IGCSE CIE Maths Probability Revision | IGCSE CIE 数学：概率考点精讲

📚 IGCSE CIE Maths Probability Revision | IGCSE CIE 数学：概率考点精讲

Probability is a fascinating and exam-relevant topic in IGCSE CIE Mathematics. It helps you measure the chance of an event happening, from flipping a coin to drawing a card. Mastering probability will boost your confidence in both Core and Extended papers, as questions on tree diagrams, expected frequency, and combined events appear frequently. This revision guide walks you through every key concept with clear explanations and worked examples.

概率是 IGCSE CIE 数学中既有趣又高频的考点。它帮助你衡量事件发生的可能性，从抛硬币到抽纸牌都离不开它。掌握好概率，能让你在核心和扩展卷中从容应对，因为树形图、期望频率和组合事件的题目经常出现。这篇考点精讲将带你逐一梳理所有重要概念，并配以清晰的解释和例题。

1. What is Probability? | 什么是概率？

Probability is a measure of how likely an event is to occur. It is always a number between 0 and 1 inclusive. An event with probability 0 is impossible, while an event with probability 1 is certain. You can also express probability as a fraction, decimal or percentage. For example, a fair coin landing on heads has a probability of 0.5, or ½.

概率是衡量某个事件发生可能性的量度。它的取值范围总是在 0 到 1 之间（含 0 和 1）。概率为 0 表示不可能发生，概率为 1 表示必然发生。你也可以用分数、小数或百分数来表示概率。例如，一枚均匀的硬币正面朝上的概率就是 0.5，或 ½。

The basic formula is: P(event) = (number of favourable outcomes) ÷ (total number of possible outcomes). This works when all outcomes are equally likely.

基本公式是：P(事件) = (有利结果的数量) ÷ (所有可能结果的总数)。这适用于所有结果发生可能性相等的情形。

2. Probability Scale | 概率标度

The probability scale is a visual way to place events on a line from 0 to 1. Words such as impossible, unlikely, even chance, likely and certain correspond to positions on this line. An even chance is exactly 0.5. Events closer to 1 are more likely; events closer to 0 are less likely.

概率标度是一种直观的方法，把事件放在从 0 到 1 的直线上。不可能、不太可能、等可能性、很可能、一定这些词语就对应着这条线上的位置。等可能性恰好是 0.5。越接近 1 的事件越可能发生；越接近 0 的事件越不可能发生。

For example, picking a red ball from a bag containing 50 red and 50 blue balls has a probability of 0.5, sitting right in the middle of the scale.

例如，从一个装有 50 个红球和 50 个蓝球的袋子里摸出红球，概率就是 0.5，位于标度正中间。

3. Sample Space | 样本空间

A sample space is the set of all possible outcomes of an experiment. You can list outcomes in curly brackets, or use a two-way table for two events. For tossing a coin, the sample space is {Head, Tail}. For rolling a fair six-sided die, it is {1, 2, 3, 4, 5, 6}.

样本空间是指一个试验所有可能结果的集合。你可以用大括号列出所有结果，或者对两个事件使用二维表格。抛一枚硬币的样本空间是 {正面, 反面}。掷一个均匀六面骰子的样本空间是 {1, 2, 3, 4, 5, 6}。

When two events take place, like rolling two dice, you can draw a 6 by 6 grid to show all 36 ordered pairs. This grid helps count favourable outcomes quickly.

当有两个事件发生，比如掷两个骰子，你可以画一个 6×6 的网格来表示全部 36 个有序数对。这个网格能帮你快速数出有利结果。

4. Relative Frequency | 相对频率

Relative frequency is an estimate of probability based on actual trials or experiments. You calculate it as: relative frequency = (number of times event occurs) ÷ (total number of trials). The more trials you carry out, the closer relative frequency tends to get to the theoretical probability. This idea is also known as experimental probability.

相对频率是根据实际试验或实验结果对概率的估计。它的计算方法是：相对频率 = (事件发生的次数) ÷ (总试验次数)。你进行的试验次数越多，相对频率就越趋向于理论概率。这个思想也叫做实验概率。

For example, if you flip a coin 100 times and get 62 heads, the relative frequency of heads is 62/100 = 0.62. You would expect it to approach 0.5 with more flips.

例如，如果你抛硬币 100 次，得出 62 次正面，那么正面的相对频率就是 62/100 = 0.62。随着抛掷次数增多，你会预期它接近 0.5。

5. Mutually Exclusive Events | 互斥事件

Two events are mutually exclusive if they cannot happen at the same time. For instance, when rolling a die, getting a 3 and getting a 5 are mutually exclusive. The probability of either event happening is found by adding their individual probabilities: P(A or B) = P(A) + P(B).

如果两个事件不能同时发生，它们就是互斥事件。例如，掷一个骰子时，掷出 3 点和掷出 5 点就是互斥的。任一事件发生的概率是它们各自概率的和：P(A 或 B) = P(A) + P(B)。

This is the addition rule for mutually exclusive events. If events are not mutually exclusive, you must subtract the intersection: P(A or B) = P(A) + P(B) – P(A and B).

这就是针对互斥事件的加法法则。如果事件不是互斥的，你必须减去它们的交集：P(A 或 B) = P(A) + P(B) – P(A 且 B)。

6. Independent Events | 独立事件

Independent events are those where the outcome of one event does not affect the outcome of the other. Tossing a coin and rolling a die are independent. For independent events, the probability of both occurring is the product of their individual probabilities: P(A and B) = P(A) × P(B).

独立事件是指一个事件的结果不会影响另一个事件结果的情况。抛硬币和掷骰子就是相互独立的。对于独立事件，两者同时发生的概率是它们各自概率的乘积：P(A 且 B) = P(A) × P(B)。

This is the multiplication rule for independent events. Always check whether events are truly independent before applying it.

这是针对独立事件的乘法法则。在应用之前，一定要先检查事件是否真的相互独立。

7. Tree Diagrams | 树形图

Tree diagrams are extremely useful for showing all possible outcomes of two or more sequential events. Each branch represents a possible outcome with its probability written along it. To find the overall probability of a particular path, you multiply the probabilities along the branches. If there are multiple paths leading to the same final outcome, you add their probabilities.

树形图在展示两个或两个以上相继事件的所有可能结果时非常有用。每条分支代表一个可能的结果，概率写在分支上。要找出某条路径的整体概率，只需把沿路各分支的概率相乘。如果有多个路径通向同一个最终结果，再把它们的概率相加。

Always check that probabilities on branches from the same point add up to 1. CIE exams often require you to complete a partially drawn tree diagram and use it to find combined probabilities.

务必检查从同一个节点分出的分支上概率之和是否等于 1。CIE 考试经常让你补全一个部分绘制的树形图，并用它求组合概率。

8. Conditional Probability | 条件概率

Conditional probability is the probability of an event A occurring given that event B has already occurred. It is written as P(A|B). This occurs in situations without replacement, such as picking two balls from a bag one after another without putting the first one back. The probabilities on the second set of branches change because the sample space has been altered.

条件概率是指在已知事件 B 已经发生的条件下，事件 A 发生的概率。写作 P(A|B)。这出现在不放回的情形中，比如从一个袋子里一个接一个地摸出两个球，而不把第一个球放回去。由于样本空间发生了变化，第二层分支上的概率也随之改变。

For example, if a bag contains 4 red and 6 blue balls and you pick two without replacement, the probability of a red on the second draw depends on the colour of the first ball drawn. Tree diagrams clearly display these changing probabilities.

例如，如果一个袋子装有 4 个红球和 6 个蓝球，你无放回地抽两次，第二个球是红色的概率就取决于第一个球的颜色。树形图能清晰地展示这些变化着的概率。

9. Probability from Tables | 从表格求概率

Two-way tables and Venn diagrams are common tools for organising data and calculating probabilities. A two-way table shows frequencies for two categories, allowing you to easily find total outcomes and favourable outcomes. The probability of an event is the number in the desired cell divided by the overall total.

双向表格和韦恩图是整理数据和计算概率的常用工具。双向表格展示了两个分类的频数，让你能够轻松求出总结果数和有利结果数。事件的概率就是目标单元格里的数字除以总计。

Venn diagrams show sets as overlapping circles. The probability of being in a set or its complement can be read from the diagram. Remember to use set notation symbols like ∪ (union) and ∩ (intersection) where required.

韦恩图用相交的圆表示集合。属于某个集合或其补集的概率可以从图上读出。记得在需要时使用 ∪ (并集) 和 ∩ (交集) 等集合符号。

10. Expected Frequency | 期望频数

Expected frequency tells you how many times you would expect an event to happen in a given number of trials. It is calculated as: expected frequency = probability × total number of trials. This is not a guarantee but a long-term average prediction.

期望频数告诉你，在一系列给定次数的试验中，你预期某个事件发生多少次。它的计算方法是：期望频数 = 概率 × 总试验次数。这不是一个保证，而是一个长期的平均预测。

For instance, if the probability of rain on any day is 0.3, then in 200 days you would expect it to rain on 0.3 × 200 = 60 days. CIE questions frequently ask you to compare expected frequencies with observed data.

例如，如果任何一天下雨的概率是 0.3，那么在 200 天里，你预期下雨的天数是 0.3 × 200 = 60 天。CIE 考题经常让你将期望频数与实际观测数据做比较。

11. The sum of probabilities | 概率之和

The sum of probabilities of all mutually exclusive and exhaustive outcomes of an experiment is always 1. This means if you know the probabilities of some outcomes, you can find the missing one by subtracting from 1. This is particularly useful in ‘not’ events: P(not A) = 1 – P(A).

一个试验中所有互斥且穷举的结果的概率之和总是等于 1。这意味着，如果你知道某些结果的概率，就可以用 1 减去它们来求出缺失的部分。这对“非”事件尤为有用：P(非 A) = 1 – P(A)。

For example, if the probability of a traffic light being green is 0.4 and yellow is 0.15, the probability of it being red is 1 – (0.4 + 0.15) = 0.45.

例如，如果交通灯为绿灯的概率是 0.4，黄灯是 0.15，那么红灯的概率就是 1 – (0.4 + 0.15) = 0.45。

12. Exam Tips and Common Pitfalls | 应试技巧与常见错误

First, always read the question carefully – check whether items are replaced or not in multiple picks. Second, do not confuse mutually exclusive with independent events; they are different concepts. Third, show your working on tree diagrams and write probabilities clearly. If you use a calculator, leave probabilities as fractions rather than decimals unless asked otherwise.

首先，认真审题——多次抽取物品时，要看清楚是否放回。其次，不要把互斥事件与独立事件混淆，它们是不同的概念。第三，在树形图上展示解题过程，并清晰地写出概率。如果使用计算器，除非题目另有要求，最好把概率保留为分数形式，而不是小数。

Also, before using the addition rule, check if the events overlap. If they do, you must subtract the overlap. When drawing a tree diagram, make sure the branches from a single point sum to 1.

此外，在应用加法法则之前，检查事件是否有重叠。若有，你必须减去重叠部分。绘制树形图时，确保从同一节点分出的各分支概率之和等于 1。

Finally, practice past paper questions to become familiar with CIE wording and typical mark allocations. Probability questions often combine several concepts, so build a strong foundation step by step.

最后，通过练习历年真题来熟悉 CIE 的提问方式和典型分值分布。概率题常常综合多个概念，所以要一步一步打好扎实的基础。

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IGCSE CIE Maths Probability Revision | IGCSE CIE 数学：概率 考点精讲