📚 Laws of Probability | 概率法则
Probability is the branch of mathematics that deals with the chance or likelihood of an event occurring. Understanding the laws of probability allows you to quantify uncertainty and make predictions based on known information. For IGCSE Edexcel Mathematics, you are expected to master the fundamental rules, including how to handle single events, combined events, mutually exclusive outcomes, independent events, and conditional probability using tree diagrams and formulas.
概率是数学中研究事件发生可能性或几率的学科。理解概率法则能让你量化不确定性，并根据已知信息做出预测。在 Edexcel IGCSE 数学考试中，你需要掌握基本规则，包括如何处理单一事件、组合事件、互斥结果、独立事件以及用树状图和公式计算条件概率。
1. What is Probability? | 什么是概率?
Probability is a measure of the likelihood that a particular outcome will happen. It is always a number between 0 and 1 (inclusive), where 0 means the event is impossible and 1 means the event is certain. For example, when flipping a fair coin, the probability of getting heads is 0.5. Probabilities can also be written as fractions, decimals, or percentages. A probability of 0.2 is the same as 1/5 or 20%.
概率是衡量某个特定结果发生可能性的指标。它总是一个介于 0 和 1 之间（含端点）的数字，0 表示事件不可能发生，1 表示事件必然发生。例如，抛一枚质地均匀的硬币时，正面朝上的概率是 0.5。概率也可以用分数、小数或百分比表示。概率0.2 等同于1/5 或20%。
ber from 1 to 6, and the total probability is P(1) + P(2) + P(3) + P(4) + P(5) + P(6) = 1.
一项实验中所有可能结果的概率之和始终为1。例如，掷一颗均匀六面骰子，每个数字出现的概率可以是1到6任一，且总概率为 P(1) + P(2) + P(3) + P(4) + P(5) + P(6) = 1。
2. The Probability Scale | 概率尺度
Visualising probability on a number line from 0 to 1 helps build intuition. Impossible events sit at 0, certain events at 1, and equally likely events at 0.5. For example, drawing a blue ball from a bag containing only red balls has probability 0, while the probability that the sun will rise tomorrow is 1. Events with probabilities closer to 1 are more likely to occur.
在从0到1的数轴上直观地表示概率有助于培养直觉。不可能的事件位于0，必然事件位于1，等可能的事件位于0.5。例如，从一个只有红球的袋子中摸出蓝球的概率为0，而明天太阳升起的概率为1。概率越接近1的事件越可能发生。
Uncertainty is often expressed in words, but IGCSE questions require numerical expressions. Terms like 'likely', 'unlikely', or 'even chance' can be mapped onto the probability scale: 'even chance' is 0.5, 'certain' is 1, and 'impossible' is 0. Always convert descriptions into exact fractions or decimals where possible.
不确定性通常用词语描述，但 IGCSE 题目要求数值表达。像“可能”、“不可能”或“均等机会”这类词语可以映射到概率尺度上：“均等机会”为0.5，“必然”为1，“不可能”为0。尽可能将描述转换为精确的分数或小数。
3. Basic Probability Formula | 基本概率公式
The probability of an event A is given by:
P(A) = n(A) / n(S)
roll a fair die, n(S) = 6. The probability of rolling a 4 is P(4) = 1/6, because n(A) = 1.
事件 A 的概率由下式给出：
P(A) = n(A) / n(S)
其中 n(A) 是事件 A 的有利结果数目，n(S) 是样本空间中所有等可能结果的总数。样本空间 S 是该实验所有可能结果的集合。例如，掷一粒公平的骰子，n(S) = 6。掷出4点的概率为 P(4) = 1/6，因为 n(A) = 1。
This formula works when all outcomes are equally likely. If the coin is biased, the outcomes are not equally likely, and probability must be determined from given relative frequencies or experimental data. Always check the fairness assumption before applying this simple formula.
当所有结果等可能时，这个公式才成立。如果硬币有偏差，结果不再是等可能的，概率必须根据给定的相对频率或实验数据来确定。在应用这个简单公式之前，务必检查是否满足等可能的假设。
4. Sample Spaces | 样本空间
A sample space is the list or table of all possible outcomes of an experiment. For a single event, it can be a simple list: flipping a coin gives sample space {Heads, Tails}. For two events, a two-way table or a list of ordered pairs is often used. For instance, rolling two dice gives a sample space of 36 ordered pairs such as (1,1), (1,2), …, (6,6).
样本空间是实验所有可能结果的列表或表格。对于单一事件，它可以是一个简单的列表：掷硬币的样本空间是{正面, 反面}。对于两个事件，常常使用双向表或有序对列表。例如，掷两粒骰子会产生36个有序对的样本空间，如 (1,1), (1,2), …, (6,6)。
Constructing a sample space systematically helps you avoid missing outcomes. When you have combined events, always consider the order if it matters. For example, picking two counters from a bag without replacement creates a sample space where the second outcome depends on the first. Use a tree diagram or a grid to represent all possibilities.
系统性地构建样本空间有助于避免遗漏结果。处理组合事件时，如果顺序重要，一定要考虑顺序。例如，从袋中不放回地取两个筹码，就产生了一个样本空间，其中第二个结果依赖于第一个结果。使用树状图或网格来表示所有可能的情况。
5. Complementary Events | 互补事件
The complement of an event A, written as A', is the event that A does not happen. Since A and A' cover all possible outcomes, their probabilities sum to 1:
P(A) + P(A') = 1
or equivalently, P(A' if the probability of passing a test is 0.9, the probability of failing is 1 - 0.9 = 0.1.
事件 A 的补集，记为 A'，是指 A 不发生的事件。由于 A 和 A' 覆盖了所有可能的结果，它们的概率之和为1：
P(A) + P(A') = 1
或者等价地，P(A') = 1 - P(A)。当计算补集的概率并做减法更容易时，这个规则就非常有用。例如，如果通过考试的概率是0.9，那么不通过的概率就是 1 - 0.9 = 0.1。
In IGCSE problems, the complement rule often simplifies complex 'at least one' problems. Instead of summing probabilities for one success, two successes, etc., you can compute 1 - P(none). This approach saves time and reduces errors in exam papers.
在 IGCSE 题目中，补集规则常被用来简化复杂的“至少一个”问题。与其将一次成功、两次成功等情况的概率相加，不如计算 1 - P(零个) 。这种方法在考卷中节省时间并减少错误。
6. Mutually Exclusive Events & the Addition Law | 互斥事件与加法法则
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 because a single roll cannot produce both outcomes simultaneously. When events A and B are mutually exclusive, the probability of A or B occurring is given by the addition law:
P(A or B) = P(A) + P(B)
两个事件如果不能同时发生，就说它们是互斥的。例如，掷一粒骰子时，得到2点和得到5点是互斥的，因为一次掷骰不可能同时产生两个结果。当事件 A 和 B 互斥时，A 或 B 发生的概率由加法法则给出：
P(A 或 B) = P(A) + P(B)
of drawing a King or a Queen from a full deck is P(King) + P(Queen) = 4/52 + 4/52 = 8/52.
这个规则可以推广到两个以上的互斥事件。对于三个互斥事件 A, B 和 C：P(A 或 B 或 C) = P(A) + P(B) + P(C)。一个典型的 IGCSE 例子是随机抽取一张牌：从一副完整的扑克牌中抽出老K或老皮蛋的概率为 P(老K) + P(老皮蛋) = 4/52 + 4/52 = 8/52。
7. Non-Mutually Exclusive Events & the General Addition Law | 非互斥事件与一般加法法则
When two events can occur at the same time, they are not mutually exclusive. For example, in a class, a student can be both tall and wear glasses. When calculating P(A or B) for non-mutually exclusive events, you must subtract the overlap to avoid double-counting:
P(A or B) = P(A) + P(B) - P(A and B)
两个事件如果可能同时发生，它们就不是互斥的。例如，在一个班级里，一个学生可以同时个子高和戴眼镜。在为非互斥事件计算 P(A 或 B) 时，必须减去重叠部分以避免重复计数：
P(A 或 B) = P(A) + P(B) - P(A 和 B)
Notice that the word 'or' in probability means union, or A ∪ ly exclusive). Exam questions often provide a Venn diagram or a two-way table to help you find the intersection P(A and B).
注意，概率中的“或”表示并集，即 A ∪ B。通用加法法则对所有事件都成立，当 P(A 和 B) = 0 （互斥）时，它就退化为简单加法法则。考题常提供韦恩图或双向表来帮助你找出交集 P(A 和 B)。
ies both is 0.3, then the probability of studying Art or Biology is 0.6 + 0.5 - 0.3 = 0.8.
通过填充表格来练习。例如，若随机选一名学生，学美术的概率是0.6，学生物的概率是0.5，两门都学的概率是0.3，那么学美术或生物的概率为 0.6 + 0.5 - 0.3 = 0.8。
8. Independent Events & the Multiplication Law | 独立事件与乘法法则
Two events are independent if the outcome of one does not affect the probability of the other. For independent events A and B, the probability that both A and B occur is the product of their individual probabilities:
P(A and B) = P(A) × P(B)
如果两个事件中一个的结果不影响另一个的概率，则它们是独立的。对于独立事件 A 和 B，两者都发生的概率是它们各自概率的乘积：
P(A 和 B) = P(A) × P(B)
xample, the probability of getting two heads when flipping a fair coin twice is ½ × ½ = ¼.
这个规则可以推广：如果 A, B 和 C 相互独立，那么 P(A 和 B 和 C) = P(A) × P(B) × P(C)。独立性常常出现在抛硬币、掷骰子和放回抽取等情境中。例如，抛一枚公平硬币两次，得到两次正面的概率是 ½ × ½ = ¼。
Be careful: the multiplication law only applies if events are independent. If events are not independent (e.g., drawing cards without replacement), you must use conditional probability or tree diagrams. Always read the question to determine whether replacement occurs or whether there is a clear statement of independence.
注意：乘法法则仅在事件独立时适用。如果事件不独立（例如不放回地抽牌），则必须使用条件概率或树状图。一定要仔细读题，判断是否有放回或题目是否明确说明事件独立。
9. Conditional Probability | 条件概率
Conditional probability is the probability of an event occurring given that another event has already occurred. It is written as P(A | B), meaning 'probability of A given B'. The formula is:
P(A | B) = P(A and B) / P(B)
ty that it is also a heart (event A) is P(heart and red) / P(red) = (13/52) / (26/52) = ½.
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。它写作 P(A | B)，表示“在 B 发生的条件下 A 的概率”。公式为：
P(A | B) = P(A 和 B) / P(B)
其中 P(B) > 0。当我们掌握部分信息时，这个公式能帮助我们更新概率。例如，若我们知道从一副牌中抽出的牌是红色（事件 B），则它也是红桃（事件 A）的概率为 P(红桃且红色) / P(红色) = (13/52) / (26/52) = ½。
In IGCSE, conditional probability often appears with tree diagrams. Each branch after the first is labelled with a conditional probability. You may be asked to compute the overall probability of a path by multiplying the first probability by the conditional probability on the second branch, then using the addition law for combined outcomes.
在 IGCSE 中，条件概率常与树状图一同出现。第一个分支之后的每一分支都标有条件概率。你可能会被要求通过将第一个概率乘以第二个分支上的条件概率来计算一条路径的整体概率，然后再用加法法则计算组合结果。
10. Tree Diagrams for Probability | 用于概率的树状图
A tree diagram is a visual tool that helps organise outcomes of multi-stage experiments, especially when events are not independent. Each branch represents a possible outcome, and the probabilities on the branches from a node sum to 1. To find the probability of a sequence of events, multiply the probabilities along the branches. For combined events that can arise in different ways, add the end probabilities of those paths.
树状图是一种可视化工具，有助于组织多阶段实验的结果，尤其当事件不独立时。每个分支代表一个可能的结果，从节点出发的分支概率之和为1。要找出事件序列的概率，就将各分支上的概率相乘。对于可以通过不同方式出现的组合事件，则将这些路径的末端概率相加。
nce (red then blue) has probability 3/5 × 2/4 = 6/20. All outcomes can be found similarly.
例如，袋中有3个红筹码和2个蓝筹码。你不放回地抽取两个筹码。第一次抽取的概率为 3/5 红和 2/5 蓝。若第一次是红，第二次抽取会受到影响：剩余为 2/4 红和 2/4 蓝。序列（先红后蓝）的概率为 3/5 × 2/4 = 6/20。所有结果都可以类似求出。
Tree diagrams are especially powerful for conditional probability questions. If asked for a conditional probability like P(second is red | first is blue), you can read the relevant branch directly or apply the formula using path probabilities. Always label probabilities clearly and check that probabilities at each node add to 1.
树状图对于条件概率问题特别有效。如果要求条件概率，如 P(第二个红 | 第一个蓝)，你可以直接读出相关分支的概率，或运用公式和路径概率进行计算。务必清晰标注概率，并检查每个节点上的概率之和是否为1。
11. Expected Frequency and Relative Frequency | 期望频数与相对频数
The expected frequency of an event is the number of times you would expect it to occur if you repeated an experiment many times. It is calculated as:
Expected frequency = probability × number of trials
s not guarantee exactly 50 sixes, but tells you the average outcome over many repetitions.
事件的期望频数是指当你多次重复一个实验时，预期该事件会发生的次数。计算公式为：
期望频数 = 概率 × 试验次数
例如，掷一粒公平骰子300次，预期出现六点的次数为 (1/6) × 300 = 50。这并不保证精确出现50次，但告诉你多次重复后的平均结果。
he number of trials increases, relative frequency tends to stabilise around the theoretical probability, a concept known as the law of large numbers.
相对频数是概率的实验估计：它是事件在实际试验中发生的次数比例。如果你抛硬币100次并得到47次正面，正面的相对频数就是 47/100 = 0.47。随着试验次数增加，相对频数会围绕理论概率趋于稳定，这个概念称为大数定律。
IGCSE questions often ask you to compare experimental relative frequency with theoretical probability and discuss possible reasons for differences, such as bias in the equipment or too few trials. Always use precise terms when explaining.
IGCSE 题目常常要求你比较实验相对频数和理论概率，并讨论差异的可能原因，比如设备偏差或试验次数太少。解释时要使用准确的术语。
12. Summary of Key Probability Laws | 关键概率法则总结
To excel in IGCSE probability, you must internalise these core laws:
- Basic probability: P(A) = n(A) / n(S), for equally likely outcomes.
- Complement rule: P(A') = 1 – P(A).
- Mutually exclusive addition: P(A ∪ B) = P(A) + P(B).
- General addition: P(A ∪ B) = P(A) + P(B) – P(A ∩ B).
- Independent multiplication: P(A ∩ B) = P(A) × P(B) when independent.
- Conditional probability: P(A | B) = P(A ∩ B) / P(B).
- Tree diagrams: Multiply along branches for sequences, add across branches for combined events.
在 IGCSE 概率部分取得优异成绩，你必须内化以下核心法则：
- 基本概率： P(A) = n(A) / n(S)，适用于等可能结果。
- 补集规则： P(A') = 1 – P(A)。
- 互斥事件加法： P(A ∪ B) = P(A) + P(B)。
- 一般加法： P(A ∪ B) = P(A) + P(B) – P(A ∩ B)。
- 独立事件乘法： 当独立时，P(A ∩ B) = P(A) × P(B)。
- 条件概率： P(A | B) = P(A ∩ B) / P(B)。
- 树状图： 沿分支相乘得到序列概率，跨分支相加得到组合事件概率。
Always identify the type of events first: Are they mutually exclusive? Independent? Then select the right formula or representation. Draw a Venn diagram or tree diagram if needed. With systematic practice, the laws of probability become a reliable toolkit for solving any likelihood problem on your IGCSE paper.
始终先判断事件类型：它们是互斥的吗？独立吗？然后选用正确的公式或表示法。必要时画出韦恩图或树状图。通过系统的练习，概率法则将成为你在 IGCSE 试卷上解决任何可能性问题的可靠工具。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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