📚 Introduction to Probability | 概率入门
Probability is the branch of mathematics that tells us how likely an event is to happen. At KS3 level, you will learn to describe chance using words like impossible, unlikely, even chance, likely, and certain, then move on to using numbers between 0 and 1. This topic is essential for understanding statistics, games, risk, and making predictions in everyday life. In this article, we will explore the core ideas of probability, how to calculate simple probabilities, and how to work with probability scales, sample spaces, and mutually exclusive events – all aligned with the Cambridge KS3 curriculum.
概率是数学中一个分支，它告诉我们一个事件发生的可能性有多大。在 KS3 阶段，你将学会用不可能、不太可能、等可能、可能、一定等词语描述机会，然后过渡到用 0 到 1 之间的数字来表示。这个主题对于理解统计、游戏、风险以及日常生活中的预测都至关重要。在本文中，我们将探讨概率的核心概念、如何计算简单概率，以及如何使用概率标度、样本空间和互斥事件——所有这些都与剑桥 KS3 课程大纲保持一致。
1. What is Probability? | 概率是什么？
Probability measures the chance that a particular event will occur. It is always a number between 0 and 1 (inclusive). A probability of 0 means the event is impossible – it can never happen. A probability of 1 means the event is certain – it will definitely happen. The closer the probability is to 1, the more likely the event is to occur.
概率衡量的是某个特定事件发生的机会。它总是 0 到 1（包括端点）之间的一个数。概率为 0 意味着该事件是不可能发生的——它永远不会发生。概率为 1 意味着该事件是必然的——它一定会发生。概率越接近 1，事件发生的可能性就越大。
We often express probability as a fraction, a decimal, or a percentage. For example, when flipping a fair coin, the probability of getting heads is 1/2, which is 0.5 as a decimal and 50% as a percentage. This is known as an even chance.
我们通常用分数、小数或百分数来表示概率。例如，抛一枚公平硬币时，得到正面的概率是 1/2，即小数 0.5，百分数 50%。这被称为等可能的机会。
vourable outcomes / Total number of possible outcomes, provided all outcomes are equally likely.
事件概率 = 有利结果数 / 所有可能结果总数，前提是所有结果都是等可能的。
2. The Probability Scale | 概率标度
The probability scale is a line from 0 to 1 that helps us visualise how likely events are. We place words and numbers on it to show the range of possibilities.
概率标度是一条从 0 到 1 的线段，帮助我们直观地看出事件的可能性大小。我们在上面放置词语和数字来表示可能性的范围。
On the scale:
0 – Impossible (e.g. a pig flying)
0.25 – Unlikely (e.g. rolling a 6 on a fair die – actually 1/6 ≈ 0.167, but here it's an example)
0.5 – Even chance (e.g. getting heads on a coin toss)
0.75 – Likely (e.g. picking a red ball from a bag with 3 red and 1 blue)
1 – Certain (e.g. the sun rising tomorrow)
在标度上：
0 – 不可能（如：猪会飞）
0.25 – 不太可能（如：掷公平骰子得 6 ——实际 1/6 ≈ 0.167，此处仅作示例）
0.5 – 等可能（如：抛硬币得正面）
0.75 – 可能（如：从装有 3 红 1 蓝的袋子中抽到红色）
1 – 一定（如：明天太阳升起）
We can mark the probability of any event on this scale to compare likelihoods. For instance, rolling a number less than 7 on a standard die has probability 1, because all outcomes {1,2,3,4,5,6} are less than 7 – it is certain.
我们可以将任何事件的概率标记在这个标度上，以便比较可能性。例如，在一个标准骰子上掷出小于 7 的数的概率为 1，因为所有结果 {1,2,3,4,5,6} 都小于 7 ——它是必然的。
3. Calculating Basic Probability | 计算基本概率
To find the probability of an event A, we use the formula:
P(A) = Number of ways A can happen / Total number of equally likely outcomes
要计算事件 A 的概率，我们使用公式：
P(A) = A 能够发生的方式数 / 等可能结果的总数
Example 1: A bag contains 5 red sweets, 3 blue sweets, and 2 green sweets. All sweets are identical except colour. What is the probability of picking a red sweet at random?
Total number of sweets = 5 + 3 + 2 = 10. Number of red sweets = 5. So P(red) = 5/10 = 1/2.
示例 1：一个袋子里有 5 颗红色糖果、3 颗蓝色糖果和 2 颗绿色糖果。所有糖果除颜色外完全相同。随机抽取一颗，抽到红色糖果的概率是多少？
糖果总数 = 5 + 3 + 2 = 10。红色糖果数 = 5。因此 P(红色) = 5/10 = 1/2。
Example 2: A fair six-sided die is rolled. Find the probability of rolling an even number.
Even numbers on a die: {2,4,6} → 3 outcomes. Total outcomes = 6. So P(even) = 3/6 = 1/2.
示例 2：掷一枚公平六面骰子。求掷出偶数的概率。
骰子上的偶数：{2,4,6} → 3 个结果。总结果数 = 6。所以 P(偶数) = 3/6 = 1/2。
4. Using Fractions, Decimals, and Percentages | 使用分数、小数和百分数
Probability can be written in three ways. You must be able to convert between them quickly.
概率可以用三种方式书写。你必须能够快速转换它们。
xample: P(winning) = 3/8. As a decimal, 3 ÷ 8 = 0.375. As a percentage, 0.375 × 100 = 37.5%.
示例：P(获胜) = 3/8。化为小数：3 ÷ 8 = 0.375。化为百分数：0.375 × 100 = 37.5%。
Sometimes, answers are simpler as fractions. Always simplify fractions when possible. For instance, 4/8 simplifies to 1/2.
有时，答案用分数表示更简洁。要尽可能化简分数。例如，4/8 化简为 1/2。
are easy to compare. If P(rain) = 0.3 and P(sun) = 1/4, converting to percentages gives 30% and 25%, so rain is more likely.
在比较概率时，使用百分数可能更有帮助，因为它们容易比较。如果 P(下雨) = 0.3，P(晴天) = 1/4，转换为百分数得到 30% 和 25%，因此下雨的可能性更大。
5. Sample Space and Listing Outcomes | 样本空间与列出结果
A sample space is a list of all possible outcomes of an experiment. It is essential for calculating probabilities accurately. You can show the sample space as a simple list, a two-way table, or a tree diagram.
样本空间是实验所有可能结果的清单。它是准确计算概率的基础。你可以用简单列表、双向表格或树状图来表示样本空间。
Example: When rolling a fair die and flipping a coin, the sample space can be listed as:
H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}. There are 2 × 6 = 12 equally likely outcomes.
示例：掷一枚公平骰子并抛一枚硬币时，样本空间可以列为：
{H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}。共有 2 × 6 = 12 个等可能的结果。
ber, we count the outcomes that fit: H2, H4, H6 (3 outcomes). So probability = 3/12 = 1/4.
如果我们需要求得到正面和偶数的概率，我们就数出符合条件的结果：H2, H4, H6（3 个结果）。因此概率 = 3/12 = 1/4。
Using a table is another helpful method, especially for two-step experiments.
使用表格是另一种有用的方法，尤其适用于两步实验。
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, getting a 2 and getting a 5 are mutually exclusive because a single roll cannot show both numbers at once.
如果两个事件不可能同时发生，则它们是互斥的。例如，掷骰子时，得到点数为 2 和得到点数为 5 是互斥的，因为单次投掷不可能同时出现两个数字。
When events are mutually exclusive, the probability of either event A or event B occurring is simply the sum of their individual probabilities:
P(A or B) = P(A) + P(B)
当事件互斥时，事件 A 或事件 B 发生的概率就是它们各自概率的和：
P(A 或 B) = P(A) + P(B)
Example: A spinner has four equal sections numbered 1, 2, 3, 4. What is the probability of spinning a 1 or a 3?
P(1) = 1/4, P(3) = 1/4. Since they are mutually exclusive, P(1 or 3) = 1/4 + 1/4 = 1/2.
示例：一个旋转盘分为四个相等区域，分别标有 1、2、3、4。转到 1 或 3 的概率是多少？
P(1) = 1/4，P(3) = 1/4。由于它们互斥，P(1 或 3) = 1/4 + 1/4 = 1/2。
If events are not mutually exclusive, you cannot simply add the probabilities because there may be overlap. That is covered in more advanced work.
如果事件不是互斥的，就不能简单相加，因为会有重叠部分。这在更高级的内容中会学习到。
7. The Sum of Probabilities of All Outcomes | 所有结果概率之和
For any experiment, the sum of the probabilities of all mutually exclusive and exhaustive outcomes must equal 1. This is a fundamental rule in probability.
对于任何实验，所有互斥且穷举的结果的概率之和必须等于 1。这是概率的基本规则。
f landing heads of 0.6. Since the only possible outcomes are heads and tails, P(tails) = 1 – 0.6 = 0.4.
示例：一枚偏置硬币落地正面的概率是 0.6。由于唯一可能的结果是正面和反面，因此 P(反面) = 1 – 0.6 = 0.4。
This rule is very useful for finding missing probabilities in probability distributions or when given a partially described situation.
这个规则在寻找概率分布中缺失的概率，或在给出部分描述的情况下非常有用。
ag contains red, blue, and green balls, and P(red) = 0.5, P(blue) = 0.3, then P(green) = 1 – (0.5 + 0.3) = 0.2.
如果一个袋子里有红色、蓝色和绿色球，且 P(红色) = 0.5，P(蓝色) = 0.3，那么 P(绿色) = 1 – (0.5 + 0.3) = 0.2。
8. Experimental vs Theoretical Probability | 实验概率与理论概率
Theoretical probability is what we expect to happen based on mathematics, like the 1/2 chance of heads when flipping a fair coin. Experimental probability (also called relative frequency) is based on actual trials or experiments.
理论概率是我们基于数学预期发生的概率，比如抛一枚公平硬币得到正面的 1/2 机会。实验概率（也称为相对频率）是基于实际试验或实验的。
Formula for experimental probability:
Experimental probability = Number of times event occurs / Total number of trials
实验概率公式：
实验概率 = 事件发生的次数 / 总试验次数
we do, the closer the experimental probability tends to get to the theoretical probability. This is known as the Law of Large Numbers.
示例：一枚骰子掷了 100 次，出现 6 点 22 次。得到 6 点的实验概率是 22/100 = 0.22，而理论概率是 1/6 ≈ 0.167。我们进行的试验次数越多，实验概率往往越接近理论概率。这被称为大数定律。
9. Probability and Two-Way Tables | 概率与双向表格
Two-way tables help organise data for two categories, making it easier to find probabilities. They are especially useful for questions involving conditional or combined events at KS3 introduction level.
双向表格有助于整理两个类别的数据，使求概率更容易。在 KS3 入门阶段，它们对于涉及条件概率或联合事件的问题特别有用。
Example: A survey of 50 students about their favourite subject:
| Maths | Science | Total | |
|---|---|---|---|
| Boys | 12 | 8 | 20 |
| Girls | 10 | 20 | 30 |
| Total | 22 | 28 | 50 |
What is the probability that a randomly chosen student is a girl who prefers Science? From the table, 20 girls prefer Science, total 50 students, so probability = 20/50 = 2/5.
随机选出一名学生是喜欢科学的女孩的概率是多少？从表格可知，20 名女孩喜欢科学，学生总数 50 人，因此概率 = 20/50 = 2/5。
or the probability that a boy prefers Maths = 12/20 (if we already know a boy is chosen).
这类表格还能帮助求出诸如 P(男孩) = 20/50、P(科学) = 28/50，或已知选出一名男孩时他喜欢数学的概率 = 12/20 等。
10. Probability Tree Diagrams (Introduction) | 概率树状图（入门）
Tree diagrams show all possible outcomes of two or more events happening one after another. At KS3, you will mostly draw them and list outcomes, without multiplying probabilities along branches yet (that comes later).
树状图展示两个或多个事件相继发生时所有可能的结果。在 KS3 阶段，你主要是画出它们并列出结果，尚不用沿着分支乘概率（那是后来的内容）。
For example, a bag contains 3 red and 2 blue marbles. You pick one marble, note its colour, put it back, and pick again. Draw a tree diagram to show the sample space.
例如，一个袋子里有 3 个红色和 2 个蓝色弹珠。你摸出一个，记下颜色，放回，然后再次摸取。画树状图来展示样本空间。
First pick branches: Red (R) and Blue (B). Second pick branches from each of those again: R and B. Outcomes: RR, RB, BR, BB. Note: because the marble is replaced, probabilities for each branch stay the same, but at KS3 you only need to list outcomes and count them.
第一次摸取分支：红色 (R) 和蓝色 (B)。从每种颜色的第二次摸取分支再次分出 R 和 B。结果：RR、RB、BR、BB。注意：因为弹珠被放回，每个分支的概率保持不变，但在 KS3 你只需要列出结果并计数。
11. Common Misconceptions in Probability | 概率中的常见误解
Many students believe that if you flip a coin and get heads five times in a row, the next flip is more likely to be tails. This is called the "gambler's fallacy". In reality, each flip is independent – the probability remains 1/2 each time.
许多学生认为，如果你抛硬币连续五次得到正面，那么下一次抛更可能是反面。这被称为“赌徒谬误”。实际上，每次抛掷都是独立的——每次的概率仍然是 1/2。
Another common mistake is adding probabilities when events are not mutually exclusive, or forgetting that the total probability must sum to 1. Always check whether outcomes overlap before using the addition rule.
另一个常见错误是当事件不互斥时相加概率，或是忘记总概率必须为 1。在运用加法规则前，一定要检查结果是否重叠。
Also, students sometimes confuse "odds" with probability. Odds are expressed as a ratio, e.g., odds of 3:1 against, while probability is a fraction like 1/4.
此外，学生有时混淆“赔率”与概率。赔率用比率表示，例如 3:1 的反对赔率，而概率是像 1/4 这样的分数。
12. Real-Life Applications and Revision Tips | 实际应用与复习建议
Probability is not just for exams – it is used in weather forecasting, insurance, sports analytics, and medical studies. Understanding probability helps you make informed decisions based on likelihood and risk.
概率不只用于考试——它还应用于天气预报、保险、体育分析和医学研究。理解概率有助于你根据可能性和风险做出明智的决定。
For revision, practise writing probabilities as fractions in simplest form, listing sample spaces systematically, and using two-way tables. Try questions involving spinners, dice, and coloured counters to build confidence.
复习时，练习将概率写成最简分数、系统地列出样本空间并使用双向表格。尝试涉及转盘、骰子和彩色计数器的题目，以建立信心。
Always remember these key points:
- Probability is between 0 and 1.
- P(event) = favourable / total (when equally likely).
- Sum of probabilities of all outcomes = 1.
- For mutually exclusive events, P(A or B) = P(A) + P(B).
请始终记住这些要点：
- 概率介于 0 和 1 之间。
- P(事件) = 有利结果数 / 总数（当等可能时）。
- 所有结果的概率之和 = 1。
- 对于互斥事件，P(A 或 B) = P(A) + P(B)。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图