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Introduction to Probability – 概率入门

Probability is one of the most practical and fascinating topics in KS3 Mathematics. It helps us understand chance, make predictions, and evaluate risk in everyday situations – from weather forecasts to game strategies. In the Cambridge Lower Secondary Mathematics curriculum, probability is introduced gradually through Key Stage 3, building from simple experiments to more sophisticated calculations involving combined events. This article provides a comprehensive guide to probability as taught in the Cambridge KS3 syllabus, with clear explanations, worked examples, and practice problems.

概率是 KS3 数学中最实用、最引人入胜的主题之一。它帮助我们理解随机性、做出预测、评估日常情景中的风险 – 从天气预报到游戏策略。在剑桥初中数学课程中，概率通过 Key Stage 3 逐步引入，从简单的实验过渡到涉及组合事件的更复杂计算。本文提供了剑桥 KS3 教学大纲中概率教学的全面指南，包含清晰的解释、例题和练习题。

What is Probability? – 什么是概率？

Probability is a measure of how likely an event is to occur. It is expressed as a number between 0 and 1, where 0 means the event is impossible and 1 means the event is certain. Probability can also be written as a fraction, a decimal, or a percentage. For example, the probability of flipping a fair coin and getting heads is 1/2, 0.5, or 50%.

概率是衡量事件发生可能性大小的量度。它用 0 到 1 之间的数字表示，其中 0 表示事件不可能发生，1 表示事件必然发生。概率也可以写成分数、小数或百分比。例如，抛一枚均匀硬币得到正面的概率是 1/2、0.5 或 50%。

The basic formula for probability is:

概率的基本公式是：

Probability = Number of favourable outcomes / Total number of possible outcomes

概率 = 有利结果的数量 / 所有可能结果的总数

This formula works when all outcomes are equally likely, which is the assumption we start with in KS3. Understanding this fundamental relationship is the key to solving most probability problems at this level.

当所有结果等可能时，这个公式适用，这也是我们在 KS3 阶段的基础假设。理解这一基本关系是解决该阶段大多数概率问题的关键。

The Probability Scale – 概率标尺

The probability scale is a visual tool that helps students understand where events fall on the spectrum from impossible to certain. On a line from 0 to 1, we can place different events according to their likelihood. This is particularly helpful for developing intuition before moving into calculations.

概率标尺是一个可视化工具，帮助学生理解事件在从不可能到必然的谱系中的位置。在 0 到 1 的线段上，我们可以根据不同事件的可能性大小放置它们。这在进入计算之前，特别有助于培养直觉。

Here are the key markers on the probability scale:

以下是概率标尺上的关键标记：

0 (Impossible) – The sun rising in the west. An event with no chance of occurring.

0 (不可能) – 太阳从西边升起。完全不可能发生的事件。

1/4 (Unlikely) – Rolling a 6 on a fair six-sided die. There is one favourable outcome out of six possibilities.

1/4 (不太可能) – 掷一个均匀的六面骰子得到 6。六个可能结果中只有一个有利结果。

1/2 (Even chance) – Getting heads when flipping a fair coin. Two equally likely outcomes.

1/2 (等可能) – 抛一枚均匀硬币得到正面。两个等可能的结果。

3/4 (Likely) – Not rolling a 6 on a fair six-sided die. Five favourable outcomes out of six.

3/4 (很可能) – 掷一个均匀的六面骰子不得 6。六个可能结果中有五个有利结果。

1 (Certain) – The sun will set tonight. An event that is guaranteed to happen.

1 (必然) – 太阳今晚会落山。一定会发生的事件。

Sample Spaces and Outcomes – 样本空间与结果

A sample space is the set of all possible outcomes of an experiment. In KS3 Cambridge Mathematics, students learn to list sample spaces systematically using tables, lists, and diagrams. Being able to identify and enumerate all possible outcomes is the foundation for calculating accurate probabilities.

样本空间是实验所有可能结果的集合。在 KS3 剑桥数学中，学生学习使用表格、列表和图来系统地列出样本空间。能够识别并枚举所有可能结果是计算准确概率的基础。

For example, when rolling a fair six-sided die, the sample space is {1, 2, 3, 4, 5, 6}. When flipping a coin, the sample space is {Heads, Tails}. When doing both simultaneously, the sample space expands to include all combinations:

例如，掷一个均匀的六面骰子时，样本空间是 {1, 2, 3, 4, 5, 6}。抛一枚硬币时，样本空间是 {正面, 反面}。当同时进行两者时，样本空间扩展到包含所有组合：

{(H,1), (H,2), (H,3), (H,4), (H,5), (H,6), (T,1), (T,2), (T,3), (T,4), (T,5), (T,6)}

{(H,1), (H,2), (H,3), (H,4), (H,5), (H,6), (T,1), (T,2), (T,3), (T,4), (T,5), (T,6)}

This is a two-way table, and it contains 12 equally likely outcomes. Understanding how to construct such tables is essential for solving problems involving two events.

这是一个双向表，包含 12 个等可能的结果。理解如何构建这样的表格对于解决涉及两个事件的问题至关重要。

Experimental vs Theoretical Probability – 实验概率与理论概率

There are two main approaches to probability in the Cambridge KS3 curriculum. Theoretical probability is what we calculate using the formula – it is based on what should happen in theory, assuming fair conditions. Experimental probability, also called relative frequency, is based on actual trials and observations.

剑桥 KS3 课程中有两种主要的概率方法。理论概率是我们使用公式计算出来的 – 它基于理论上应该发生的情况，假设条件是公平的。实验概率，也称为相对频率，基于实际的试验和观察。

The formula for experimental probability is:

实验概率的公式是：

Experimental Probability = Number of times the event occurs / Total number of trials

实验概率 = 事件发生的次数 / 总试验次数

For example, if you flip a coin 100 times and get heads 47 times, the experimental probability of heads is 47/100 = 0.47. This is close to the theoretical probability of 0.5, but not exactly equal. As the number of trials increases, the experimental probability tends to get closer to the theoretical probability – this is called the Law of Large Numbers.

例如，如果你抛硬币 100 次，得到 47 次正面，那么正面的实验概率是 47/100 = 0.47。这接近理论概率 0.5，但并不完全相等。随着试验次数的增加，实验概率趋向于接近理论概率 – 这被称为大数定律。

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, getting a 3 and getting a 5 are mutually exclusive – you cannot roll both numbers on a single throw. Understanding mutual exclusivity is important because it affects how we add probabilities.

如果两个事件不能同时发生，则它们是互斥的。例如，掷骰子时，得到 3 和得到 5 是互斥的 – 你不可能在一次投掷中同时掷出两个数字。理解互斥性很重要，因为它影响我们如何相加概率。

For mutually exclusive events A and B, the probability that either A or B occurs is:

对于互斥事件 A 和 B，A 或 B 发生的概率是：

P(A or B) = P(A) + P(B)

P(A 或 B) = P(A) + P(B)

This is called the Addition Rule for mutually exclusive events. For example, the probability of rolling either a 2 or a 4 on a fair die is 1/6 + 1/6 = 2/6 = 1/3.

这被称为互斥事件的加法法则。例如，在均匀骰子上掷出 2 或 4 的概率是 1/6 + 1/6 = 2/6 = 1/3。

Independent Events – 独立事件

Two events are independent if the outcome of one does not affect the outcome of the other. For example, flipping a coin and rolling a die are independent – the coin result does not influence the die result. This concept is introduced in the later stages of KS3 and is fundamental to understanding combined probability.

如果一个事件的结果不影响另一个事件的结果，则这两个事件是独立的。例如，抛硬币和掷骰子是独立的 – 硬币的结果不影响骰子的结果。这个概念在 KS3 后期引入，是理解组合概率的基础。

For independent events A and B, the probability that both A and B occur is:

对于独立事件 A 和 B，A 和 B 同时发生的概率是：

P(A and B) = P(A) x P(B)

P(A 且 B) = P(A) x P(B)

This is called the Multiplication Rule for independent events. For example, the probability of getting heads on a coin AND rolling a 6 on a die is 1/2 x 1/6 = 1/12.

这被称为独立事件的乘法法则。例如，抛硬币得到正面并且掷骰子得到 6 的概率是 1/2 x 1/6 = 1/12。

Tree Diagrams – 树状图

Tree diagrams are powerful visual tools for representing sequences of events and calculating combined probabilities. In KS3 Cambridge Mathematics, students learn to draw tree diagrams for two or more independent events. Each branch represents a possible outcome, and probabilities are written along the branches.

树状图是表示事件序列和计算组合概率的强大可视化工具。在 KS3 剑桥数学中，学生学习为两个或更多独立事件绘制树状图。每个分支代表一个可能的结果，概率写在分支旁边。

To find the probability of a particular sequence of outcomes, multiply the probabilities along the branches that lead to that sequence. To find the total probability of an event that can occur in multiple ways, add the probabilities of all the relevant paths.

要找出特定结果序列的概率，将通向该序列的各分支上的概率相乘。要找出可以通过多种方式发生的事件总概率，将所有相关路径的概率相加。

For example, consider flipping a coin twice. The tree diagram has two levels, each with two branches (Heads, Tails). The probability of getting two heads in a row is 1/2 x 1/2 = 1/4. The probability of getting exactly one head (HT or TH) is 1/4 + 1/4 = 1/2.

例如，考虑抛硬币两次。树状图有两层，每层有两个分支（正面、反面）。连续两次得到正面的概率是 1/2 x 1/2 = 1/4。恰好得到一次正面的概率（先正后反或先反后正）是 1/4 + 1/4 = 1/2。

Probability in Real Life – 现实生活中的概率

Probability is not just an abstract mathematical concept – it has countless real-world applications that make it one of the most relevant topics in the KS3 curriculum. Understanding probability helps students become more informed decision-makers in their daily lives.

概率不仅仅是抽象的数学概念 – 它有无数的现实世界应用，使其成为 KS3 课程中最相关的主题之一。理解概率有助于学生在日常生活中成为更明智的决策者。

Weather forecasting relies heavily on probability. When the Met Office says there is a “70% chance of rain,” they are expressing a probability of 0.7 based on computer models and historical data. Insurance companies use probability to calculate premiums – they assess the likelihood of accidents, illnesses, and natural disasters. In medicine, probability helps doctors interpret test results and determine the most likely diagnosis. Even in sports, probability is used to analyse player performance, predict match outcomes, and develop game strategies.

天气预报严重依赖概率。当气象局说”70% 的降雨概率”时，他们基于计算机模型和历史数据表达了一个 0.7 的概率。保险公司使用概率来计算保费 – 他们评估事故、疾病和自然灾害的可能性。在医学中，概率帮助医生解读检测结果并确定最可能的诊断。甚至在体育中，概率被用来分析球员表现、预测比赛结果和制定比赛策略。

Worked Examples – 例题解析

Let us work through some typical KS3 Cambridge probability problems to see these concepts in action.

让我们通过一些典型的 KS3 剑桥概率问题来感受这些概念的实际运用。

Example 1: A bag contains 4 red marbles, 3 blue marbles, and 2 green marbles. One marble is drawn at random. What is the probability of drawing (a) a red marble, (b) a blue marble, (c) a marble that is not green?

例 1：一个袋子里有 4 颗红色弹珠、3 颗蓝色弹珠和 2 颗绿色弹珠。随机抽取一颗弹珠。求抽到 (a) 红色弹珠、(b) 蓝色弹珠、(c) 非绿色弹珠的概率。

Solution: Total marbles = 4 + 3 + 2 = 9. (a) P(red) = 4/9. (b) P(blue) = 3/9 = 1/3. (c) Marbles that are not green = 4 + 3 = 7, so P(not green) = 7/9.

解答：总弹珠数 = 4 + 3 + 2 = 9。(a) P(红色) = 4/9。(b) P(蓝色) = 3/9 = 1/3。(c) 非绿色弹珠 = 4 + 3 = 7，所以 P(非绿色) = 7/9。

Example 2: A fair six-sided die is rolled. What is the probability of rolling (a) an even number, (b) a number greater than 4, (c) a prime number?

例 2：掷一个均匀的六面骰子。求掷出 (a) 偶数、(b) 大于 4 的数、(c) 质数的概率。

Solution: Sample space = {1, 2, 3, 4, 5, 6}, total = 6. (a) Even numbers: {2, 4, 6}, so P(even) = 3/6 = 1/2. (b) Numbers greater than 4: {5, 6}, so P(>4) = 2/6 = 1/3. (c) Prime numbers: {2, 3, 5}, so P(prime) = 3/6 = 1/2.

解答：样本空间 = {1, 2, 3, 4, 5, 6}，总数 = 6。(a) 偶数：{2, 4, 6}，所以 P(偶数) = 3/6 = 1/2。(b) 大于 4 的数：{5, 6}，所以 P(大于 4) = 2/6 = 1/3。(c) 质数：{2, 3, 5}，所以 P(质数) = 3/6 = 1/2。

Example 3: A spinner has 8 equal sections numbered 1 to 8. It is spun once. Find the probability that the number is (a) a multiple of 3, (b) a factor of 8, (c) an odd number less than 6.

例 3：一个转盘有 8 个相等的部分，编号 1 到 8。转动一次。求数字是 (a) 3 的倍数、(b) 8 的因数、(c) 小于 6 的奇数的概率。

Solution: Total outcomes = 8. (a) Multiples of 3: {3, 6}, so P = 2/8 = 1/4. (b) Factors of 8: {1, 2, 4, 8}, so P = 4/8 = 1/2. (c) Odd numbers less than 6: {1, 3, 5}, so P = 3/8.

解答：总结果数 = 8。(a) 3 的倍数：{3, 6}，所以 P = 2/8 = 1/4。(b) 8 的因数：{1, 2, 4, 8}，所以 P = 4/8 = 1/2。(c) 小于 6 的奇数：{1, 3, 5}，所以 P = 3/8。

Common Mistakes to Avoid – 常见错误

Students often make several predictable mistakes when learning probability. Being aware of these pitfalls can help you avoid them in exams and assessments. Here are the most common errors seen in KS3 Cambridge probability work:

学生在学习概率时经常会犯一些可预测的错误。了解这些陷阱可以帮助你在考试和评估中避免它们。以下是 KS3 剑桥概率中最常见的错误：

First, confusing the Addition Rule and the Multiplication Rule. Remember: OR means ADD (for mutually exclusive events), AND means MULTIPLY (for independent events). Many students mix these up, especially under exam pressure. Take a moment to identify whether the question is asking for “or” or “and” before choosing your method.

第一，混淆加法法则和乘法法则。记住：OR 意味着相加（对于互斥事件），AND 意味着相乘（对于独立事件）。许多学生会混淆这两者，尤其是在考试压力下。在选择方法之前，先花点时间确定问题是问”或”还是”且”。

Second, probabilities must always be between 0 and 1 inclusive. If your calculated probability is greater than 1 or negative, you have made an error. Always check that your answer is a number between 0 and 1, and if you are expressing it as a percentage, it must be between 0% and 100%.

第二，概率必须在 0 到 1（含）之间。如果你计算出的概率大于 1 或为负数，那你就犯了错误。始终检查你的答案是否在 0 到 1 之间，如果你用百分比表示，它必须在 0% 到 100% 之间。

Third, assuming events are independent when they are not. For example, drawing two cards from a deck without replacement – the second draw’s probability depends on what was drawn first. These are dependent events and require a different approach. In KS3, most problems involve either replacement (independent) or explicitly stated independence, but it is important to be aware of the distinction.

第三，假设事件是独立的而实际并非如此。例如，从一副牌中不放回地抽两张牌 – 第二次抽取的概率取决于第一次抽到了什么。这些是相关事件，需要不同的方法。在 KS3 中，大多数问题要么涉及放回（独立），要么明确说明了独立性，但意识到这一区别很重要。

Fourth, forgetting to simplify fractions. In Cambridge exams, probabilities should be given in their simplest form. Writing 4/8 instead of 1/2, or 6/10 instead of 3/5, will lose marks even if the underlying calculation is correct.

第四，忘记化简分数。在剑桥考试中，概率应以最简形式给出。写 4/8 而不是 1/2，或写 6/10 而不是 3/5，即使底层计算正确，也会丢分。

Venn Diagrams and Probability – 维恩图与概率

Venn diagrams are another visual tool used in probability to show relationships between sets of outcomes. In KS3 Cambridge Mathematics, students learn to use Venn diagrams to represent sample spaces and calculate probabilities involving overlapping events. A Venn diagram typically uses circles to represent different events, with overlapping regions showing outcomes that belong to both events.

维恩图是概率中使用的另一种可视化工具，用于显示结果集合之间的关系。在 KS3 剑桥数学中，学生学习使用维恩图来表示样本空间，并计算涉及重叠事件的概率。维恩图通常用圆形表示不同事件，重叠区域显示同时属于两个事件的结果。

For example, consider a class of 30 students where 18 study French, 15 study Spanish, and 8 study both languages. The Venn diagram would show 10 students studying only French (18 – 8), 7 students studying only Spanish (15 – 8), 8 students studying both, and 5 students studying neither (30 – 10 – 7 – 8). From this, we can calculate probabilities such as P(studies at least one language) = 25/30 = 5/6, or P(studies only French) = 10/30 = 1/3.

例如，一个 30 名学生的班级，其中 18 人学法语，15 人学西班牙语，8 人两种语言都学。维恩图将显示 10 名学生只学法语 (18 – 8)，7 名学生只学西班牙语 (15 – 8)，8 名学生两种都学，5 名学生两种都不学 (30 – 10 – 7 – 8)。由此，我们可以计算诸如 P(至少学一门语言) = 25/30 = 5/6，或 P(只学法语) = 10/30 = 1/3 等概率。

Conditional Probability Basics – 条件概率基础

Conditional probability is introduced towards the end of KS3 and explores how the probability of an event changes when we know that another event has already occurred. The notation P(A|B) means “the probability of A given that B has happened.” While formal conditional probability formulas are typically left for GCSE, the concept is introduced in KS3 through practical scenarios.

条件概率在 KS3 后期引入，探讨当我们知道另一个事件已经发生时，事件的概率如何变化。符号 P(A|B) 表示”在 B 已发生的情况下 A 的概率”。虽然正式的条件概率公式通常留到 GCSE，但该概念在 KS3 通过实际场景引入。

A simple example: if you have a bag with 3 red and 2 blue marbles, and you draw one marble without replacement, the probability of drawing a red marble first is 3/5. If you did draw a red marble, the probability of drawing another red marble is now 2/4 = 1/2, because there are now 2 reds left out of 4 total marbles. This change in probability illustrates the core idea behind conditional probability.

一个简单的例子：如果你有一个装有 3 颗红色和 2 颗蓝色弹珠的袋子，你不放回地抽取一颗弹珠，第一次抽到红色的概率是 3/5。如果你确实抽到了一颗红色，那么再抽一颗红色的概率现在是 2/4 = 1/2，因为现在 4 颗弹珠中剩下 2 颗红色。这种概率的变化说明了条件概率背后的核心思想。

Relative Frequency and Long-Run Behaviour – 相对频率与长期行为

In the Cambridge KS3 curriculum, students are expected to conduct probability experiments and record results. This hands-on approach helps bridge the gap between theoretical understanding and practical application. When you toss a coin 10 times, you might get 7 heads and 3 tails – an experimental probability of 0.7 for heads, far from the theoretical 0.5. But as you increase the number of tosses to 100, 500, or 1000, the relative frequency typically converges towards 0.5.

在剑桥 KS3 课程中，学生需要进行概率实验并记录结果。这种动手实践的方法有助于弥合理论理解与实际应用之间的差距。当你抛硬币 10 次时，你可能得到 7 次正面和 3 次反面 – 正面的实验概率为 0.7，与理论值 0.5 相差甚远。但当你将抛掷次数增加到 100、500 或 1000 次时，相对频率通常会收敛到 0.5。

This principle, known as the Law of Large Numbers, is a cornerstone of probability theory. It explains why casinos always win in the long run (the odds are in their favour, and over thousands of games, the experimental probability closely matches the theoretical probability) and why insurance companies can accurately predict claim rates across large populations even though individual accidents are unpredictable.

这一原则被称为大数定律，是概率论的基石。它解释了为什么赌场长期来看总是赢（赔率对他们有利，在数千场游戏中，实验概率与理论概率非常接近），以及为什么保险公司可以准确预测大规模人群的理赔率，尽管个体事故是不可预测的。

Expected Number of Outcomes – 期望结果数

Once students understand probability, they can calculate the expected number of times an event will occur in a given number of trials. This is an important skill that connects probability to prediction:

一旦学生理解了概率，他们就可以计算在给定试验次数下事件期望发生的次数。这是一项将概率与预测联系起来的重要技能：

Expected number = Probability of event x Total number of trials

期望次数 = 事件概率 x 总试验次数

For example, if you roll a fair die 300 times, how many times would you expect to roll a 5? Since P(5) = 1/6, the expected number is 300 x 1/6 = 50 times. Similarly, if a basketball player has a free-throw success rate of 0.75 (75 percent), in 40 attempts you would expect 40 x 0.75 = 30 successful shots.

例如，如果你掷一个均匀的骰子 300 次，你期望掷出几次 5？由于 P(5) = 1/6，期望次数是 300 x 1/6 = 50 次。同样，如果一名篮球运动员的罚球命中率是 0.75 (75%)，在 40 次尝试中，你期望有 40 x 0.75 = 30 次命中。

This concept is widely used in quality control in manufacturing, where companies test samples of products and use probability to estimate defect rates across entire production runs. It is also used in opinion polling, where a survey of 1000 people is used to estimate the views of millions.

这个概念广泛应用于制造业的质量控制中，公司测试产品样本并使用概率来估计整个生产批次中的缺陷率。它也用于民意调查，通过对 1000 人的调查来估计数百万人的观点。

Additional Worked Examples – 更多例题

Example 4: Two fair dice are rolled. Find the probability that (a) the sum is 7, (b) the sum is greater than 10, (c) both dice show the same number.

例 4：掷两个均匀的骰子。求 (a) 和为 7、(b) 和大于 10、(c) 两个骰子显示相同数字的概率。

Solution: Total outcomes = 6 x 6 = 36. (a) Pairs summing to 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) – 6 outcomes. P(sum=7) = 6/36 = 1/6. (b) Pairs summing > 10: (5,6), (6,5), (6,6) – 3 outcomes. P(sum>10) = 3/36 = 1/12. (c) Same numbers: (1,1), (2,2), (3,3), (4,4), (5,5), (6,6) – 6 outcomes. P(same) = 6/36 = 1/6.

解答：总结果数 = 6 x 6 = 36。(a) 和为 7 的组合：(1,6), (2,5), (3,4), (4,3), (5,2), (6,1) – 6 种结果。P(和为 7) = 6/36 = 1/6。(b) 和大于 10 的组合：(5,6), (6,5), (6,6) – 3 种结果。P(和大于 10) = 3/36 = 1/12。(c) 相同数字：(1,1), (2,2), (3,3), (4,4), (5,5), (6,6) – 6 种结果。P(相同) = 6/36 = 1/6。

Example 5: A card is drawn at random from a standard deck of 52 playing cards. Find the probability that the card is (a) a heart, (b) a face card (Jack, Queen, or King), (c) a red face card.

例 5：从一副标准的 52 张扑克牌中随机抽取一张。求抽到 (a) 红心、(b) 人头牌（J、Q 或 K）、(c) 红色人头牌的概率。

Solution: (a) There are 13 hearts in a deck, so P(heart) = 13/52 = 1/4. (b) There are 12 face cards in total (3 per suit x 4 suits), so P(face) = 12/52 = 3/13. (c) Red face cards are the face cards from hearts and diamonds, which is 3 + 3 = 6 cards, so P(red face) = 6/52 = 3/26.

解答：(a) 一副牌中有 13 张红心，所以 P(红心) = 13/52 = 1/4。(b) 总共有 12 张人头牌（每花色 3 张 x 4 种花色），所以 P(人头) = 12/52 = 3/13。(c) 红色人头牌是红心和方块中的人头牌，共 3 + 3 = 6 张，所以 P(红色人头) = 6/52 = 3/26。

Systematic Listing Strategies – 系统列举策略

When working with probability problems involving multiple events, it is essential to list all possible outcomes systematically. Missing even one outcome can lead to an incorrect probability calculation. The Cambridge KS3 curriculum emphasises several structured approaches to ensure complete and accurate listings. A probability space diagram, also called a sample space diagram, is one of the most useful tools for this purpose.

在处理涉及多个事件的概率问题时，系统地列出所有可能结果至关重要。漏掉哪怕一个结果都可能导致概率计算错误。剑桥 KS3 课程强调了几种结构化的方法，以确保完整准确的列举。概率空间图，也称为样本空间图，是为此目的最有用的工具之一。

Consider rolling two dice and adding the scores. Instead of trying to list outcomes randomly, students should create a 6 by 6 grid with the first die’s results along the rows and the second die’s results along the columns. Each cell represents one of the 36 equally likely outcomes. This structured approach makes it easy to count favourable outcomes for any event, such as “the sum is less than 5” or “the product is even.”

考虑掷两个骰子并求和。与其随意列举结果，学生应该创建一个 6×6 的网格，第一颗骰子的结果沿行排列，第二颗骰子的结果沿列排列。每个单元格代表 36 个等可能结果中的一个。这种结构化方法使得计算任何事件的有利结果变得容易，例如”和小于 5″或”积为偶数”。

Another systematic approach is the use of outcome tables for combined events. For example, when flipping a coin and spinning a four-colour spinner (red, blue, green, yellow) simultaneously, a simple 2 by 4 table with 8 cells shows all possible outcomes clearly. This method is particularly useful when the two events have different numbers of possible outcomes.

另一种系统方法是使用组合事件的结果表。例如，当同时抛一枚硬币并旋转一个四色转盘（红、蓝、绿、黄）时，一个简单的 2×4 表格，共 8 个单元格，清晰地展示了所有可能的结果。当两个事件的可能结果数量不同时，这种方法特别有用。

Probability from Frequency Tables – 从频率表中计算概率

In many real-world situations, we do not have a theoretical model to calculate probabilities from. Instead, we must work with data collected from observations or surveys. Frequency tables organise this data, and from them we can calculate experimental probabilities. This skill is explicitly assessed in the Cambridge KS3 mathematics examinations.

在许多现实世界的情境中，我们没有理论模型来计算概率。相反，我们必须使用从观察或调查中收集的数据。频率表将这些数据组织起来，我们可以从中计算实验概率。这一技能在剑桥 KS3 数学考试中明确考查。

For instance, a survey of 200 KS3 students about their favourite sport might produce the following frequency table: Football 65, Basketball 45, Swimming 30, Tennis 25, Athletics 20, Other 15. From this, we can calculate that the experimental probability a randomly selected student prefers Basketball is 45/200 = 9/40 = 0.225 or 22.5 percent. The probability they prefer either Swimming or Tennis is (30 + 25)/200 = 55/200 = 11/40.

例如，对 200 名 KS3 学生关于他们最喜欢的运动的调查可能产生以下频率表：足球 65，篮球 45，游泳 30，网球 25，田径 20，其他 15。由此，我们可以计算随机选择的学生偏好篮球的实验概率是 45/200 = 9/40 = 0.225 或 22.5%。他们偏好游泳或网球的概率是 (30 + 25)/200 = 55/200 = 11/40。

When working with grouped frequency tables, where data is organised into intervals rather than individual values, students must be careful to identify which groups contain favourable outcomes. The total number of outcomes is the sum of all frequencies, and the number of favourable outcomes is the sum of frequencies in the relevant groups.

当使用分组频率表时，数据按区间而非单个值组织，学生必须小心识别哪些组包含有利结果。结果总数是所有频率之和，有利结果数是相关组中频率之和。

Comparing and Ordering Probabilities – 比较和排序概率

A key skill assessed in KS3 Cambridge Mathematics is the ability to compare probabilities expressed in different forms. A student might be given probabilities as fractions (3/5), decimals (0.45), and percentages (80 percent), and asked to order events from least likely to most likely. This requires fluency in converting between these representations.

KS3 剑桥数学中评估的一项关键技能是比较以不同形式表达的概率的能力。学生可能被给予分数 (3/5)、小数 (0.45) 和百分比 (80%) 形式的概率，并被要求将事件从最不可能到最可能排序。这需要熟练掌握在这些表示形式之间进行转换。

To compare fractions, find a common denominator or convert to decimals. For example, to compare 3/5, 2/3, and 7/10: convert to decimals (0.6, 0.667, 0.7) or to a common denominator of 30 (18/30, 20/30, 21/30). Ordering from least to greatest: 3/5, then 2/3, then 7/10. This skill is particularly tested in multi-step probability questions where different parts of the question produce probabilities in different formats.

要比较分数，找到公分母或转换为小数。例如，比较 3/5、2/3 和 7/10：转换为小数 (0.6, 0.667, 0.7) 或转换为分母 30 (18/30, 20/30, 21/30)。从小到大排序：3/5，然后 2/3，然后 7/10。这一技能在多步骤概率问题中特别会被考查，因为问题的不同部分可能以不同格式产生概率。

Key Vocabulary for Probability – 概率关键词汇

Mastering the language of probability is essential for understanding exam questions and communicating mathematical reasoning clearly. The Cambridge KS3 curriculum expects students to use precise probability vocabulary. Here is a summary of the most important terms:

掌握概率的语言对于理解考试题目和清晰地交流数学推理至关重要。剑桥 KS3 课程要求学生使用精确的概率词汇。以下是重要术语的总结：

Random: Each outcome has an equal chance of occurring. A fair die produces random outcomes.

随机：每个结果有相等的发生机会。一个均匀的骰子产生随机结果。

Bias: When outcomes are not equally likely. A weighted die is biased.

偏差：当结果不是等可能时。一个加重了的骰子是有偏差的。

Fair: All outcomes are equally likely. A fair coin has P(Heads) = P(Tails) = 1/2.

公平/均匀：所有结果等可能。一枚均匀硬币有 P(正面) = P(反面) = 1/2。

Impossible: An event with probability 0. Rolling a 7 on a six-sided die is impossible.

不可能：概率为 0 的事件。在六面骰子上掷出 7 是不可能的。

Certain: An event with probability 1. Rolling a number less than 7 on a six-sided die is certain.

必然：概率为 1 的事件。在六面骰子上掷出小于 7 的数是必然的。

Even chance: An event with probability exactly 1/2. Getting heads on a fair coin toss.

等可能：概率恰好为 1/2 的事件。抛一枚均匀硬币得到正面。

Complement: The complement of event A (written as A’) is the event that A does not happen. P(A’) = 1 – P(A).

补集：事件 A 的补集（写作 A’）是 A 不发生的事件。P(A’) = 1 – P(A)。

Summary – 总结

Probability is a core topic in KS3 Cambridge Mathematics that builds a foundation for more advanced study at GCSE and A-Level. The key concepts covered in this article include the probability scale, sample spaces, theoretical and experimental probability, mutually exclusive and independent events, and the use of tree diagrams for combined probability problems. Mastery of these concepts requires practice with a variety of problem types. Work through the examples carefully, create your own practice problems, and always check that your final answer lies between 0 and 1. With consistent practice, probability becomes not just manageable but genuinely enjoyable – it is one of the few areas of mathematics where you can directly see its relevance to the real world around you.

概率是 KS3 剑桥数学的核心主题，为 GCSE 和 A-Level 的更高级学习奠定了基础。本文涵盖的关键概念包括概率标尺、样本空间、理论概率和实验概率、互斥事件和独立事件，以及使用树状图解决组合概率问题。掌握这些概念需要练习各种题型。仔细完成例题，自己创建练习题，并始终检查最终答案是否在 0 到 1 之间。通过持续练习，概率不仅变得易于掌握，而且会真正令人愉快 – 这是数学中为数不多的能让你直接看到它与周围现实世界关联的领域之一。

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