📚 Mutually Exclusive Events and Expected Frequency | 互斥事件与期望频率
Probability allows us to describe how likely an event is to happen. In this article we focus on two key ideas: mutually exclusive events – which cannot happen at the same time – and expected frequency, which helps us predict how many times an outcome should occur when an experiment is repeated many times.
概率让我们能够描述一个事件发生的可能性有多大。本文聚焦两个关键概念：互斥事件——即不可能同时发生的事件，以及期望频率，它帮助我们预测当实验重复很多次时某个结果应该出现多少次。
1. Understanding Probability | 理解概率
Probability is a number between 0 and 1 (or 0% and 100%) that measures the chance of an event happening. A probability of 0 means the event is impossible, 1 means it is certain, and 0.5 means it has an even chance.
概率是介于 0 到 1（或 0% 到 100%）之间的一个数，用来衡量一个事件发生的机会。概率为 0 表示事件不可能发生，1 表示必然发生，0.5 表示发生的可能性是 50%。
ally likely outcomes, P(A) = (number of favourable outcomes) ÷ (total number of outcomes).
事件 A 的概率记为 P(A)。对于等可能的结果，P(A) = （有利结果的数量）÷ （所有可能结果的总数）。
2. The Probability Scale | 概率标度
We can place probabilities on a scale from 0 to 1. Words like 'impossible', 'unlikely', 'even chance', 'likely' and 'certain' map onto this scale. For example, rolling a 7 on a fair six‑sided dice is impossible (P = 0), while rolling a number less than 7 is certain (P = 1).
我们可以把概率放在 0 到 1 的标度上。像“不可能”、“不太可能”、“均等机会”、“很可能”和“必然”这些词都可以对应到这个标度上。例如，掷一个公平的六面骰子得到 7 是不可能的（P = 0），而掷出小于 7 的数是必然的（P = 1）。
Here is a quick reference table for the probability scale:
以下是概率标度的快速参考表：
| Probability P | Meaning / 含义 |
|---|---|
| P = 0 | Impossible / 不可能 |
| 0 < P < 0.5 | Unlikely / 不太可能 |
| P = 0.5 | Even chance / 均等机会 |
| 0.5 < P < 1 | Likely / 很可能 |
| P = 1 | Certain / 必然发生 |
3. 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. For example, when you toss a fair coin, getting 'heads' and getting 'tails' are mutually exclusive – you cannot get both on a single toss.
如果两个事件不能同时发生，它们就是互斥的。例如，当你抛一枚公平硬币时，得到“正面”和得到“反面”是互斥事件——你不可能在一次抛掷中同时得到两者。
Similarly, when rolling a normal six‑sided dice, the events 'roll a 2' and 'roll an odd number' are mutually exclusive, because a number cannot be both 2 and odd at the same time.
同样，掷一个普通的六面骰子时，“掷出 2”和“掷出奇数”这两个事件是互斥的，因为一个数不可能同时是 2 又是奇数。
4. The Addition Rule for Mutually Exclusive Events | 互斥事件的加法法则
When two events A and B are mutually exclusive, the probability that either A or B happens is the sum of their individual probabilities. This is written as:
当两个事件 A 和 B 互斥时，A 或 B 发生的概率就是它们各自概率的和。写成公式就是：
P(A or B) = P(A) + P(B)
This only works because there is no overlap – A and B never happen together, so we don't need to subtract anything.
这个法则之所以成立，是因为没有重叠——A 和 B 永远不会同时发生，因此不需要减去任何重叠的部分。
For example, a bag contains 3 red, 2 blue and 5 green sweets. Events 'picking a red sweet' and 'picking a blue sweet' ly are mutually exclusive. P(red) = 3/10, P(blue) = 2/10, so P(red or blue) = 3/10 + 2/10 = 5/10 = 1/2.
例如，一个袋子里有 3 颗红色糖果、2 颗蓝色糖果和 5 颗绿色糖果。事件“拿到红色糖果”和“拿到蓝色糖果”是互斥的。P(红色) = 3/10，P(蓝色) = 2/10，所以 P(红色或蓝色) = 3/10 + 2/10 = 5/10 = 1/2。
5. Identifying Mutually Exclusive Events | 识别互斥事件
To check whether events are mutually exclusive, ask: can both happen at the same time? If the answer is 'yes', they are not mutually exclusive.
要判断事件是否互斥，问自己：两者能同时发生吗？如果答案是“能”，它们就不是互斥的。
Consider drawing one card from a standard pack. The events 'draw a King' and 'draw a Heart' are not mutually exclusive, because you could draw the King of Hearts. However, 'draw a King' and 'draw a Queen' are mutually exclusive – a single card cannot be both a King and a Queen.
考虑从一副标准扑克牌中抽取一张牌。事件“抽到一张 K”和“抽到一张红桃”不是互斥的，因为你可能抽到红桃 K。而“抽到一张 K”和“抽到一张 Q”是互斥的——一张牌不可能既是 K 又是 Q。
In mutually exclusive cases, the addition rule P(A or B) = P(A) + P(B) applies perfectly.
在互斥的情况下，加法法则 P(A 或 B) = P(A) + P(B) 完全适用。
6. The Sum of All Mutually Exclusive Outcomes | 所有互斥结果的概率之和
mes of rolling a fair dice: getting 1, 2, 3, 4, 5 or 6. They are mutually exclusive (you can' et two numbers on one roll), and one of them must happen, so P(1) + P(2) + ... + P(6) = 1.
如果一组可能的结果既是互斥的，又涵盖了所有可能性，那么它们的概率之和总是 1。例如，掷公平骰子的结果：得到 1、2、3、4、5 或 6。它们是互斥的（一次不能掷出两个点数），并且其中必有一个发生，所以 P(1) + P(2) + ... + P(6) = 1。
This fact is useful when finding missing probabilities in a table or list.
这一事实在从表格或列表中寻找缺失概率时非常有用。
7. The Complement of an Event | 事件的补集
For any event A, the event 'not A' is called the complement of A, written A'. A and A' are always mutually exclusive because an event cannot both happen and not happen at the same time. Their probabilities add to 1:
对于任何事件 A，“非 A”这个事件称为 A 的补集，记作 A'。A 和 A' 总是互斥的，因为一件事不可能既发生又不发生。它们的概率之和为 1：
P(A) + P(A') = 1, so P(A') = 1 – P(A)
probability that it will rain tomorrow is 0.3, then the probability it will not rain is 1 – 0.3 = 0.7. This complement rule is widely used in probability calculations.
如果明天下雨的概率是 0.3，那么不下雨的概率就是 1 – 0.3 = 0.7。这个补集法则在概率计算中被广泛使用。
8. Expected Frequency – The Idea | 期望频率——基本概念
Expected frequency tells us how many times we would expect an event to happen if we repeat a trial many times. It is calculated by multiplying the probability of the event by the number of trials.
期望频率告诉我们，如果我们将一个试验重复很多次，我们预期某事件会发生多少次。计算方法是用事件的概率乘以试验次数。
Expected frequency = P(event) × number of trials
air coin 200 times, P(heads) = 0.5, so the expected frequency of heads is 0.5 × 200 = 100.
例如，如果你抛一枚公平硬币 200 次，P(正面) = 0.5，所以正面的期望频率是 0.5 × 200 = 100。
Remember: the expected frequency is a theoretical prediction, not a guarantee – the actual result may be different, especially for a small number of trials.
记住：期望频率是理论上的预测，并非保证——实际结果可能会不同，特别是在试验次数较少时。
9. Using Expected Frequency with Mutually Exclusive Events | 将期望频率用于互斥事件
quency for red is 0.3 × 400 = 120, for blue 0.5 × 400 = 200, and for green 0.2 × 400 = 80.
我们经常将互斥事件和期望频率结合起来。假设一个不均匀的转盘有如下概率：红色 0.3，蓝色 0.5，绿色 0.2。这些颜色是互斥的，因为转盘不可能同时停在两种颜色上。如果我们转动 400 次，红色的期望频率是 0.3 × 400 = 120，蓝色是 0.5 × 400 = 200，绿色是 0.2 × 400 = 80。
the total number of spins. This is a good way to check your calculations when dealing with a complete set of mutually exclusive outcomes.
注意，期望频率之和（120 + 200 + 80 = 400）等于总转动次数。在处理一组完备的互斥结果时，这种检查方法非常有效。
10. Real‑World Applications | 实际应用
Mutually exclusive events and expected frequency are used in many areas: weather forecasting ('rain' and 'no rain' are mutually exclusive and complementary), quality control in factories (expected number of defective items), and games of chance (calculating fair price of raffle tickets).
互斥事件和期望频率被用于许多领域：天气预报（“下雨”和“不下雨”既是互斥的，又是互补的）、工厂的质量控制（次品的期望数量），以及概率游戏（计算抽奖券的公平价格）。
Understanding these concepts gives you powerful tools to make predictions and analyse everyday situations involving chance.
理解这些概念能让你拥有分析日常生活中涉及机会情况的强大工具，帮助你做出预测。
11. Practice Tip – Spotting the Keywords | 练习技巧——识别关键词
When reading probability questions, look for phrases like 'cannot happen at the same time', 'either ... or ...', or 'at the same time'. These signal you should consider mutually exclusive events. If you see 'expected number of times' or 'predict how many', think expected frequency.
在阅读概率问题时，注意像“不能同时发生”、“要么……要么……”或“同时”这样的短语。它们提示你该考虑互斥事件。如果你看到“期望的次数”或“预测有多少次”，就要想到期望频率。
Always check whether events are mutually exclusive before adding their probabilities. If the events can overlap, you need a more general addition rule (for KS4).
在将概率相加之前，一定要检查事件是否互斥。如果事件可能重叠，就需要用到更一般的加法法则（那是 KS4 的内容）。
12. Summary – Key Points to Remember | 总结——记忆要点
- Mutually exclusive events: cannot happen together; P(A or B) = P(A) + P(B).
互斥事件：不能同时发生；P(A 或 B) = P(A) + P(B)。 - Complement: A' means not A; P(A') = 1 – P(A).
补集：A' 表示非 A；P(A') = 1 – P(A)。 - Probability sum: For all mutually exclusive outcomes covering the sample space, probabilities add to 1.
概率之和：对于覆盖样本空间的所有互斥结果，概率之和为 1。 - Expected frequency expected frequency = probability × number of trials; it is a prediction, not exact result.
期望频率：期望频率 = 概率 × 试验次数；它是一种预测，并非精确结果。 - Check your work: For a complete set of mutually exclusive events, the sum of expected frequencies equals the total number of trials.
检查计算：对于一组完备的互斥事件，期望频率之和等于总试验次数。
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