Probability: Mutually Exclusive Events | 概率：互斥事件

📚 Probability: Mutually Exclusive Events | 概率：互斥事件

This article covers a key concept from the Cambridge KS3 Mathematics syllabus, aligned with exercises found on page 133. We will explore mutually exclusive events, their probability rules, and how to apply them in exam-style questions. Understanding this topic helps you build a strong foundation for more advanced probability work in IGCSE and beyond.

本文涵盖了剑桥KS3数学课程大纲中的一个关键概念，与第133页的练习相对应。我们将探讨互斥事件、它们的概率规则，以及如何在考试风格的问题中应用这些知识。理解这个主题将为你未来在IGCSE及更高阶段学习更复杂的概率知识打下坚实基础。

1. What are Mutually Exclusive Events? | 什么是互斥事件？

Two events are mutually exclusive if they cannot happen at the same time. When one event occurs, the other event cannot occur. In other words, the occurrence of one event excludes the possibility of the other event happening. This is a fundamental idea in probability theory that we use to simplify many calculations.

两个事件如果不可能同时发生，就称为互斥事件。当一个事件发生时，另一个事件就不可能发生。换句话说，一个事件的发生排除了另一个事件发生的可能性。这是概率论中的一个基本概念，我们用它来简化许多计算。

For example, imagine tossing a fair coin once. The two possible outcomes are ‘heads’ and ‘tails’. These outcomes are mutually exclusive because you cannot get both heads and tails on a single toss. If the coin lands heads, it definitely did not land tails. Connecting this idea to numerical probability, we assign a probability of ½ to heads and ½ to tails.

例如，想象抛掷一枚公平的硬币一次。可能的结果是“正面”和“反面”。这些结果是互斥的，因为你不可能在一次抛掷中同时得到正面和反面。如果硬币落地是正面，那它就绝对不可能是反面。联系到数值概率，我们给正面分配的概率是½，反面也是½。

2. The Addition Rule for Mutually Exclusive Events | 互斥事件的加法法则

The probability that either event A or event B occurs is given by the addition rule. When A and B are mutually exclusive, the probability of A or B happening is simply the sum of their individual probabilities.

事件A或事件B发生的概率由加法法则给出。当A与B互斥时，A或B发生的概率就是它们各自概率的简单相加。

P(A or B) = P(A) + P(B)

P(A 或 B) = P(A) + P(B)

This works because there is no overlap between the two events. If they had any outcomes in common, we would be double-counting, but here the sets are completely separate. The rule is often written using the union symbol: P(A ∪ B) = P(A) + P(B). Remember, this simplified version only applies when events cannot occur together.

这之所以成立，是因为两个事件之间没有重叠。如果它们有任何共同的结果，我们就会重复计算，但这里集合是完全分开的。这个规则通常用并集符号写成：P(A ∪ B) = P(A) + P(B)。请记住，这个简化版本仅适用于事件不可能同时发生的情况。

3. Rolling a Fair Six-Sided Die | 掷一个公平的六面骰子

Let’s apply the rule with a standard dice. The possible outcomes are {1, 2, 3, 4, 5, 6}, each with probability 1/6. Consider event A: rolling an even number {2, 4, 6}, and event B: rolling a 5. Can you roll a number that is both even and 5 at the same time? No. So they are mutually exclusive.

让我们用一个标准骰子来应用这个规则。可能的结果是{1, 2, 3, 4, 5, 6}，每个结果的概率都是1/6。考虑事件A：掷出偶数{2, 4, 6}，以及事件B：掷出5。你能同时掷出一个既是偶数又是5的数吗？不能。因此它们是互斥的。

P(A) = 3/6 = ½, and P(B) = 1/6. Using the addition rule, P(even or 5) = ½ + 1/6 = 3/6 + 1/6 = 4/6, which simplifies to 2/3. This matches the intuitive count: there are four favourable outcomes {2,4,6,5} out of six equally likely outcomes.

P(A) = 3/6 = ½，P(B) = 1/6。应用加法法则，P(偶数或5) = ½ + 1/6 = 3/6 + 1/6 = 4/6，化简后为2/3。这与直观计数相符：在六个等可能结果中，有四个有利结果{2,4,6,5}。

4. Mutually Exclusive vs. Non-Mutually Exclusive Events | 互斥事件与非互斥事件

It is equally important to recognise when events are not mutually exclusive. If it is possible for two events to occur at the same time, the simple addition rule would overcount their shared probability. A classic example is drawing a card from a standard deck of 52 playing cards. Let event C be ‘drawing a King’, and event D be ‘drawing a Heart’. There is one card that is both a King and a Heart: the King of Hearts.

同样重要的是要识别事件不互斥的情况。如果两个事件可能同时发生，简单的加法规则就会多算它们共享的概率。一个经典的例子是从一副标准的52张扑克牌中抽一张牌。设事件C为“抽到一张K”，事件D为“抽到一张红心”。有一张牌既是K又是红心：红心K。

If we simply added P(C) = 4/52 and P(D) = 13/52, we would get 17/52, but that counts the King of Hearts twice. The correct probability of ‘King or Heart’ uses the general addition rule: P(C or D) = P(C) + P(D) – P(C and D) = 4/52 + 13/52 – 1/52 = 16/52 = 4/13. Mutually exclusive events are a special case where P(C and D) = 0, so the formula reduces to the simple sum.

如果我们简单地将P(C)=4/52与P(D)=13/52相加，会得到17/52，但这把红心K计算了两次。“K或红心”的正确概率使用一般加法法则：P(C或D) = P(C) + P(D) – P(C和D) = 4/52 + 13/52 – 1/52 = 16/52 = 4/13。互斥事件是一种特殊情况，此时P(C和D)=0，因此公式简化为简单相加。

5. Using Venn Diagrams to Visualise | 利用韦恩图进行可视化

A Venn diagram is an excellent tool for understanding the relationship between events. For mutually exclusive events, the circles representing each event do not overlap at all. They sit side by side, showing that the events share no outcomes. The probability space is clearly divided into separate parts.

韦恩图是理解事件之间关系的一个极好工具。对于互斥事件，代表每个事件的圆圈完全不相交。它们并排放置，表示事件没有共享任何结果。概率空间被清晰地分割成独立的部分。

In a Venn diagram for a single dice roll, circle A (even numbers) would contain {2,4,6} and circle B (rolling a 5) would contain {5}. These circles do not touch. The total probability within both circles is the sum of their probabilities. When events are non-mutually exclusive, the circles overlap, and the overlapping region represents the outcomes they have in common, which must be subtracted to avoid double-counting.

在单个骰子掷出的韦恩图中，圆圈A（偶数）包含{2,4,6}，圆圈B（掷出5）包含{5}。这些圆圈互不接触。两个圆圈内总的概率就是它们概率之和。当事件非互斥时，圆圈会重叠，重叠区域代表它们共同拥有的结果，必须减去这些结果以避免重复计算。

6. Everyday Examples and KS3 Applications | 日常例子与KS3应用

Mutually exclusive events appear in many real-world contexts that KS3 students can easily relate to. When you select a student from your class at random, the events ‘student is 12 years old’ and ‘student is 14 years old’ are mutually exclusive because one person cannot be two different ages at once. Similarly, if you spin a colour spinner with sectors red, blue and green, the event ‘spinner lands on red’ and ‘spinner lands on blue’ are mutually exclusive.

互斥事件出现在许多KS3学生容易联想到的现实情境中。当你从班上随机选择一名学生时，“学生12岁”和“学生14岁”这两个事件是互斥的，因为一个人不可能同时是两个不同的年龄。同样，如果你旋转一个带有红、蓝、绿扇区的颜色转盘，“转盘停在红色”和“转盘停在蓝色”这两个事件也是互斥的。

Another helpful KS3 example is choosing a random day from the days of the week. The event ‘the day starts with the letter T’ (Tuesday, Thursday) and the event ‘the day is the first day of the weekend’ (Saturday) are mutually exclusive. These references help build intuition before moving to more formal notation.

另一个有帮助的KS3例子是从一周的日子中随机选一天。“日子以字母T开头”（星期二、星期四）和“日子是周末的第一天”（星期六）这两个事件是互斥的。这些参考案例有助于在进入更正式的符号之前建立直觉。

7. Practice Problem Walkthrough 1 | 练习题讲解 1

Question: A bag contains 5 red marbles, 3 blue marbles and 2 green marbles. One marble is picked at random. State whether the following pairs of events are mutually exclusive: (a) picking a red marble and picking a blue marble; (b) picking a red marble and picking a marble that is not green. Explain your answers.

问题：一个袋子里有5颗红色弹珠、3颗蓝色弹珠和2颗绿色弹珠。随机摸出一颗弹珠。判断下列成对事件是否互斥：(a) 摸到红色弹珠和摸到蓝色弹珠；(b) 摸到红色弹珠和摸到一颗不是绿色的弹珠。解释你的答案。

(a) A single pick cannot be both red and blue at the same time. The sets of outcomes {red} and {blue} have no intersection. Therefore these events are mutually exclusive. (b) The event ‘not green’ means the marble is either red or blue. So the possible outcomes for this event are {red, blue}. The event ‘red’ and the event ‘not green’ share the outcome red. Because it is possible to pick a red marble which is also not green, the events can occur together. Hence they are not mutually exclusive.

(a) 单次摸取不可能既是红色又是蓝色。结果集{红}和{蓝}没有交集。因此这些事件互斥。(b) “不是绿色”意味着弹珠是红色或蓝色。所以该事件的可能结果为{红,蓝}。事件“红色”与事件“不是绿色”共享结果红色。因为有可能摸到一颗红色弹珠同时它也不是绿色，所以事件可以同时发生。因此它们非互斥。

8. Practice Problem Walkthrough 2 with Calculations | 练习题讲解 2 带计算

Question: The probability that it will rain on Monday is 0.3. The probability that it will snow on Monday is 0.1. Assuming rain and snow are mutually exclusive events, what is the probability that it will rain or snow on Monday? What assumption did you make, and is it necessarily true?

问题：星期一下雨的概率是0.3。星期一下雪的概率是0.1。假设下雨和下雪是互斥事件，星期一下雨或下雪的概率是多少？你作出了什么假设？这个假设一定成立吗？

Using the addition rule for mutually exclusive events: P(rain or snow) = P(rain) + P(snow) = 0.3 + 0.1 = 0.4. So the probability is 0.4 or 40%. We assumed that rain and snow cannot occur at the same time, which is reasonable for a simple probability model, but in reality it is possible to have a mix of rain and snow (sleet) or alternating conditions. The assumption is a simplification that makes the calculation straightforward and is helpful for learning the basic rule.

应用互斥事件的加法法则：P(下雨或下雪) = P(下雨) + P(下雪) = 0.3 + 0.1 = 0.4。所以概率是0.4或40%。我们假设下雨和下雪不可能同时发生，这对于一个简单的概率模型来说是合理的，但在现实中，雨夹雪（雨雪混合）或交替天气是可能存在的。这个假设是一种简化，使计算变得直接，并且有助于学习基本规则。

9. Common Mistakes and How to Avoid Them | 常见错误及其避免方法

A frequent mistake is applying the simple addition rule P(A) + P(B) to events that are not mutually exclusive. Always ask yourself: Can these two outcomes happen together? If the answer is yes, you must use the general formula and subtract the overlap. Another error is confusing mutually exclusive events with independent events. Mutually exclusive means they cannot occur at the same time; independent means the occurrence of one does not affect the probability of the other. They are distinct concepts.

一个常见的错误是将简单的加法规则P(A) + P(B)应用于非互斥事件。要时刻问自己：这两个结果能同时发生吗？如果答案是肯定的，就必须使用通用公式并减去重叠部分。另一个错误是将互斥事件与独立事件混淆。互斥意味着它们不能同时发生；独立意味着一个事件的发生不影响另一个事件的概率。它们是不同的概念。

Students also sometimes forget to check that the sum of probabilities for all mutually exclusive outcomes in a sample space must equal 1. When given partial information, you can often find a missing probability by subtracting the known probabilities from 1. Practising with a variety of contexts, such as spinners, dice, cards and everyday scenarios, helps embed the correct reasoning.

学生们有时还会忘记检查样本空间中所有互斥结果的概率之和必须等于1。当给出部分信息时，通常可以用1减去已知概率来求出缺失的概率。在各种情境下练习，如转盘、骰子、纸牌和日常场景，有助于巩固正确的推理方式。

10. Exam-Style Question and Solution | 考试风格题目与解答

Question: A fair eight-sided spinner with sections labelled 1 to 8 is spun once. Event X is scoring a factor of 8. Event Y is scoring a multiple of 3. (a) Write down the set of outcomes for X and Y. (b) Explain why X and Y are mutually exclusive. (c) Find P(X or Y).

问题：一个标有数字1到8的公平八边形转盘被旋转一次。事件X是得分为8的因数。事件Y是得分为3的倍数。(a) 写出X和Y的结果集。(b) 解释为什么X和Y互斥。(c) 求出P(X或Y)。

Solution: (a) Factors of 8 from 1 to 8 are {1, 2, 4, 8}. So X = {1, 2, 4, 8}. Multiples of 3 from 1 to 8 are {3, 6}. Thus Y = {3, 6}. (b) The sets X and Y have no numbers in common. There is no outcome that is both a factor of 8 and a multiple of 3 on this spinner, so the events are mutually exclusive. (c) P(X) = 4/8 = ½, P(Y) = 2/8 = ¼. Since X and Y are mutually exclusive, P(X or Y) = ½ + ¼ = ¾. Alternatively, the combined favourable outcomes are {1,2,4,8,3,6}, which gives 6/8 = ¾.

解答：(a) 在1到8中，8的因数是{1, 2, 4, 8}。所以X = {1, 2, 4, 8}。在1到8中，3的倍数是{3, 6}。因此Y = {3, 6}。(b) 集合X和Y没有共同数字。在这个转盘上，没有结果既是8的因数又是3的倍数，因此事件互斥。(c) P(X) = 4/8 = ½，P(Y) = 2/8 = ¼。因为X与Y互斥，P(X或Y) = ½ + ¼ = ¾。或者，合并的有利结果集为{1,2,4,8,3,6}，得出6/8 = ¾。

11. Key Points Summary | 重点总结

Definition: Mutually exclusive events cannot occur at the same time.
定义：互斥事件不可能同时发生。
Addition Rule: For mutually exclusive events A and B, P(A or B) = P(A) + P(B).
加法法则：对于互斥事件A和B，P(A或B) = P(A) + P(B)。
Non-mutually exclusive: Use P(A or B) = P(A) + P(B) – P(A and B).
非互斥：使用P(A或B) = P(A) + P(B) – P(A和B)。
Venn diagrams: Non-overlapping circles show mutual exclusivity.
韦恩图：不相交的圆圈表示互斥性。
Check total probability: Sum of probabilities of all exclusive outcomes in sample space = 1.
检查总概率：样本空间中所有互斥结果的概率之和 = 1。

12. Explore Further and Keep Practising | 进一步探索并保持练习

Once you are confident with mutually exclusive events, try tackling problems that combine this idea with relative frequency or tree diagrams. Look for questions where you must list all possible outcomes systematically using a sample space diagram. The skills you build now will directly support probability topics in IGCSE, where you will study conditional probability and combined events in greater depth.

一旦你对互斥事件充满信心，就可以尝试解决将这一概念与相对频率或树状图结合起来的问题。寻找那些需要使用样本空间示意图系统列出所有可能结果的题目。你现在培养的技能将直接支持IGCSE中的概率主题，届时你将更深入地学习条件概率和复合事件。

Remember, probability is not just about formulas; it is about logical reasoning and carefully considering what the question is asking. Always define your events clearly and decide whether they overlap before choosing the correct rule. Keep practising with aleveler.com resources to strengthen your understanding.

请记住，概率不仅仅关乎公式；它还关乎逻辑推理和仔细思考问题所求的是什么。始终要清晰地定义你的事件，并在选择正确的规则之前判断它们是否重叠。继续使用aleveler.com的资源进行练习，以强化你的理解。

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Probability: Mutually Exclusive Events | 概率：互斥事件