📚 P301 Revision: Probability | p301 复习：概率
This article revisits the key probability concepts covered in the Cambridge Lower Secondary Mathematics course, based on the revision material on page 301. You will explore theoretical and experimental probability, sample spaces, tree diagrams, and how to apply these ideas to solve typical problems. Each idea is illustrated with examples and followed by a clear explanation to strengthen your understanding.
本文回顾剑桥初中数学课程中第 301 页复习材料所涵盖的关键概率概念。你将探讨理论概率、实验概率、样本空间、树状图以及如何运用这些概念解决典型问题。每个概念都配有实例和清晰解释，以加深你的理解。
1. What is Probability? | 什么是概率？
Probability is a branch of mathematics that measures how likely an event is to happen. It gives us a numerical way to describe uncertainty, ranging from situations that are impossible to those that are absolutely certain.
概率是数学的一个分支，用于衡量事件发生的可能性。它用数字方式描述不确定性，覆盖从不可能到绝对确定的各种情形。
In everyday language, we often use words like ‘likely’, ‘unlikely’, ‘even chance’ or ‘certain’. Probability replaces these words with precise numbers between 0 and 1, making it easier to compare risks and make decisions.
在日常用语中，我们常用 “可能”、”不太可能”、”机会均等” 或 “必然” 这样的词。概率用 0 到 1 之间的精确数字取代这些词语，使风险比较和决策更加容易。
A probability of 0 means an event is impossible (it can never happen), while a probability of 1 means the event is certain (it will definitely happen). Most interesting probabilities lie somewhere in between.
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。大多数有趣的概率值介于两者之间。
2. The Probability Scale | 概率尺度
The probability scale provides a visual way to position the likelihood of events. It goes from 0 to 1, and we can also express probabilities as percentages (0% to 100%) or fractions.
概率尺度能直观地展示事件的可能性大小。它从 0 到 1，我们也可以用百分数（0% 到 100%）或分数来表示概率。
The table below shows common descriptions, their numerical values and equivalent expressions.
下表列出了常见的描述、对应的数值和等效表达。
| Description | Value | 中文描述 |
|---|---|---|
| Impossible | 0 | 不可能 |
| Very unlikely | close to 0 | 非常不可能 |
| Unlikely | 1/4 or 0.25 | 不太可能 |
| Even chance | 1/2 or 0.5 | 机会均等 |
| Likely | 3/4 or 0.75 | 很可能 |
| Very likely | close to 1 | 非常可能 |
| Certain | 1 | 必然 |
When you see a probability expressed as a fraction like 3/5, you can think of it as “3 out of 5 chances”. The scale helps you decide whether an event is more or less likely than another.
当你看到像 3/5 这样的分数概率时，可以将其理解为 “5 次中有 3 次的机会”。这个尺度能帮助你判断一个事件比另一个事件更可能还是更不可能发生。
3. Theoretical Probability | 理论概率
Theoretical probability is what we expect to happen when all outcomes are equally likely. It is calculated without actually performing an experiment, using the idea of symmetry or fairness.
理论概率是在所有结果等可能的前提下，我们预期发生的可能性。它不需要真正做实验，而是基于对称性或公平性进行计算。
The formula for theoretical probability is:
理论概率的计算公式为：
P(event) = (number of favourable outcomes) ÷ (total number of possible outcomes)
We use the capital letter P to denote “probability of”, followed by the event in brackets. All outcomes must be equally likely for this formula to be valid.
我们用大写字母 P 表示 “某事件的概率”，后面跟括号内的事件。只有当所有结果等可能时，该公式才有效。
Example: When rolling a fair six-sided die, the probability of rolling a number greater than 4 is P(greater than 4) = 2/6 = 1/3, because the favourable outcomes are 5 and 6.
举例：掷一个均匀的六面骰子，掷出大于 4 的数的概率为 P(大于4) = 2/6 = 1/3，因为有利结果是 5 和 6。
4. Experimental Probability | 实验概率
Experimental probability (also called relative frequency) is found by actually carrying out trials or by observing past data. It can be different from the theoretical probability, especially when the number of trials is small.
实验概率（也叫相对频率）是通过实际进行试验或观察以往数据得出的。它可能与理论概率不同，在试验次数较少时尤其明显。
The formula is:
其公式为：
Experimental probability = (number of times the event occurs) ÷ (total number of trials)
For instance, if you toss a coin 50 times and get heads 28 times, the experimental probability of heads is 28/50 = 0.56, while the theoretical probability is 0.5. The more trials you do, the closer the experimental probability usually gets to the theoretical value. This fact is called the law of large numbers.
例如，如果你抛一枚硬币 50 次，出现正面 28 次，那么正面的实验概率为 28/50 = 0.56，而理论概率是 0.5。你做的试验次数越多，实验概率通常越接近理论值。这一事实被称为大数定律。
In many real-world situations, we cannot list equally likely outcomes, so experimental probability gives us a practical way to estimate risk.
在许多现实情况下，我们无法列出等可能结果，因此实验概率为我们提供了一种实用的风险估计方法。
5. Sample Spaces | 样本空间
A sample space is the set of all possible outcomes of an experiment. Listing the sample space clearly is the first step to solving many probability problems.
样本空间是指一次实验所有可能结果的集合。清晰地列出样本空间是解决许多概率问题的第一步。
For a single coin toss, the sample space is {Heads, Tails}. For rolling a die, it is {1, 2, 3, 4, 5, 6}. When two events happen together, we list all ordered pairs or combinations.
对于单次抛硬币，样本空间是 {正面，反面}。对于掷骰子，则是 {1, 2, 3, 4, 5, 6}。当两个事件同时发生时，我们要列出所有有序对或组合。
Consider tossing two fair coins. The sample space can be written as:
考虑抛两枚均匀硬币。其样本空间可以写为：
- The sample space is {HH, HT, TH, TT}, where H stands for heads and T for tails.
- 样本空间是 {正正, 正反, 反正, 反反}，其中正表示正面，反表示反面。
From the sample space, we see there are four equally likely outcomes, so the probability of getting exactly one head is P(one head) = 2/4 = 1/2, because HT and TH satisfy the condition.
从样本空间可以看出，共有四个等可能结果，因此恰好出现一个正面的概率为 P(恰有一个正面) = 2/4 = 1/2，因为 HT 和 TH 满足条件。
A well-organised sample space, often in the form of a list or a table, helps you avoid missing any outcomes.
一个组织良好的样本空间（通常以列表或表格形式呈现）能帮助你避免遗漏任何结果。
6. Mutually Exclusive Events | 互斥事件
Two events are mutually exclusive if they cannot happen at the same time. For example, when rolling a die, the events “getting a 2” and “getting a 5” are mutually exclusive because a single roll cannot show both.
如果两个事件不能同时发生，则称它们是互斥的。例如，掷一个骰子时，”掷出 2″ 和 “掷出 5” 这两个事件是互斥的，因为一次投掷不可能同时出现两个数字。
For mutually exclusive events A and B, the probability of A or B happening is found by adding their individual probabilities:
对于互斥事件 A 和 B，发生 A 或 B 的概率等于各自概率相加：
P(A or B) = P(A) + P(B)
This is called the addition rule for mutually exclusive events. If events are not mutually exclusive, the rule needs adjustment to avoid double-counting the overlap, but at KS3 level, most “or” questions involve mutually exclusive events.
这被称为互斥事件的加法法则。如果事件不是互斥的，就需要调整规则以避免重复计算重叠部分，但在 KS3 阶段，大多数 “或” 的问题都涉及互斥事件。
On page 301, you may see questions like: “A
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