📚 Probability Essentials (Unit 26.1) | 概率要点（第26.1单元）
Probability is a way of measuring how likely an event is to happen. It ranges from 0 (impossible) to 1 (certain) and helps us understand chance in everyday situations, from flipping a coin to predicting the weather. In this article, we explore key probability concepts covered in Cambridge KS3 Mathematics Unit 26.1, building a strong foundation for future study.
概率是衡量事件发生可能性大小的一种方法。它的值介于 0（不可能）到 1（必然）之间，能帮助我们理解日常生活中的随机现象，从抛硬币到天气预报。本文将梳理剑桥 KS3 数学第 26.1 单元的核心概念，为今后的学习打下坚实基础。
1. What is Probability? | 什么是概率？
Probability is a number that describes the chance of something happening. It can be written as a fraction, a decimal or a percentage. The word 'event' is used for the outcome we are interested in. For example, rolling a 4 on a fair six-sided die is an event.
概率是一个描述某事发生机会的数值，可以用分数、小数或百分数表示。我们关注的结果称为“事件”。例如，在一枚均匀六面骰子上掷出 4 点就是一个事件。
In mathematics, we write P(A) to mean 'the probability that event A occurs'. All probabilities lie between 0 and 1 inclusive. A probability of 0 means the event will never happen, and a probability of 1 means it is certain to happen. The closer the probability is to 1, the more likely the event.
数学中用 P(A) 表示“事件 A 发生的概率”。所有概率都在 0 到 1 之间（含端点）。概率为 0 表示事件永远不会发生，概率为 1 表示事件必然发生。概率越接近 1，事件越可能出现。
2. The Probability Scale | 概率标度
The probability scale is a number line from 0 to 1. We label the ends 'impossible' at 0 and 'certain' at 1. The midpoint, 0.5 (or ½), means an event has an even chance of happening, like getting heads on a fair coin toss. Words such as 'likely', 'unlikely', 'very likely' and 'very unlikely' can be placed along the scale.
概率标度是一条从 0 到 1 的数轴。0 端标为“不可能”，1 端标为“必然”。中点 0.5（即 ½）表示事件发生的机会均等，例如抛一枚均匀硬币得到正面。我们可以将“很可能”、“不太可能”、“极有可能”和“极不可能”等词语标注在标度上。
It is important to remember that even if an event is 'very likely', it is not certain. A probability of 0.99 still leaves a 1% chance the event does not occur. The scale helps us compare risks and make informed decisions.
务必记住，即使一个事件“极有可能”，也并非必然发生。0.99 的概率仍留有 1% 的不发生机会。概率标度帮助我们比较风险并做出合理决策。
3. Outcomes and Events | 结果与事件
An outcome is a single possible result of a probability experiment. When you roll a die, the possible outcomes are 1, 2, 3, 4, 5 and 6. The set of all possible outcomes is called the sample space. An event is a set of one or more outcomes. For example, rolling an even number is the event {2, 4, 6}.
一个结果是指概率实验的单个可能结果。掷一枚骰子时，可能的结果有 1、2、3、4、5 和 6。所有可能结果的集合称为样本空间。事件则是一个或多个结果的集合。例如，掷出偶数是事件 {2, 4, 6}。
We often list outcomes in curly brackets when the sample space is small. If the outcomes are equally likely, we can easily calculate probabilities. Understanding the difference between an outcome and an event is essential for setting up probability calculations correctly.
当样本空间不大时，我们通常用花括号列出结果。如果每个结果等可能发生，就能够方便地计算概率。分清结果和事件的区别，对正确建立概率计算至关重要。
4. Theoretical Probability | 理论概率
For situations where all outcomes are equally likely, the theoretical probability of an event A is given by:
P(A) = Number of favourable outcomes / Total number of outcomes
对于所有结果等可能的情况，事件 A 的理论概率由下式给出：
P(A) = 有利结果的数量 / 结果的总数
on states otherwise. Probabilities can also be written as decimals, so 1/3 ≈ 0.333 or 33.3%.
例如，掷一枚均匀骰子时，得到大于 4 的数的概率是 P(5 或 6) = 2/6 = 1/3。除非题目另有要求，分数通常应化简。概率也常写成小数，因此 1/3 ≈ 0.333 或 33.3%。
The theoretical probability assumes perfect randomness and fairness of the objects used (dice, coins, spinners). It gives the expected likelihood before any actual experiments are carried out.
理论概率假设所用物体（骰子、硬币、转盘）完美随机且公平。它给出了进行任何实际实验之前预期的可能性。
5. Experimental Probability | 实验概率
Experimental probability is found by carrying out an experiment or survey. It is also called relative frequency and is calculated as:
Experimental probability = Number of times event occurs / Total number of trials
实验概率通过进行实验或调查获得，也称相对频率，计算公式为：
实验概率 = 事件发生的次数 / 试验总次数
ay differ slightly from the theoretical probability of 0.5 due to random variation. With more trials, the experimental probability tends to get closer to the theoretical probability.
如果你抛硬币 100 次，得到正面 48 次，则正面的实验概率为 48/100 = 0.48。由于随机波动，这可能与 0.5 的理论概率略有不同。随着试验次数增加，实验概率会趋近于理论概率。
It is important to distinguish between experimental and theoretical probability, especially when analysing data or questioning whether a die might be biased.
区分实验概率与理论概率非常重要，尤其在分析数据或质疑骰子是否可能存在偏斜时。
6. Complementary Events | 互补事件
The complement of event A is the event that A does not happen. It is denoted by A' (or sometimes Ā). Since an event either occurs or does not occur, the sum of their probabilities is always 1:
P(A) + P(A') = 1
事件 A 的补事件是指 A 不发生的事件，记作 A'（有时也记作 Ā）。因为事件要么发生要么不发生，两者的概率之和总是 1：
P(A) + P(A') = 1
arly, if a spinner lands on red with probability 2/5, then the probability of not landing on red is 3/5.
这个关系非常有用。如果 P(明天下雨) = 0.3，那么 P(明天不下雨) = 1 − 0.3 = 0.7。同理，如果转盘指针落在红色区域的概率为 2/5，那么不落在红色区域的概率就是 3/5。
Complementary events help simplify calculations. Instead of finding the probability of 'at least one' event happening, it is often easier to find the probability of its complement and subtract from 1.
互补事件有助于简化计算。与其直接求“至少发生一次”的概率，通常先求出补事件的概率再被 1 减会更简单。
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 a die, the events 'roll a 3' and 'roll a 5' are mutually exclusive. If events A and B are mutually exclusive, then:
P(A or B) = P(A) + P(B)
如果两个事件不可能同时发生，它们就互斥。例如，掷一枚骰子时，“掷出 3 点”和“掷出 5 点”互斥。若事件 A、B 互斥，则有：
P(A 或 B) = P(A) + P(B)
/3. However, if events are not mutually exclusive, we cannot simply add probabilities because we would double-count the overlap.
此即互斥事件的加法法则。对于均匀六面骰子，P(3 或 5) = 1/6 + 1/6 = 1/3。但如果事件不互斥，简单相加会导致重复计算重叠部分。
Being able to identify mutually exclusive events is key to using the addition rule correctly. Overlapping events require a different approach, often involving Venn diagrams or subtracting the intersection.
能识别互斥事件是正确使用加法法则的关键。有重叠的事件需要采用不同方法，通常借助维恩图或扣除交集。
8. Sample Spaces | 样本空间
A sample space is the list of all possible outcomes of an experiment. For a single coin toss, the sample space is {Heads, Tails}. For rolling a die, it is {1, 2, 3, 4, 5, 6}. When more than one event is combined, we use tables or diagrams to systematically list all outcomes.
样本空间是实验所有可能结果的列表。对于单次抛硬币，样本空间是 {正面, 反面}。对于掷一枚骰子，样本空间为 {1, 2, 3, 4, 5, 6}。当组合多个事件时，我们借助表格或图表系统性地列出所有结果。
head is 2/4 = ½. Organising outcomes carefully prevents missing or repeating combinations.
两枚硬币同时抛掷时，样本空间可以表示为 {HH, HT, TH, TT}。如果每个结果等可能，则恰好出现一次正面的概率为 2/4 = ½。仔细整理结果可以避免遗漏或重复组合。
Sample space diagrams are particularly useful for solving problems involving two dice, two spinners, or combined events where the order matters.
样本空间图对于解决涉及两个骰子、两个转盘或顺序重要的组合事件尤其有用。
9. Using Probability to Predict | 利用概率进行预测
Probability can be used to predict how many times an event will occur in a given number of trials. If the probability of an event is p and we carry out n trials, the expected frequency is:
Expected frequency = n × p
概率可以用来预测在给定试验次数中某事件发生多少次。如果事件概率为 p，试验次数为 n，则期望频数是：
期望频数 = n × p
6 = 10 times. This is only a long-run average; actual results will vary because of chance.
例如，均匀骰子被掷 60 次，我们预期每个点数出现约 60 × 1/6 = 10 次。这仅是长期平均值，实际结果会因随机性而波动。
Expected frequency helps us check whether an experimental result is surprising. If a die is rolled 300 times and a '6' appears only 30 times, the gap from the expected 50 might suggest the die is biased.
期望频数有助于判断实验结果是否反常。若将骰子掷 300 次，“6”仅出现 30 次，与预期的 50 次相差较大，可能意味着骰子有偏斜。
10. Key Points to Remember | 核心记忆要点
All probabilities lie between 0 and 1 inclusive; they can be expressed as fractions, decimals or percentages.
所有概率都在 0 到 1 之间（含端点）；可以用分数、小数或百分数表示。
For equally likely outcomes, P(Event) = favourable outcomes ÷ total outcomes.
对于等可能结果，P(事件) = 有利结果数 ÷ 总结果数。
f successes ÷ number of trials; it gets closer to theoretical probability with more trials.
实验概率 = 成功次数 ÷ 试验次数；试验次数越多，越接近理论概率。
P(A) + P(not A) = 1; use this shortcut to save time.
P(A) + P(非 A) = 1；利用这一捷径可节省时间。
For mutually exclusive events, simply add their probabilities to find P(A or B). Be careful of overlapping events.
对于互斥事件，直接相加求 P(A 或 B)；注意有重叠的事件。
Always check that your probability makes sense – it must never be negative or exceed 1.
务必检查所求概率是否合理——绝不能为负数或大于 1。
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