📚 Understanding Probability: From Basics to Application | 理解概率：从基础到应用
Probability is a branch of mathematics that helps us measure how likely an event is to happen. It is used everywhere – from predicting the weather and testing new medicines to playing card games and designing fair competitions. For KS3 Cambridge Mathematics students, mastering probability means learning to describe chance, calculate simple probabilities, and interpret data from experiments. In this article, we will explore key concepts step by step, with clear examples and straightforward explanations.
概率是数学中帮助我们衡量事件发生可能性的一个分支。它无处不在——从预测天气、测试新药到玩纸牌游戏和设计公平的竞赛。对于学习剑桥初中数学的学生来说，掌握概率意味着学会描述可能性、计算简单的概率并解释实验数据。在本文中，我们将逐步探讨关键概念，并提供清晰的例子和直白的解释。
1. What is Probability? | 什么是概率？
Probability is a number that expresses the chance of a particular event occurring. It always lies between 0 and 1. An event that cannot happen has a probability of 0, and an event that is absolutely certain has a probability of 1. For instance, when you flip a fair coin, the probability of getting a head is between 0 and 1 – specifically 0.5, because heads is one of two equally likely outcomes.
概率是表示某一特定事件发生机会的数值。它始终介于0和1之间。不可能发生的事件概率为0，绝对确定的事件概率为1。例如，当你抛一枚公平硬币时，得到正面的概率介于0和1之间——具体是0.5，因为正面是两个等可能结果中的一个。
The word ‘probability’ is often used in everyday language. In mathematics, however, we give it a precise numerical value. Whether we write it as a fraction, a decimal, or a percentage, the meaning stays the same: the measure of how likely something is to occur. Rolling a standard six-sided die gives a probability of 1/6 for each face, because all six faces are equally likely.
“概率”一词常在日常语言中使用。然而在数学中，我们赋予它一个精确的数值。无论我们将其写成分数、小数还是百分数，其含义不变：衡量某事发生的可能性。掷一个标准六面骰子，每一面出现的概率都是1/6，因为所有六个面等可能发生。
2. The Probability Scale | 概率尺度
The probability scale is a visual way of placing the chance of any event on a line from 0 to 1. Words often used to describe probability are ‘impossible’, ‘unlikely’, ‘even chance’, ‘likely’ and ‘certain’. These words can be matched with approximate numerical values. An event with an even chance has a probability of 0.5, while something very unlikely might have a probability of 0.1 or 10%.
概率尺度是一种直观的方式，将任何事件的可能性放在从0到1的直线上。常用来描述概率的词语有“不可能”“不太可能”“对等机会”“很可能”和“一定发生”。这些词语可以与近似的数值对应。具有对等机会的事件概率为0.5，而非常不可能发生的事件概率可能为0.1或10%。
| Word / 词语 | Probability value / 概率值 | Example / 例子 |
|---|---|---|
| Impossible / 不可能 | 0 | Getting a 7 on a normal die / 掷普通骰子得7点 |
| Unlikely / 不太可能 | close to 0.2 / 接近0.2 | Picking a red ball from a bag with 1 red and 9 blue / 从1红9蓝的袋中取出红球 |
| Even chance / 对等机会 | 0.5 | Flipping a coin and getting heads / 抛硬币得正面 |
| Likely / 很可能 | close to 0.8 / 接近0.8 | Drawing a blue ball from a bag with 8 blue and 2 red / 从8蓝2红的袋中取出蓝球 |
| Certain / 一定发生 | 1 | The sun rising tomorrow / 明天太阳升起 |
Using the scale helps you check whether a calculated probability makes sense. If you get a probability of 1.5 or –0.2, you know something has gone wrong, because probabilities cannot be less than 0 or greater than 1. This simple check is one of the most useful habits in probability work.
利用尺度可以帮助检查计算出的概率是否合理。如果你得出1.5或–0.2的概率，你就知道出了错，因为概率不能小于0或大于1。这个简单的检查是概率学习中最有用的习惯之一。
3. Basic Probability Formula | 基本概率公式
When all outcomes of an experiment are equally likely, probability can be found using a simple formula. The probability of an event A is the number of ways A can happen divided by the total number of possible outcomes.
当实验的所有结果等可能发生时，可以用一个简单公式求概率。事件A的概率等于A可能发生的方式数除以所有可能结果的总数。
P(A) = Number of favourable outcomes / Total number of possible outcomes
For example, when rolling a fair six-sided die, the total number of possible outcomes is 6. If event A is ‘rolling an even number’, the favourable outcomes are 2, 4 and 6, so there are 3 favourable outcomes. Therefore, P(even) = 3/6 = 1/2 or 0.5.
例如，掷一个公平的六面骰子时，可能结果的总数是6。如果事件A是“掷出偶数”，那么有利的结果是2、4和6，即有3个有利结果。因此，P(偶数) = 3/6 = 1/2 或 0.5。
This formula only works when each outcome is equally likely. If a die is biased, some faces appear more often, and we cannot simply count favourable outcomes without knowing the weightings. In KS3, we mainly deal with fair, equally likely situations – coins, dice, spinners and well-shuffled cards.
这个公式仅在每个结果等可能时才成立。如果骰子不均匀，某些面出现得更频繁，我们就不能简单地计算有利结果数而不了解权重。在初中阶段，我们主要处理公平、等可能的情形——硬币、骰子、转盘和洗匀的纸牌。
4. Complementary Events | 互补事件
The complement of an event A is the event that A does not happen. It is often written as ‘not A’. Since either A happens or does not happen, the sum of their probabilities is always 1. This gives a very useful relationship: P(not A) = 1 – P(A).
一个事件A的补事件是指A没有发生的事件。它通常写作“非A”。由于要么A发生，要么不发生，它们的概率之和总是1。这就得出了一个非常有用的关系：P(非A) = 1 – P(A)。
For instance, if the probability it will rain tomorrow is 0.3, then the probability it will not rain is 1 – 0.3 = 0.7. When calculating probabilities, the complement rule can save time: sometimes it is easier to find the probability that something does not happen and subtract from 1.
例如，明天下雨的概率是0.3，那么不下雨的概率就是1 – 0.3 = 0.7。在计算概率时，补事件规则可以节省时间：有时先求出某事不会发生的概率，再从1中减去，反而更简便。
In a bag with 5 red, 3 blue and 2 green counters, the probability of picking a red counter is 5/10 = 0.5. The probability of not picking a red counter (i.e., picking blue or green) is 1 – 0.5 = 0.5. This matches the direct count of 5 non-red out of 10, showing the rule works.
在一个装有5个红、3个蓝和2个绿筹码的袋子中，抽到红色筹码的概率是5/10 = 0.5。没有抽到红色（即抽到蓝色或绿色）的概率为1 – 0.5 = 0.5。这与直接计数10个中有5个非红的结果一致，验证了该规则。
5. Simple Experiments and Outcomes | 简单实验与结果
The simplest probability experiments involve flipping coins, rolling dice or spinning spinners with equal sections. An ‘outcome’ is a single possible result. The set of all possible outcomes is called the ‘sample space’. For a coin toss, the sample space is {Head, Tail}. For rolling a die, it is {1, 2, 3, 4, 5, 6}.
最简单的概率实验包括抛硬币、掷骰子或旋转分段相等的转盘。“结果”是指单个可能的结果。所有可能结果的集合称为“样本空间”。抛一枚硬币的样本空间是{正面，反面}。掷骰子的样本空间是{1, 2, 3, 4, 5, 6}。
When you carry out a simple experiment once, the result is called an outcome; when you repeat it many times, you can compare the observed results with the theoretical probability. KS3 students often use frequency tables to record outcomes.
当你进行一次简单实验时，结果称为一个结果；当你重复多次时，就可以将观察到的结果与理论概率进行比较。初中生常用频数表来记录结果。
6. Sample Spaces and Systematic Listing | 样本空间与系统列举
For two or more simple events, it is important to list all possible outcomes in a structured way so that none are missed. A sample space diagram can be a list, a table or a grid. When you flip two coins, the sample space can be written as {HH, HT, TH, TT}, where H stands for head and T for tail.
对于两个或更多简单事件，重要的是以有结构的方式列出所有可能结果，以免遗漏。样本空间图可以是一个列表、一个表格或一个网格。当你抛两枚硬币时，样本空间可以写成{HH, HT, TH, TT}，其中H代表正面，T代表反面。
When rolling two dice, a 6 by 6 grid is often used to show the 36 equally likely outcomes. If you are interested in the sum of the two dice, you can count how many outcomes give each sum. For example, a sum of 7 occurs in 6 outcomes: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). So P(sum of 7) = 6/36 = 1/6.
掷两个骰子时，常用一个6乘6的网格来表示36个等可能的结果。如果你关心两个骰子的点数之和，可以数出产生每个和的结果有多少个。例如，和为7的情况有6种：(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)。因此P(和为7) = 6/36 = 1/6。
Listing outcomes systematically is a key skill for KS3. You might be asked to find all combinations of shirts and trousers from given choices, or all possible meal deals from a menu. The same principle applies: make an ordered list or a two-way table.
系统地列举结果是初中阶段的关键技能。你可能会被要求找出给定衬衫和裤子的所有搭配，或者菜单上所有可能的套餐组合。原理相同：制作有序列表或双向表。
7. Calculating Probabilities for Combined Events | 计算组合事件的概率
When events are combined, you can often use the sample space to find probabilities. For instance, if you roll a fair die and flip a fair coin, there are 12 equally likely outcomes (6 × 2). To find the probability of getting an even number and a head, you count the favourable outcomes: (2,H), (4,H), (6,H) – that is 3 outcomes. So P(even and head) = 3/12 = 1/4.
当事件组合时，你通常可以利用样本空间求概率。例如，如果你掷一个公平骰子并抛一枚公平硬币，共有12个等可能结果（6 × 2）。要计算得到一个偶数和正面的概率，需要数出有利结果：(2,H), (4,H), (6,H)——共3个结果。因此P(偶数且正面) = 3/12 = 1/4。
In KS3 we do not multiply probabilities for independent events formally, but we can see the multiplication principle through counting: total outcomes multiply. The key is always to count equally likely outcomes carefully.
在初中阶段我们不正式用独立事件的概率相乘，但可以通过计数看到乘法原理：结果总数相乘。关键始终是仔细数出等可能的结果。
8. Experimental Probability and Relative Frequency | 实验概率与相对频率
Theoretical probability tells us what we expect to happen in an ideal world. Experimental probability (also called relative frequency) is what actually happens when we carry out an experiment. It is calculated as: Experimental probability = Number of times the event occurs / Total number of trials.
理论概率告诉我们在理想世界中预期会发生什么。实验概率（也称相对频率）是我们真正进行实验时实际发生的情况。计算公式为：实验概率 = 事件发生的次数 / 试验总次数。
If you toss a coin 100 times and get heads 54 times, the experimental probability of heads is 54/100 = 0.54. The theoretical probability remains 0.5. As you increase the number of trials, the experimental probability usually gets closer to the theoretical value – this is called the law of large numbers.
如果你抛硬币100次，得到54次正面，那么正面的实验概率为54/100 = 0.54。理论概率仍然是0.5。当你增加试验次数时，实验概率通常会趋近于理论值——这称为大数定律。
KS3 students may carry out experiments with dice, counters or spinners and compare their results with predictions. This builds understanding that probability describes long-term behaviour, not short-term runs.
初中生可能会用骰子、筹码或转盘进行实验，并将结果与预测进行比较。这有助于理解概率描述的是长期行为，而非短期结果。
9. Using Two-Way Tables | 使用双向表
A two-way table is an excellent tool for organising data when there are two categories. For example, a class survey might ask students whether they like football and whether they like basketball. The table shows the number of students in each combination.
当存在两个分类时，双向表是整理数据的绝佳工具。例如，一项班级调查可能询问学生是否喜欢足球和是否喜欢篮球。表格显示了每种组合的学生人数。
Suppose 40 students are surveyed. The two-way table might look like this:
假设有40名学生接受了调查。双向表可能如下：
| Like football / 喜欢足球 | Do not like football / 不喜欢足球 | Total / 总计 | |
|---|---|---|---|
| Like basketball / 喜欢篮球 | 12 | 8 | 20 |
| Do not like basketball / 不喜欢篮球 | 6 | 14 | 20 |
| Total / 总计 | 18 | 22 | 40 |
If a student is chosen at random, the probability that they like football is 18/40 = 9/20. The probability that they like both football and basketball is 12/40 = 3/10. Two-way tables make it easy to pick out the numbers you need for these calculations.
如果随机选择一名学生，他喜欢足球的概率是18/40 = 9/20。他既喜欢足球又喜欢篮球的概率是12/40 = 3/10。双向表使挑选这些计算所需的数据变得容易。
10. Mutually Exclusive Events | 互斥事件
Two events are mutually exclusive if they cannot happen at the same time. For example, when you roll a die once, the events ‘rolling a 3’ and ‘rolling a 5’ are mutually exclusive. When events are mutually exclusive, the probability of either one happening is the sum of their individual probabilities: P(A or B) = P(A) + P(B).
如果两个事件不能同时发生，则它们是互斥的。例如，掷一次骰子时，“掷出3”和“掷出5”是互斥事件。当事件互斥时，两者中任一发生的概率是它们各自概率的和：P(A 或 B) = P(A) + P(B)。
For a standard die, P(3) = 1/6 and P(5) = 1/6, so P(3 or 5) = 1/6 + 1/6 = 2/6 = 1/3. This rule works because there is no overlap between the two sets of outcomes.
对于标准骰子，P(3) = 1/6，P(5) = 1/6，所以P(3或5) = 1/6 + 1/6 = 2/6 = 1/3。这条规则起作用是因为两组结果之间没有重叠。
If events are not mutually exclusive, we cannot simply add, because we would count the overlap twice. At KS3, you will mostly use the addition rule for mutually exclusive events.
如果事件不互斥，我们就不能简单相加，因为会重复计算重叠部分。在初中阶段，你主要会在互斥事件中使用加法规则。
11. Expected Frequency | 期望频数
Expected frequency is the number of times you would expect an event to occur in a certain number of trials, based on theoretical probability. It is found by multiplying the probability by the number of trials: Expected frequency = Probability x Number of trials.
期望频数是指基于理论概率，在特定试验次数中你预期事件发生的次数。它通过概率乘以试验次数求得：期望频数 = 概率 × 试验次数。
For example, if you roll a fair die 300 times, the expected frequency of rolling a 4 is (1/6) × 300 = 50. This does not mean you will get exactly 50 fours, but if you repeated the experiment many times, the average number of fours would be close to 50.
例如，如果你掷一个公平骰子300次，掷出4点的期望频数是 (1/6) × 300 = 50。这并不意味着你一定恰好得到50次4点，但如果你多次重复这个实验，平均得到的4点次数会接近50。
Expected frequency is very useful for predicting outcomes in larger samples and for checking whether observed results are unusual.
期望频数对于预测较大样本的结果以及检查观测结果是否异常非常有用。
12. Key Points Summary | 要点总结
Probability always lies between 0 and 1. It can be expressed as a fraction, decimal or percentage. The basic formula P(event) = favourable outcomes / total outcomes applies when all outcomes are equally likely. Complementary events sum to 1, so P(not A) = 1 – P(A). A systematic approach to listing outcomes, such as sample space diagrams and two-way tables, helps avoid mistakes. Experimental probability can differ from theoretical probability, but the two get closer with more trials. Mutually exclusive events can be combined by adding their probabilities. Finally, expected frequency = probability x number of trials, which connects theoretical calculations to practical forecasts.
概率总是介于0和1之间。它可以用分数、小数或百分数表示。基本公式 P(事件) = 有利结果数 / 总结果数 适用于所有结果等可能的情形。互补事件概率之和为1，所以 P(非A) = 1 – P(A)。使用样本空间图和双向表等系统化方法列出结果有助于避免错误。实验概率可能与理论概率不同，但试验次数增多时两者会更接近。互斥事件可以通过概率相加来组合。最后，期望频数 = 概率 × 试验次数，它将理论计算与实际预测联系起来。
By working through plenty of examples and checking your answers with the probability scale, you will build confidence and skill in handling probability problems – an essential part of KS3 Cambridge Mathematics and a foundation for later statistical work.
通过练习大量例题并用概率尺度检验答案，你会建立处理概率问题的信心和技巧——这是剑桥初中数学的重要部分，也是后续统计学习的基础。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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