📚 Understanding Probability: Single Events and Sample Spaces | 理解概率：简单事件与样本空间
Probability is the branch of mathematics that measures how likely an event is to happen. In KS3, you learn to calculate probabilities for single events, use the probability scale from 0 to 1, and describe sample spaces to analyse all possible outcomes. This article will guide you through the key concepts of probability, helping you master simple event calculations, complementary events, and the use of diagrams such as two‑way tables.
概率是衡量事件发生可能性的数学分支。在 KS3 阶段，你将学习计算简单事件的概率，使用从 0 到 1 的概率尺度，并通过样本空间来分析所有可能的结果。本文将引导你掌握概率的核心概念，帮助你熟练计算简单事件概率、认识互补事件以及使用双向表等图表。
1. What is Probability? | 什么是概率？
Probability is a measure of how likely an event is to occur, expressed as a number between 0 and 1. A probability of 0 means the event is impossible, and a probability of 1 means it is certain. For example, if you toss a fair coin, the probability of getting heads is 1/2, because there are two equally likely outcomes.
概率是衡量事件发生可能性的度量，用 0 到 1 之间的数字表示。概率为 0 表示事件不可能发生，概率为 1 表示事件必然发生。例如，抛一枚公平硬币时，得到正面的概率是 1/2，因为有两种等可能的结果。
Probabilities can be written as fractions, decimals or percentages. The statement 'the chance of rain is 0.8' communicates the same likelihood as 4/5 or 80%. In probability we always assume that the outcomes we consider come from well‑defined trials with clear, repeatable conditions.
概率可以用分数、小数或百分数表示。'降雨概率为 0.8' 这一表述与 4/5 或 80% 传达的信息相同。在概率中，我们总是假定所考虑的结果来自定义明确的试验，并且条件清晰、可重复。
2. The Probability Scale | 概率尺度
The probability scale is a number line from 0 to 1 that helps us visualise how likely events are. Impossible events sit at 0, certain events at 1, and an even chance is placed at 1/2 (or 0.5). Values such as 1/4 (0.25) indicate an event that is less likely to happen, while 3/4 (0.75) means it is more likely to happen than not.
概率尺度是一条从 0 到 1 的数轴，可帮助我们直观地理解事件的可能性。不可能事件位于 0 点，必然事件位于 1 点，而均等的机会放在 1/2（或 0.5）处。像 1/4 (0.25) 这样的值表示事件不太可能发生，而 3/4 (0.75) 则表示事件发生的可能性大于不发生的可能性。
outcomes, so the probability is 3/6 = 1/2. This sits exactly at the midpoint of the scale.
你可以在概率尺度上标记公平骰子掷出偶数的概率。六种可能结果中有三个偶数 (2, 4, 6)，因此概率为 3/6 = 1/2，恰好落在尺度的中点。
3. Calculating Theoretical Probability | 计算理论概率
When all outcomes of an experiment are equally likely, the theoretical probability of an event E is found by comparing favourable outcomes to the total number of outcomes. The formula is:
P(E) = Number of favourable outcomes / Total number of possible outcomes
当试验的所有结果等可能时，事件 E 的理论概率通过比较有利结果与结果总数来求得。公式为：
P(E) = 有利结果数 / 可能结果总数
lity of selecting a yellow ball is 0/10 = 0, because there are no yellow balls in the bag.
例如，一个袋子里有 5 个红球、3 个蓝球和 2 个绿球。如果随机取出一个球，取到蓝球的概率是 3 / (5+3+2) = 3/10。取到黄球的概率为 0/10 = 0，因为袋中没有黄球。
Always check that your probability fraction lies between 0 and 1. If you obtain a fraction greater than 1, you have probably miscounted the outcomes.
永远要检查你的概率分数是否介于 0 和 1 之间。如果得到的分数大于 1，很可能是数错了结果数目。
4. Sample Spaces and Listing Outcomes | 样本空间与结果列举
,4), (H,5), (H,6), (T,1), (T,2), ..., (T,6). There are 2 × 6 = 12 equally likely outcomes.
样本空间是一次试验所有可能结果的集合。当你同时进行两个简单实验——例如抛硬币和掷骰子——你可以系统地列出样本空间。方法之一是写出所有数对：(H,1), (H,2), (H,3), (H,4), (H,5), (H,6), (T,1), (T,2), …, (T,6)，共有 2 × 6 = 12 个等可能的结果。
a number larger than 4 is 2/12 = 1/6, because only (H,5) and (H,6) satisfy the condition.
当结果不够直观时，树状图或表格会有所帮助。在 KS3 阶段，你需要列出两个事件的结果，并利用列表求概率。例如，得到正面且数字大于 4 的概率是 2/12 = 1/6，因为只有 (H,5) 和 (H,6) 满足条件。
5. Complementary Events | 互补事件
Complementary events are pairs of events that cover all possible outcomes and do not overlap. If A is an event, the complement of A (often written as A' or 'not A') is the event that A does not happen. The probabilities of an event and its complement always add up to 1:
P(A) + P(A') = 1
互补事件是一对覆盖所有可能结果且互不重叠的事件。如果 A 是一个事件，那么 A 的互补事件（常记作 A' 或 '非 A'）就是 A 不发生的事件。一个事件与其互补事件的概率之和总是等于 1：
P(A) + P(A') = 1
This relation is useful when it is easier to calculate the probability of the complement.
例如，一颗种子发芽的概率为 0.92，那么它不发芽的概率就是 1 − 0.92 = 0.08。当计算互补事件的概率更容易时，这一关系非常有用。
6. Experimental Probability and Relative Frequency | 实验概率与相对频率
Sometimes we cannot assume equally likely outcomes and must estimate probability by experiment. The experimental probability (or relative frequency) of an event is given by:
Experimental probability = Number of times the event occurred / Total number of trials
有时我们无法假定结果等可能，必须通过实验来估计概率。事件的实验概率（或相对频率）由下式给出：
实验概率 = 事件发生的次数 / 试验总次数
If a drawing pin is dropped 150 times and lands point‑up 87 times, the experimental probability of 'point‑up' e experiment with more trials, the relative frequency tends to settle around a fixed value – an idea known as the long‑run relative frequency.
如果将一枚图钉丢下 150 次，其中 87 次钉尖朝上，那么 '钉尖朝上' 的实验概率为 87/150 ≈ 0.58。如果增加试验次数进行重复实验，相对频率会趋于一个固定值——这一思想被称为长期相对频率。
Remember that experimental probability is an estimate. It may differ from the theoretical probability, especially when the number of trials is small.
请记住，实验概率只是一种估计，尤其是当试验次数较少时，它可能与理论概率存在差异。
7. Mutually Exclusive Events | 互斥事件
Two events are mutually exclusive if they cannot happen at the same time. For example, when rolling one die, the events 'obtaining a 5' and 'obtaining an even number' are not mutually exclusive? Actually, 5 is odd, so they cannot occur together – they are mutually exclusive. The events 'obtaining an even number' and 'obtaining a number greater than 4' are not mutually exclusive because 6 satisfies both.
如果两个事件不能同时发生，则称它们为互斥事件。例如，掷一粒骰子时，事件 '掷出 5' 和 '掷出偶数' 是互斥的，因为 5 是奇数。而事件 '掷出偶数' 和 '掷出大于 4 的数' 则不是互斥的，因为 6 同时满足两者。
For mutually exclusive events A and B, the probability that either A or B happens is the sum of their individual probabilities:
P(A or B) = P(A) + P(B)
对于互斥事件 A 和 B，A 或 B 发生的概率等于各自概率之和：
P(A 或 B) = P(A) + P(B)
If a spinner is divided into 8 equal sections labelled 1 to 8, the events 'landing on a 1' and 'landing on a 2' mutually exclusive. The probability of obtaining a 1 or a 2 equals 1/8 + 1/8 = 2/8 = 1/4.
如果转盘被均分为 8 个标有 1 到 8 的扇形，事件 '停在 1' 和 '停在 2' 是互斥的。得到 1 或 2 的概率是 1/8 + 1/8 = 2/8 = 1/4。
8. Using Two‑Way Tables | 使用双向表
A two‑way table organises data about two attributes and makes it easy to calculate probabilities. Each cell shows the frequency for a combination of categories. The row and column totals give you the information to find marginal probabilities.
双向表将关于两个属性的数据加以整理，方便计算概率。每个单元格展示某种类别组合的频数，行总数与列总数则提供了求边缘概率所需的信息。
Consider the following table showing the favourite sport of students by gender:
| Football | Basketball | Tennis | Total | |
| Boys | 12 | 8 | 4 | 24 |
| Girls | 6 | 10 | 8 | 24 |
| Total | 18 | 18 | 12 | 48 |
4. The probability that the student prefers football is 18/48 = 3/8, regardless of gender.
如果随机选取一名学生，选出喜欢篮球的女生的概率是 10/48 = 5/24。不论性别，选出喜欢足球的学生的概率是 18/48 = 3/8。
9. Probability from Venn Diagrams | 从韦恩图看概率
A Venn diagram is a visual way to show relationships between sets. It can be used to display a sample space with overlapping events. The rectangle represents the whole sample space, and circles represent events. Numbers written in each region show how many outcomes belong there.
韦恩图是展示集合关系的直观方式，也可用来呈现有重叠事件的样本空间。矩形表示整个样本空间，圆圈代表事件。每个区域中标出的数字表示有多少个结果属于该区域。
ou can fill a Venn diagram with the two overlapping circles. The intersection holds 7, the 'Art only' region holds 18 − 7 = 11, the 'Music only' he probability a randomly chosen student studies exactly one subject is (11+8)/30 = 19/30.
例如，在一个 30 名学生的班级中，18 人学习美术，15 人学习音乐，7 人两者都学。你可以用两个重叠的圆填充韦恩图。交集部分放 7，'只学美术' 的区域放 18 − 7 = 11，'只学音乐' 的区域放 15 − 7 = 8，两圆之外有 30 − (11+7+8) = 4 名学生。随机选出一名学生只学习一门科目的概率为 (11+8)/30 = 19/30。
10. Real‑Life Applications of Probability | 概率的实际应用
Probability is not just an abstract topic – it is used everywhere around us. Weather forecasts are expressed as probabilities (e.g. '90% chance of rain
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