📚 Introduction to Probability: From Impossible to Certain | 概率入门：从不可能到必然
Probability is a fascinating branch of mathematics that measures uncertainty. Whether you are rolling a die, flipping a coin, or checking the weather forecast, probability provides a way to quantify how likely an event is to occur. In KS3 Cambridge mathematics, you will learn the basics of probability, including the probability scale, how to calculate theoretical probabilities, and how to use diagrams such as sample space tables and tree diagrams to represent combinations of outcomes. Mastering these fundamentals will help you think logically about chance and make informed predictions.
概率是数学中一个迷人的分支，用来衡量不确定性。无论你是在掷骰子、抛硬币还是查看天气预报，概率都提供了一种量化事件发生可能性的方法。在 KS3 剑桥数学中，你将学习概率的基础知识，包括概率尺度、如何计算理论概率，以及如何使用样本空间表和树状图等图表来表示结果的组合。掌握这些基础知识将帮助你逻辑地思考随机现象并做出有根据的预测。
This article follows the KS3 Cambridge Mathematics framework and builds the topic step by step: vocabulary, the probability scale, the theoretical probability formula, sample space tables, tree diagrams, experimental probability, expected outcomes, and the mistakes that cost marks in tests. Every idea is explained in English first and then in Chinese, so you can revise in either language.
本文按照 KS3 剑桥数学的框架，循序渐进地构建议题：术语、概率尺度、理论概率公式、样本空间表、树状图、实验概率、期望结果，以及考试中容易丢分的错误。每个知识点都先以英文讲解，再以中文讲解，方便你用任意一种语言复习。
1. What is Probability? | 什么是概率？
Probability is a measure of the chance that a particular event will happen. It is always expressed as a number between 0 and 1, inclusive. An event with probability 0 is impossible - it will never happen. An event with probability 1 is certain - it will definitely happen. Most everyday events have probabilities somewhere between these extremes, reflecting the uncertainty we experience in real life.
概率是对某一特定事件发生几率的度量。它总是用一个介于 0 和 1 之间（包括 0 和 1）的数字来表示。概率为 0 的事件是不可能事件——它永远不会发生。概率为 1 的事件是必然事件——它一定会发生。日常生活中的大多数事件，其概率都介于这两个极端之间，反映了我们在现实中经历的不确定性。
In probability theory, an experiment (also called a trial) is any process that produces a well-defined outcome, such as tossing a coin, rolling a die, spinning a spinner, or drawing a card from a deck. The set of all possible results is called the sample space, and any collection of outcomes we are interested in is called an event.
在概率论中，实验（也叫试验）是指任何能够产生明确结果的过程，例如抛一枚硬币、掷一颗骰子、转动一个转盘，或者从一副牌中抽出一张牌。所有可能结果构成的集合称为样本空间，而我们感兴趣的任意一组结果称为事件。
percentage. For example, the probability of getting a head on a fair coin is ½ = 0.5 = 50%. Being able to move between these forms fluently is an essential skill in KS3 and beyond.
概率有三种等价的书写形式：分数、小数和百分数。例如，抛一枚均匀硬币得到正面的概率是 ½ = 0.5 = 50%。能够在这些形式之间自如转换，是 KS3 及以后阶段的一项基本技能。
2. The Probability Scale | 概率尺度
The probability scale runs from 0 to 1 and gives us a common language for describing chance. Values close to 0 mean an event is very unlikely; values close to 1 mean it is very likely; and a value of 0.5 means an even chance, where the event is just as likely to happen as not to happen.
概率尺度从 0 延伸到 1，为我们提供了一套描述机会大小的通用语言。接近 0 的数值表示事件非常不可能发生；接近 1 的数值表示非常可能发生；而 0.5 表示可能性均等，也就是事件发生与不发生的可能性一样大。
Words and numbers should always agree. If a weather forecast says there is a 90% chance of rain, then rain is "likely" or even "very likely", not "unlikely". Examiners often ask you to place an event on the scale or to choose the correct word, so learn the standard landmarks.
文字描述与数值必须一致。如果天气预报说降雨概率为 90%，那么下雨就是"很可能"甚至"极可能"，而不是"不太可能"。考官经常要求你把某个事件放到尺度上，或者选出正确的描述词，所以要记住这些标准的刻度点。
| Probability | 概率 | Description | 描述 | Example | 例子 |
| 0 | Impossible | 不可能 | Rolling a 7 on a normal die | 普通骰子掷出 7 |
| 0.25 (¼) | Unlikely | 不太可能 | Guessing one correct answer from four options | 四选一猜对答案 |
| 0.5 (½) | Even chance | 可能性均等 | Getting a head on a fair coin | 均匀硬币掷出正面 |
| 0.75 (¾) | Likely | 很可能 | Not rolling a 1 on a fair die | 均匀骰子不掷出 1 |
| 1 | Certain | 必然 | Getting a number less than 7 on a die | 骰子掷出小于 7 的数 |
3. Key Vocabulary You Must Know | 必须掌握的核心术语
Probability questions are written in precise mathematical language, and misreading one word can change the whole answer. Before calculating anything, make sure you can identify the experiment, the outcomes, the sample space and the event.
概率题目使用的是精确的数学语言，误读一个词就可能改变整个答案。在动手计算之前，先确保你能识别实验、结果、样本空间和事件。
- Experiment / trial - a repeatable process with a well-defined result. 实验／试验——可重复进行且结果明确的过程。
- Outcome - a single possible result of an experiment, e.g. rolling a 4. 结果——实验中单个可能的结果，例如掷出 4。
- Sample space - the set of all possible outcomes, often written inside curly brackets. 样本空间——所有可能结果的集合，通常用大括号表示。
- Event - a set of outcomes you are interested in, e.g. rolling an even number. 事件——你感兴趣的一组结果，例如掷出偶数。
- Fair - every outcome is equally likely; biased - some outcomes are more likely than others. 均匀——每个结果可能性相同；有偏——某些结果比其他结果更容易出现。
- Mutually exclusive - two events that cannot happen at the same time. 互斥——两个事件不可能同时发生。
- At random - every item has an equal chance of being chosen. 随机——每个对象被选中的机会相同。
For example, when you roll a fair six-sided die, the sample space is {1, 2, 3, 4, 5, 6}. The event "rolling a prime number" is {2, 3, 5}, and the event "rolling a number greater than 5" is {6}.
例如，掷一颗均匀的六面骰子时，样本空间是 {1, 2, 3, 4, 5, 6}。事件"掷出质数"是 {2, 3, 5}，事件"掷出大于 5 的数"是 {6}。
4. Calculating Theoretical Probability | 计算理论概率
Theoretical probability is calculated from reasoning about the situation, not from doing the experiment. When all outcomes are equally likely, the formula is simple and reliable.
理论概率来自于对情境的推理，而不是实际做实验。当所有结果可能性相同时，公式既简单又可靠。
P(event) = number of favourable outcomes ÷ total number of equally likely outcomes
The answer should always be written as a fraction in its simplest form, unless the question asks for a decimal or a percentage. If your answer is greater than 1 or negative, you have made a mistake, because no probability can lie outside the range 0 to 1.
答案通常要写成最简分数，除非题目要求小数或百分数。如果你的答案大于 1 或是负数，那就一定错了，因为任何概率都不可能超出 0 到 1 的范围。
(blue) = 3/10. Notice that 7/10 + 3/10 = 1, which confirms that every possibility has been accounted for.
例题：一个袋子里有 7 个红色棋子和 3 个蓝色棋子，随机取出一个棋子。棋子总数为 10，所以 P(红) = 7/10，P(蓝) = 3/10。注意 7/10 + 3/10 = 1，这证实了所有可能情况都已被计入。
n) = 4/8 = ½, and P(greater than 6) = 2/8 = ¼, because only 7 and 8 satisfy the condition.
再举一例：一个转盘被分成 8 个相等的扇形，编号 1 到 8。P(偶数) = 4/8 = ½，P(大于 6) = 2/8 = ¼，因为只有 7 和 8 满足条件。
5. Equally Likely Outcomes and Fair Objects | 等可能结果与均匀物体
The formula in Section 4 only works if every outcome is equally likely. A fair coin, a fair six-sided die, a well-shuffled pack of cards and an equally divided spinner all satisfy this condition. A biased die, a bent coin or a spinner with unequal sectors do not.
第 4 节的公式只有在每个结果可能性都相同时才成立。均匀的硬币、均匀的六面骰子、充分洗过的扑克牌以及等分的转盘都满足这一条件；而有偏的骰子、被折弯的硬币或扇形面积不等的转盘则不满足。
bs, spades) and 13 cards per suit. So P(a heart) = 13/52 = ¼, and P(a king) = 4/52 = 1/13.
对于一副标准的 52 张扑克牌，样本空间有 52 个等可能的结果。共有 4 种花色（红桃、方块、梅花、黑桃），每种花色 13 张。因此 P(红桃) = 13/52 = ¼，P(K) = 4/52 = 1/13。
If a question says "a biased die shows a 6 with probability 0.4", you cannot use the simple counting formula. Instead you must use the information given, together with the rule that all probabilities add up to 1.
如果题目说"一颗有偏的骰子掷出 6 的概率为 0.4"，你就不能使用简单的计数公式，而必须利用题目给出的信息，并结合所有概率之和为 1 这一规则。
6. Complementary Events | 对立事件
Two events are complementary if exactly one of them must happen. The complement of event A is written A′ and read as "not A". Since the probabilities of all possible outcomes must total 1, the complement rule follows immediately.
如果两个事件中必有一个发生且仅有一个发生，它们就是对立事件。事件 A 的对立事件记作 A′，读作"A 不发生"。由于所有可能结果的概率之和必须等于 1，对立事件法则便随之而来。
P(A′) = 1 − P(A)
This rule is extremely useful because it is often much easier to calculate the probability of "not A" than the probability of A itself. This is especially true for questions containing the phrase "at least one".
这条法则非常有用，因为计算"A 不发生"的概率常常比直接计算 A 的概率容易得多，尤其是在包含"至少一个"这类字眼的问题中。
ie) = 1/6, so P(not rolling a 6) = 1 − 1/6 = 5/6. Example 3: if a football team wins with probability 0.5 and draws with probability 0.3, then it loses with probability 1 − 0.5 − 0.3 = 0.2.
例 1：明天下雨的概率是 0.35，那么不下雨的概率就是 1 − 0.35 = 0.65。例 2：P(均匀骰子掷出 6) = 1/6，所以 P(不掷出 6) = 1 − 1/6 = 5/6。例 3：若某足球队获胜概率为 0.5、平局概率为 0.3，则输球概率为 1 − 0.5 − 0.3 = 0.2。
7. Sample Space Tables and Two-Way Tables | 样本空间表与双向表
When an experiment has two stages, listing every outcome in a systematic table prevents you from missing any. A sample space table (also called a possibility space or two-way table) shows all combinations at a glance.
当一个实验包含两个阶段时，把所有结果系统地列在表格中可以避免遗漏。样本空间表（也叫可能性空间表或双向表）可以一目了然地展示所有组合。
the scores. There are 6 × 6 = 36 equally likely outcomes. The table below shows the sums.
考虑掷两颗均匀骰子并把点数相加的情形。共有 6 × 6 = 36 个等可能结果，下表列出的是点数之和。
| + | 1 | 2 | 3 | 4 | 5 | 6 |
| 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 5 | 6 | 7 | 8 | 9 | 10 | 11 |
| 6 | 7 | 8 | 9 | 10 | 11 | 12 |
The sums 2 and 12 each appear only once, so P(total = 2) = 1/36 and P(total = 12) = 1/36.
点数之和为 7 的情况出现 6 次，所以 P(总和 = 7) = 6/36 = 1/6。这是最可能出现的总和，也是骰子游戏中 7 最常见的原因。而总和为 2 和 12 各只出现一次，所以 P(总和 = 2) = 1/36，P(总和 = 12) = 1/36。
Two-way tables are also used to organise data by two categories, for example boys and girls against whether they walk or cycle to school. From such a table you can read off probabilities directly, provided you divide by the correct grand total.
双向表也用来按两个类别整理数据，例如把男生和女生与步行或骑车上学对应起来。只要除以正确的总计，就能直接从这类表格中读出概率。
8. Tree Diagrams for Combined Events | 用于组合事件的树状图
A tree diagram shows the outcomes of two or more stages of an experiment. Each branch represents one outcome, and the probability is written on the branch. There are two golden rules for using tree diagrams.
树状图展示的是一个实验两个或多个阶段的结果。每一条分支代表一个结果，概率写在分支上。使用树状图有两条黄金法则。
- Multiply along the branches to find the probability of a combined outcome. 沿分支相乘，得到组合结果的概率。
- Add between the branches to find the probability of two or more combined outcomes. 在分支之间相加，得到两个或多个组合结果的概率。
ls) = ¼, P(head then tail) = ¼ and P(tail then head) = ¼. The four probabilities add up to 1, which is a useful check.
例题：抛一枚均匀硬币两次。每次抛掷出现正面和反面的概率都是 ½。两次都是正面的概率是 ½ × ½ = ¼。同样地，P(两次反面) = ¼，P(先正后反) = ¼，P(先反后正) = ¼。四个概率相加等于 1，这是一个有用的验算方法。
appens in two ways: head then tail, or tail then head. So P(exactly one head) = ¼ + ¼ = ½.
现在求恰好出现一次正面的概率。这有两种方式：先正后反，或先反后正。因此 P(恰好一次正面) = ¼ + ¼ = ½。
Remember that the probabilities on the branches coming out of any single point must always add up to 1. If they do not, you have made an error before you even start calculating.
请记住，从任何一个节点分出的所有分支上的概率之和必须等于 1。如果它们不等于 1，那么你在开始计算之前就已经出错了。
9. Independent and Dependent Events | 独立事件与相关事件
Two events are independent if the outcome of the first does not affect the outcome of the second. Tossing a coin twice, or rolling a die and then spinning a spinner, are independent events. For independent events, the multiplication rule applies directly.
如果第一个事件的结果不影响第二个事件的结果，这两个事件就是独立的。抛两次硬币，或者先掷骰子再转转盘，都是独立事件。对于独立事件，可以直接使用乘法法则。
P(A and B) = P(A) × P(B) (independent events)
Two events are dependent if the first outcome changes the probabilities for the second. This usually happens when items are taken without replacement. In that case the denominator of the second fraction decreases.
如果第一个结果改变了第二个事件的概率，这两个事件就是相关的。这种情况通常发生在"不放回"抽取时。此时第二个分数的分母会减小。
e second fraction uses 4/7 because one red counter has already been removed, leaving 4 reds out of 7 counters.
例题：一个袋子里有 5 个红棋子和 3 个蓝棋子，共 8 个。不放回地取出两个棋子。P(先红后红) = 5/8 × 4/7 = 20/56 = 5/14。第二个分数用 4/7，是因为已经取走了一个红棋子，剩下 7 个棋子中有 4 个是红色。
ing a 5 in a single roll. If the events can overlap, you must subtract the overlap: P(A or B) = P(A) + P(B) − P(A and B).
对于互斥事件，适用加法法则：P(A 或 B) = P(A) + P(B)。互斥事件不可能同时发生，例如在一次投掷中掷出 2 或掷出 5。如果两个事件可能重叠，则必须减去重叠部分：P(A 或 B) = P(A) + P(B) − P(A 和 B)。
10. Experimental Probability and Relative Frequency | 实验概率与相对频率
Theoretical probability predicts what should happen in theory. Experimental probability, also called relative frequency, describes what actually happened when an experiment was carried out.
理论概率预测的是理论上应该发生的情况。实验概率也称相对频率，描述的是实验实际进行时真实发生的情况。
Experimental probability = number of times the event occurs ÷ total number of trials
l probability for a fair coin would be 0.5, so the results suggest the coin may be biased.
假设一枚有偏的硬币被抛了 200 次，其中 90 次正面朝上。出现正面的实验概率为
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