📚 Understanding Probability for KS3 | KS3 概率入门
Probability is a branch of maths that helps us measure how likely something is to happen. In KS3 Cambridge Mathematics, you learn to describe chance using words, fractions, decimals, and percentages, and to calculate probabilities in simple situations. This article covers the key ideas, from basic vocabulary to mutually exclusive events and expected frequency. Each section pairs an English explanation with an equivalent Chinese paragraph to support bilingual learners.
概率是数学的一个分支，帮助我们衡量某件事发生的可能性有多大。在 KS3 剑桥数学中，你将学习用词语、分数、小数和百分比来描述机会，并计算简单情况下的概率。本文涵盖从基本词汇到互斥事件和期望频数的关键概念。每个部分都提供英文解释和对应的中文段落，帮助双语学习者理解。
1. What Is Probability? | 什么是概率？
Probability is a number between 0 and 1 that tells how likely an event is. A probability of 0 means the event is impossible; a probability of 1 means it is certain. For example, the probability of rolling a 7 on a normal six‑sided dice is 0 (impossible), and the probability of getting a number from 1 to 6 is 1 (certain).
概率是一个介于 0 和 1 之间的数字，表示事件发生的可能性。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。例如，掷一个普通的六面骰子得到 7 的概率是 0（不可能），得到 1 到 6 之间的数字的概率是 1（必然）。
2. Probability Scale | 概率标度
The probability scale places events according to their likelihood. Words such as ‘impossible’, ‘unlikely’, ‘even chance’, ‘likely’, and ‘certain’ correspond to numbers: impossible = 0, even chance = ½, certain = 1. We often draw a line from 0 to 1 and mark where an event sits. For instance, picking a red card from a standard deck of playing cards has a probability of ½, so it sits in the middle of the scale.
概率标度根据事件的可能性将其放置。像“不可能”、“不太可能”、“一半机会”、“很可能”和“一定”等词语对应数字：不可能 = 0，一半机会 = ½，一定 = 1。我们经常画一条从 0 到 1 的线，标出事件的位置。例如，从一副标准扑克牌中拿出一张红牌的概率是 ½，因此它位于标度的中间。
3. Basic Formula for Probability | 概率的基本公式
The probability of an event ‘A’ is given by: Number of favourable outcomes ÷ Total number of possible outcomes. This only works when all outcomes are equally likely. For example, the probability of rolling a 2 on a fair six‑sided dice is 1/6 because there is one ‘2’ and six possible numbers.
事件 “A” 的概率公式为：有利结果的数量 ÷ 可能结果的总数。这仅在所有结果等可能时成立。例如，掷一枚均匀的六面骰子得到 2 的概率是 1/6，因为有一个“2”和六个可能的数字。
P(A) = Number of favourable outcomes / Total number of outcomes
4. The Outcomes Must Be Equally Likely | 结果必须是等可能的
The basic formula relies on every outcome having the same chance. For example, when flipping a fair coin, heads and tails are equally likely. But if the coin is biased, the formula no longer gives the true probability. In KS3, we mostly deal with fair dice, coins, spinners, and equally likely selections.
基本公式要求每个结果的机会相同。例如，抛一枚均匀硬币时，正面和反面的可能性相同。但如果硬币不均匀，公式就不再给出真实概率。在 KS3 中，我们主要处理均匀的骰子、硬币、转盘和等可能的选择。
5. Writing Probabilities as Fractions, Decimals, and Percentages | 用分数、小数和百分比表示概率
Probabilities can be written in different forms. A probability of ½ can also be written as 0.5 or 50%. You should be able to convert between these: ¼ = 0.25 = 25%, ¾ = 0.75 = 75%, and so on. For example, the probability of flipping a head on a fair coin is ½, 0.5, or 50%.
概率可以用不同形式表示。概率 ½ 也可以写成 0.5 或 50%。你应该能够在它们之间转换：¼ = 0.25 = 25%，¾ = 0.75 = 75% 等等。例如，抛一枚均匀硬币得到正面的概率是 ½、0.5 或 50%。
6. The Sum of All Possible Outcomes | 所有可能结果的概率之和
If you list all possible outcomes of an event, the sum of their probabilities must equal 1. For a fair dice, P(1) + P(2) + P(3) + P(4) + P(5) + P(6) = 1/6 × 6 = 1. This rule is very useful for checking your work. If the total is not 1, you have probably missed an outcome or double‑counted.
如果你列出某个事件的所有可能结果，它们的概率之和必须等于 1。对于一个均匀的骰子，P(1) + P(2) + P(3) + P(4) + P(5) + P(6) = 1/6 × 6 = 1。这个规则对检查你的工作非常有用。如果总和不等于 1，你可能漏掉了一个结果或者重复计算了。
7. Complementary Events | 互补事件
The complement of event A is the event that A does not happen. The probability of the complement is P(not A) = 1 – P(A). For example, if the probability of a bus being late is 0.2, then the probability it is not late is 1 – 0.2 = 0.8. This is a common way to simplify calculations.
事件 A 的互补事件是 A 不发生的事件。互补的概率为 P(not A) = 1 – P(A)。例如，如果公交车晚点的概率是 0.2，那么它不晚点的概率是 1 – 0.2 = 0.8。这是一种常见的简化计算的方法。
8. Mutually Exclusive Events | 互斥事件
Two events are mutually exclusive if they cannot happen at the same time. For mutually exclusive events A and B, the probability of either A or B occurring is P(A) + P(B). For instance, rolling a 2 or a 3 on a dice: P(2 or 3) = 1/6 + 1/6 = 2/6 = 1/3. Be careful: this formula only works when events cannot overlap.
如果两个事件不可能同时发生，它们就是互斥的。对于互斥事件 A 和 B，A 或 B 发生的概率是 P(A) + P(B)。例如，掷骰子得到 2 或 3：P(2 或 3) = 1/6 + 1/6 = 2/6 = 1/3。注意：这个公式只在事件不能重叠时才有效。
9. Sample Space Diagrams | 样本空间图
A sample space diagram lists all possible outcomes of two events. For example, rolling two dice: the sample space can be shown as a 6 × 6 table with 36 equally likely pairs. To find the probability of a total score of 7, count the pairs that sum to 7 (there are 6) and divide by 36, giving 6/36 = 1/6.
样本空间图列出了两个事件的所有可能结果。例如，掷两个骰子：样本空间可以表示为一个 6 × 6 的表格，共有 36 个等可能的一对数。要找出总点数为 7 的概率，数出和为 7 的一对数（有 6 个），除以 36，得到 6/36 = 1/6。
| + | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 5 | 6 | 7 | 8 | 9 | 10 | 11 |
| 6 | 7 | 8 | 9 | 10 | 11 | 12 |
10. Expected Frequency | 期望频数
Expected frequency tells how many times you would expect an event to occur in a number of trials. It is calculated as: Expected frequency = Probability × Number of trials. If a fair coin is flipped 200 times, the expected frequency of heads is 0.5 × 200 = 100. The actual result may differ, but the expected frequency gives a long‑run average.
期望频数是指在多次试验中，你预期某事件发生的次数。计算公式为：期望频数 = 概率 × 试验次数。如果均匀硬币抛掷 200 次，正面的期望频数是 0.5 × 200 = 100。实际结果可能不同，但期望频数给出了长期的平均值。
11. Experimental vs Theoretical Probability | 实验概率与理论概率
Theoretical probability is what you expect from equally likely outcomes (e.g., P(head) = ½). Experimental probability comes from actually doing an experiment: Number of times event occurs ÷ Total number of trials. As you do more trials, the experimental probability usually gets closer to the theoretical probability. This is called the law of large numbers.
理论概率是根据等可能结果你期望的数值（例如，P(正面) = ½）。实验概率来自实际实验：事件发生的次数 ÷ 总试验次数。随着试验次数增加，实验概率通常会越来越接近理论概率。这叫做大数定律。
12. Probability in Everyday Life | 日常生活中的概率
Probability is used in weather forecasting (‘30% chance of rain’), risk assessment, games, and even medical testing. Understanding basic probability helps you make informed decisions and recognise when claims are misleading. In KS3, you build the foundation for more advanced topics like tree diagrams and conditional probability in later years.
概率用于天气预报（“30% 的降雨概率”）、风险评估、游戏，甚至医学检测。理解基本的概率知识有助于你做出明智的决定，并识别出那些误导性的说法。在 KS3 中，你为以后学习更高级的主题（如树状图和条件概率）打下基础。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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