=2/20. Summing appropriate paths gives answers to questions like ‘balls of the same colour’.
例如，一个袋子装有 3 个红球和 2 个蓝球，不放回地抽取两次。树图第一阶段概率为 3/5 和 2/5。第二阶段，如果已取出一个红球，概率变为红球 2/4，蓝球 2/4。四条组合概率为 (3/5)×(2/4)=6/20，(3/5)×(2/4)=6/20，(2/5)×(3/4)=6/20，(2/5)×(1/4)=2/20。将适当路径相加即可回答诸如“同色球”之类的问题。
When drawing your own tree diagram for multiple stages, label branches clearly and write the event descriptions at the ends. If probabilities are given as decimals, keep at least three decimal places for accuracy. Check the sum of final probabilities equals 1.
当为多阶段绘制自己的树形图时，要清楚地标记分支并在末尾写上事件描述。如果概率以小数给出，保留至少三位小数以确保准确。检查最终概率总和是否等于 1。
8. Conditional Probability | 条件概率
Conditional probability is the probability of an event occurring given that another event has already occurred. It is denoted by P(A|B) and read as ‘the probability of A given B’. The formula is P(A|B) = P(A and B) / P(B), provided P(B) > 0. This formula is a key part of the IGCSE syllabus and appears both on its own and within harder tree diagram problems.
条件概率是指在另一个事件已经发生的条件下，某事件发生的概率。记作 P(A|B)，读作“在 B 发生的条件下 A 的概率”。公式为 P(A|B) = P(A 和 B) / P(B)，前提是 P(B) > 0。这个公式是 IGCSE 大纲中的关键部分，既会单独出现，也会出现在较难的树形图问题中。
You can rearrange the formula to give the multiplication rule for any events (not just independent): P(A and B) = P(A|B) × P(B), or equivalently P(B|A) × P(A). This is particularly useful when reading probability from a two‑way table or a tree diagram where the second stage probabilities are conditional.
你可以重新排列该公式，得出适用于任何事件（不仅独立）的乘法法则：P(A 和 B) = P(A|B) × P(B)，或等价地 P(B|A) × P(A)。这在从双向表或树形图中读取概率时尤其有用，因为其中第二阶段概率往往是条件概率。
A common exam question gives a table of 100 people classified by gender and eye colour, then asks: ‘Given that a person chosen has blue eyes, what is the probability they are male?’ Locate the column or row for blue eyes, find its total, and then find the intersection with male — divide to get the conditional probability.
常见的考试题目会给出一个 100 人按性别和眼睛颜色分类的表格，然后问：“已知选到的人有蓝眼睛，那么他是男性的概率是多少？”找出蓝眼睛对应的列或行，找到其总数，然后找到与男性的交叉点——相除即得条件概率。
9. Venn Diagrams and Probability | 韦恩图与概率
Venn diagrams are used to display sets and the relationships between them. The rectangle represents the universal set (total sample space). Circles represent events, and overlapping regions show intersections (A ∩ B). The probability that an event occurs can be written using set notation: P(A), P(A ∩ B), P(A ∪ B). The complement of A is A’, and P(A’) = 1 − P(A).
韦恩图用于展示集合以及它们之间的关系。矩形表示全集（总样本空间）。圆圈代表事件，重叠区域表示交集（A ∩ B）。事件发生的概率可以使用集合符号表示：P(A)、P(A ∩ B)、P(A ∪ B)。A 的补集是 A’，且 P(A’) = 1 − P(A)。
Given a Venn diagram with numbers in each region, you can find any probability by dividing the number in the relevant region by the total number of outcomes. For instance, if 10 students study both Maths and Physics, 20 study only Maths, 30 only Physics, and 40 neither, total = 100. P(Maths ∩ Physics) = 10/100, P(Maths ∪ Physics) = (20+10+30)/100 = 60/100.
给定一张带有数字的韦恩图，你可以用相关区域中的数字除以结果总数来求得任何概率。例如，10 名学生同时学习数学和物理，20 人只学数学，30 人只学物理，40 人两者都不学，总数为 100。P(数 ∩ 物) = 10/100，P(数 ∪ 物) = (20+10+30)/100 = 60/100。
Edexcel questions often require you to fill in missing numbers on a Venn diagram from given probabilities. Set up equations using the fact that the sum of all regions equals the total. Work systematically from the innermost intersection outward.
Edexcel 的题目常常要求你根据给定的概率填写韦恩图中缺失的数字。利用所有区域之和等于总数这一事实来建立方程。从最内层的交集向外，系统地进行运算。
10. Two‑way Tables | 双向表
A two‑way table (or contingency table) organises data by two categories. They are perfect for finding totals, ‘and’ probabilities, and conditional probabilities. Each cell shows frequency for the combination of categories. Marginal totals (row and column sums) give the frequencies for single events.
双向表（或列联表）按两个类别组织数据。它们非常适合求总数、“和”概率以及条件概率。每个单元格显示类别组合的频率。边际总数（行和与列和）给出了单一事件的频率。
To find P(A), divide the total for A by the grand total. To find P(A and B), use the inner cell. For conditional probability P(A|B), restrict attention to the column (or row) for B and then find the proportion of A within that subset. Always check that your probabilities make sense contextually.
要求 P(A)，用 A 的总数除以总计数。要求 P(A 和 B)，使用内部的单元格。对于条件概率 P(A|B)，将注意力限制在 B 所在的列（或行），然后找出该子集中 A 的比例。始终检查概率在语境中是否合理。
Example: a table shows 30 girls and 20 boys like football; 10 girls and 40 boys do not. Grand total 100. P(girl) = 40/100, P(likes football) = 50/100. P(girl | likes football) = 30/50. You must read the table carefully to avoid mixing up rows and columns.
示例：一张表显示 30 名女生和 20 名男生喜欢足球；10 名女生和 40 名男生不喜欢。总计 100。P(女生) = 40/100，P(喜欢足球) = 50/100。P(女生 | 喜欢足球) = 30/50。你必须仔细阅读表格，避免混淆行与列。
11. Probability Using Set Notation | 集合符号的概率
The Edexcel IGCSE syllabus expects you to interpret and use set notation confidently, especially in probability contexts. Symbols include: ∪ (union, ‘or’), ∩ (intersection, ‘and’), ∈ (is an element of), ∉ (is not an element of), ∅ (empty set), ξ (universal set), A’ (complement of A). You should be able to translate between plain English and symbolic statements such as ‘n(A ∩ B)’ meaning the number of elements in the intersection.
Edexcel IGCSE 大纲要求你自信地理解和使用集合符号，尤其是在概率语境中。符号包括：∪（并集，“或”）、∩（交集，“且”）、∈（是……的元素）、∉（不是……的元素）、∅（空集）、ξ（全集）、A’（A 的补集）。你应当能在日常英语与诸如 n(A ∩ B) 意为交集中元素的个数这类符号语句之间进行翻译。
Questions might ask for P(A’ ∩ B) or P(A ∪ B’). Draw a Venn diagram if you are stuck. Remember that n(A ∪ B) = n(A) + n(B) − n(A ∩ B). This counting formula is the basis for the addition rule of probability. Practise writing probabilities using set notation from a given diagram.
题目可能会要求 P(A’ ∩ B) 或 P(A ∪ B’)。如果卡住了就画韦恩图。记住 n(A ∪ B) = n(A) + n(B) − n(A ∩ B)。这个计数公式是概率加法法则的基础。练习从给定图表中用集合符号书写概率。
12. Common Pitfalls and Exam Tips | 常见错误与考试技巧
One of the most common mistakes is adding probabilities for non‑mutually exclusive events without subtracting the intersection. Always check whether events can overlap. Another error is assuming independence when items are selected without replacement – before multiplying, verify that the probabilities remain the same for each trial.
最常见的错误之一是在非互斥事件中将概率相加，但没有减去交集部分。始终检查事件是否可能重叠。另一个错误是在不放回抽样时仍假定独立性——相乘之前，要确认每次试验的概率是否保持不变。
When using tree diagrams, forgetting to update the probabilities for the second stage when dealing with ‘without replacement’ leads to wrong final answers. Write new fractions clearly on each branch. Finally, in conditional probability questions, students sometimes divide the wrong way; remember the event after the bar ‘|’ is the condition – its probability is the denominator.
使用树形图时，忘记在处理“不放回”时更新第二阶段的概率会导致错误的最终答案。要在每个分支上清晰地写下新的分数。最后，在条件概率问题中，学生们有时会用错作除法的方向；记住竖线“|”后面的事件是条件——它的概率是分母。
Exam technique: show all working, even for seemingly obvious steps. Probability answers can often be left as fractions in their simplest form unless instructed otherwise. If a question asks for a probability ‘estimate’, it is referring to relative frequency. In multi‑part questions, check that your probabilities are between 0 and 1 and that the sum of probabilities of mutually exclusive and exhaustive events equals 1.
考试技巧：展示所有解题步骤，即使看似显而易见的步骤。除非另有指示，概率答案通常可保留为最简分数形式。如果题目要求“估计”概率，它指的是相对频率。在多部分问题中，检查你的概率是否在 0 和 1 之间，且互斥且穷举的事件概率之和等于 1。
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