📚 Mutually Exclusive and Exhaustive Events | 互斥事件与穷举事件
Probability is about chance. When we roll a fair dice, we can list all the possible outcomes: 1, 2, 3, 4, 5, 6. Some events cannot happen at the same time – we call them mutually exclusive. Other events together cover every possible outcome – we call them exhaustive. Understanding these two ideas helps us build correct probability models and check our answers.
概率研究的是机会。当我们掷一个公平的骰子时，可以列出所有可能的结果：1、2、3、4、5、6。有些事件不可能同时发生，我们称之为互斥事件。另一些事件合起来覆盖了每一个可能的结果，我们称之为穷举事件。理解这两个概念可以帮助我们建立正确的概率模型，并检验我们的答案。
1. Recap of basic probability | 基础概率回顾
Probability is always a number between 0 and 1. A probability of 0 means the event is impossible; a probability of 1 means the event is certain. We calculate the theoretical probability of an event as
P(event) = number of favourable outcomes ÷ total number of equally likely outcomes
概率总是0到1之间的一个数。概率为0表示事件不可能发生；概率为1表示事件一定发生。我们计算理论概率的公式为
P(事件) = 有利结果的个数 ÷ 所有等可能结果的总数
2. What are mutually exclusive events? | 什么是互斥事件？
clusive if they cannot happen at the same time. For example, when you flip a coin, getting 'heads' and getting 'tails' not show both at once. In symbols, if A and B are mutually exclusive, then P(A and B) = 0.
如果两个事件不可能同时发生，它们就是互斥的。例如，抛一枚硬币时，得到“正面”和得到“反面”是互斥的，因为硬币无法同时显示两面。用符号表示，若A与B互斥，则 P(A 且 B) = 0。
3. Examples of mutually exclusive events | 互斥事件的例子
Consider rolling a fair six-sided dice once. The event 'rolling a 3' and the event 'rolling an even number' are mutually exclusive because 3 is not even. Another pair of mutually exclusive events is 'drawing a red card' and 'drawing a black card' from an ordinary pack of playing cards – a single card cannot be both red and black.
考虑掷一次公平的六面骰子。事件“掷出3”和事件“掷出偶数”是互斥的，因为3不是偶数。另一对互斥事件是从一副普通扑克牌中“抽到红色牌”和“抽到黑色牌”——一张牌不可能既是红色又是黑色。
4. The addition rule for mutually exclusive events | 互斥事件的加法法则
If two events A and B are mutually exclusive, the probability that either A or B occurs is simply the sum of their individual probabilities:
P(A or B) = P(A) + P(B)
or example, the probability of rolling a 2 or a 5 on a fair dice is 1/6 + 1/6 = 2/6 = 1/3.
若两个事件A和B互斥，那么发生A或B的概率就是它们各自概率之和：
P(A 或 B) = P(A) + P(B)
这条规则之所以成立，是因为没有重叠。例如，在掷公平骰子时，得到2或5的概率为 1/6 + 1/6 = 2/6 = 1/3。
5. What are exhaustive events? | 什么是穷举事件？
Events are exhaustive if together they cover every possible outcome of an experiment. When you throw a dice, the set of events {roll a 1, roll a 2, roll a 3, roll a 4, roll a 5, roll a 6} is exhaustive because no other outcomes exist. Exhaustive events do not need to be mutually exclusive, but a set of events that is both mutually exclusive and exhaustive forms a complete partition of the sample space.
如果一组事件合起来覆盖了实验的每一个可能结果，这组事件就是穷举的。掷骰子时，事件集合{掷出1, 掷出2, 掷出3, 掷出4, 掷出5, 掷出6}是穷举的，因为没有其他结果存在。穷举事件不一定要互斥，但若一组事件既是互斥的又是穷举的，它们就构成了样本空间的一个完整划分。
6. Examples of exhaustive events | 穷举事件的例子
From a normal deck of cards, the events 'draw a red card', 'draw a black card' are exhaustive because every card is either red or black. However, 'draw a heart' and 'draw a spade' are not exhaustive, as there are also diamonds and clubs. Similarly, when choosing a day of the week at random, {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday} is exhaustive.
从一副普通扑克牌中，“抽到红色牌”和“抽到黑色牌”是穷举事件，因为每张牌不是红色就是黑色。然而，“抽到红桃”和“抽到黑桃”就不是穷举的，因为还有方块和梅花。类似地，随机选择一周中的一天时，{星期一, 星期二, 星期三, 星期四, 星期五, 星期六, 星期日}是穷举的。
7. The sum of probabilities for mutually exclusive and exhaustive events | 互斥且穷举事件的概率之和
bution where the sum is not 1, you know a mistake has been made or some outcomes have been missed.
如果一组事件既互斥又穷举，那么它们的概率之和等于1。这是一个非常有用的检验方法。对于公平的骰子，P(1) + P(2) + P(3) + P(4) + P(5) + P(6) = 1/6 × 6 = 1。如果你的概率分布总和不等于1，你就知道出错了，或者遗漏了某些结果。
8. Distinguishing between mutually exclusive and exhaustive | 区分互斥与穷举
Students often confuse these two concepts. Remember: mutually exclusive means NO OVERLAP – they cannot happen together. Exhaustive means NO OUTCOMES LEFT OUT – they cover the whole sample space. It is possible for events to be mutually exclusive but not exhaustive, such as {roll a 2, roll a 5} on a dice. It is also possible to be exhaustive but not mutually exclusive: {roll an even number, roll a number greater than 4} are exhaustive? No, they do not cover 1 and 3. Let's find a better example: from a pack of cards, {red card, face card} is exhaustive? No, a red face card is counted twice, and the whole space is still covered? Actually, every card is either red or black, face or number, but 'red card' and 'face card' together do not guarantee covering every outcome, e.g. black number cards. Therefore, to be exhaustive, the union of the events must equal the whole sample space.
学生常常混淆这两个概念。请记住：互斥意味着没有重叠——它们不会同时发生。穷举意味着没有遗漏结果——它们覆盖了整个样本空间。事件可以互斥但不穷举，比如骰子上的{掷出2, 掷出5}。事件也可以穷举但不互斥。例如，从一副扑克牌中，{红色牌, 人头牌}并不穷举，因为黑色数字牌没有被包含。若要成为穷举事件，所有事件的并集必须恰好等于整个样本空间。
9. Using Venn diagrams and tables | 使用维恩图和表格
Venn diagrams can help visualise mutually exclusive events: two circles that do not overlap. For exhaustive events, the circles together fill the entire sample space rectangle. Two-way tables are also useful to list outcomes and check whether events overlap or leave gaps. For example, when throwing two dice, a table of sums helps identify whether 'sum is odd' and 'sum is even' are mutually exclusive and exhaustive (they are both).
维恩图有助于将互斥事件可视化：两个不相交的圆。对于穷举事件，这些圆一起填满整个样本空间的矩形。双向表对于列出结果、检查事件是否重叠或遗漏也非常有用。例如，投掷两枚骰子时，用和值表就可以轻松看出“和为奇数”与“和为偶数”是否既互斥又穷举（它们就是如此）。
10. Experimental probability and exhaustiveness | 实验概率与穷举性
In an experiment, you can record frequencies for a set of mutually exclusive and exhaustive outcomes. The relative frequencies should add up to 1 (allowing for very slight rounding errors). For instance, if you spin a spinner with four colours 100 times and record the results, the sum of the relative frequencies for the four colours must be exactly 1. This helps you spot recording errors.
在实验中，你可以记录一组互斥且穷举的结果的频数。这些相对频率加起来应该等于1（允许极小的舍入误差）。例如，若你转动一个有四种颜色的转盘100次并记录结果，四种颜色的相对频率之和必须恰好为1。这可以帮助你发现记录错误。
11. Common mistakes and misconceptions | 常见错误与误解
- Assuming all mutually exclusive sets are exhaustive: 'Rain' and 'no rain' are mutually exclusive and exhaustive, but 'rain' and 'sun' are not exhaustive because it could be cloudy with no rain. Always check the sample space.
- Adding probabilities for non-mutually exclusive events: proach for events that are not mutually exclusive is P(A or B) = P(A) + P(B) − P(A and B).
- Forgetting that probability sum must be 1 for exhaustive, mutually exclusive events.
- 误以为所有互斥集合都是穷举的：“下雨”和“不下雨”互斥且穷举，但“下雨”和“晴天”就不穷举，因为可能多云而不下雨。一定要检查样本空间。
- 对非互斥事件直接相加概率：若把一副牌中的P(红色)和P(K)相加(1/2 + 1/13)，你会把红色K算了两次。对于非互斥事件，正确的公式是 P(A 或 B) = P(A) + P(B) − P(A 且 B)。
- 忘记互斥且穷举事件的概率和必须为1。
12. Key points and practice ideas | 重点总结与练习思路
Always start by listing all equally likely outcomes. Then define your events clearly. Ask two questions: can these events happen at the same time? (If no, mutually exclusive.) Do these events cover every single outcome? (If yes, exhaustive.) When you answer probability questions in your Cambridge Checkpoint exam, checking these properties will improve your accuracy and reasoning.
始终从列出所有等可能结果开始。然后清晰地定义你的事件。问自己两个问题：这些事件能同时发生吗？（若不能，则互斥。）这些事件涵盖了每一个结果吗？（若能，则穷举。）在剑桥Checkpoint考试中回答概率问题时，检查这些性质会提高你的准确性和逻辑性。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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