KS3 Maths: Probability Revision Guide | KS3 数学：概率考点精讲

📚 KS3 Maths: Probability Revision Guide | KS3 数学：概率考点精讲

Probability is the branch of mathematics that deals with how likely events are to happen. In KS3, you will learn to describe probability on a scale from 0 to 1, calculate probabilities of simple events, and use different methods to list outcomes. This guide covers all the key points you need for your tests.

概率是数学中研究事件发生可能性的分支。在 KS3 阶段，你将学习用 0 到 1 的量尺来描述概率，计算简单事件的概率，并使用不同的方法列出所有可能结果。本指南涵盖了所有你需要掌握的考试要点。

1. What is Probability? | 什么是概率？

Probability is a measure of how likely an event is to happen. It is a number that describes the chance of a particular outcome occurring when you run an experiment, play a game, or observe a random process. In KS3 maths, you will learn to assign a probability value between 0 and 1 to different events.

概率是衡量某个事件发生可能性的度量。它是一个用来描述在进行实验、玩游戏或观察随机过程时，某个特定结果发生几率的数字。在 KS3 数学中，你将学习给不同的事件赋予一个介于 0 和 1 之间的概率值。

An experiment is any process that can be repeated and has a well-defined set of possible outcomes. Tossing a coin, rolling a dice, or picking a card from a shuffled deck are all examples of random experiments. The result of a single trial is called an outcome. An event is a set of one or more outcomes that you are interested in.

实验是指任何可以重复进行、并且具有一组明确可能结果的过程。抛硬币、掷骰子或者从洗好的牌中抽一张牌，都是随机实验的例子。单次试验的结果叫做一个结果。事件则是由一个或多个你所关心的结果组成的集合。

2. Probability Scale | 概率量尺

The probability scale runs from 0 to 1. An event that is impossible has a probability of 0. An event that is certain to happen has a probability of 1. All other probabilities lie somewhere in between these two extremes.

概率量尺的范围是从 0 到 1。不可能发生的事件概率为 0。必然发生的事件概率为 1。所有其他的概率都落在两个极端值之间。

An event that is just as likely to occur as not to occur has a probability of ½ (one‑half). For example, when you toss a fair coin, getting ‘heads’ has a probability of ½. The closer the probability is to 1, the more likely the event is. The closer it is to 0, the less likely it is.

发生与不发生的可能性恰好相等的事件，其概率为 ½（一半）。例如，抛一枚均匀硬币，得到“正面”的概率为 ½。概率越接近 1，事件越可能发生。概率越接近 0，事件越不可能发生。

You can describe probability using words such as impossible, unlikely, even chance, likely and certain. These words can be placed along the number line from 0 to 1 to help you visualise the likelihood of an event.

你可以使用诸如不可能、不太可能、均等机会、很可能和必然等词语来描述概率。这些词语可以沿着从 0 到 1 的数轴放置，帮助你想象事件发生的可能性大小。

3. Basic Probability Formula | 基本概率公式

When all outcomes of an experiment are equally likely, the probability of an event A can be calculated using the formula:

P(A) = Number of favourable outcomes / Total number of possible outcomes

当实验的所有结果都具有相等的可能性时，事件 A 发生的概率可以用以下公式计算：

P(A) = 有利结果的数量 / 所有可能结果的总数

For example, when you roll a fair six‑sided dice, there are 6 equally likely outcomes: 1, 2, 3, 4, 5, 6. The probability of rolling a 3 is 1/6 because there is exactly one favourable outcome (the 3) out of six possible outcomes. Similarly, the probability of rolling an even number is 3/6, which simplifies to ½, because the favourable outcomes are 2, 4 and 6.

例如，当你掷一枚均匀的六面骰子时，有 6 个等可能的结果：1、2、3、4、5、6。掷出 3 的概率是 1/6，因为六种可能结果中只有一个有利结果（即 3）。类似地，掷出偶数的概率是 3/6，化简得 ½，因为有利结果是 2、4 和 6。

Always remember to simplify your fraction where possible. You can write the answer as a fraction in its simplest form, or convert it to a decimal or percentage if the question asks for that.

记住，如果可能的话，一定要约简分数。你可以把答案写成最简分数形式，或者根据题目的要求将其转换为小数或百分比。

4. Listing Outcomes: Sample Spaces | 列出结果：样本空间

To calculate probabilities accurately, you often need to list all the possible outcomes of an experiment. This complete list is called the sample space. You can organise outcomes in a list, a table, or a diagram to make sure no outcome is missed.

为了准确计算概率，你经常需要列出实验的所有可能结果。这个完整的列表就叫做样本空间。你可以用清单、表格或图表来整理结果，以确保没有遗漏任何结果。

For a single coin toss, the sample space is {Heads, Tails}. For tossing two coins together, you can write the sample space as {HH, HT, TH, TT}, where H stands for head and T for tail. Notice that TH (tail on first coin, head on second) is different from HT, so you must list both.

抛一枚硬币的样本空间是 {正面, 反面}。同时抛两枚硬币时，你可以把样本空间写成 {HH, HT, TH, TT}，其中 H 代表正面，T 代表反面。注意 TH（第一枚反面、第二枚正面）和 HT 是不同的，所以两者都必须列出。

For two dice, using a table is very helpful. Below is a sample space table showing the sum of the numbers on two six‑sided dice. The table helps you find, for example, that there are 6 ways to roll a total of 7, giving a probability of 6/36 = 1/6.

对于两枚骰子，使用表格非常有帮助。下面是一个样本空间表格，展示了两枚六面骰子点数之和。这个表格能帮助你找到，比方说，掷出总和为 7 的情况有 6 种，因此概率为 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

There are 6 × 6 = 36 equally likely outcomes in the table.

表格中共有 6 × 6 = 36 个等可能的结果。
The probability of getting a sum of 7 is 6/36 = 1/6.

得到总和为 7 的概率是 6/36 = 1/6。
The probability of getting a sum of 12 is 1/36.

得到总和为 12 的概率是 1/36。

5. Expected Outcomes | 预期结果

If you know the probability of an event, you can predict how many times it is likely to happen over many trials. The expected number of successes is found by multiplying the probability by the number of trials.

如果你知道某个事件的概率，就可以预测在多次试验中它可能发生多少次。成功的预期次数等于概率乘以试验次数。

Expected number = P(event) × number of trials

预期次数 = P(事件) × 试验次数

For instance, if you roll a fair dice 600 times, you would expect to roll a ‘3’ on about 600 × 1/6 = 100 occasions. This does not mean you will definitely get exactly one hundred 3s, but over a large number of trials the actual count should be close to the expected value.

例如，如果你掷一枚均匀骰子 600 次，你预期掷出“3”的次数大约是 600 × 1/6 = 100 次。这并不意味着你一定恰好得到 100 个 3，但在大量试验后，实际次数应该接近预期值。

Expected outcomes are very useful in games, surveys and making predictions. Always be clear that the expected value is an average, not a guarantee.

预期结果在游戏、调查和预测中非常有用。始终要清楚，预期值是一个平均值，而不是保证的结果。

6. Relative Frequency vs Theoretical Probability | 相对频率与理论概率

Theoretical probability is calculated from the structure of the experiment, assuming all outcomes are equally likely. Relative frequency is calculated after carrying out the experiment for real, using the formula: relative frequency = number of times event occurred ÷ total number of trials.

理论概率是根据实验结构计算出来的，假设所有结果是等可能的。相对频率则是在实际进行实验之后计算出来的，使用的公式是：相对频率 = 事件发生的次数 ÷ 试验总次数。

For example, the theoretical probability of getting heads when flipping a coin is ½ = 0.5. If you flip a coin 100 times and get 53 heads, the relative frequency of heads is 53/100 = 0.53. As you repeat the experiment more and more times, the relative frequency tends to get closer and closer to the theoretical probability. This is sometimes called the law of large numbers.

例如，抛一枚硬币得到正面的理论概率是 ½ = 0.5。如果你抛硬币 100 次得到了 53 次正面，那么正面的相对频率就是 53/100 = 0.53。随着你重复实验的次数越来越多，相对频率往往会越来越接近理论概率。这有时被称为大数定律。

You may be asked to compare theoretical and experimental probabilities in KS3. Always use clear language: the experimental probability depends on the actual results, while the theoretical probability is a fixed value when outcomes are equally likely.

在 KS3 阶段，你可能会被要求比较理论概率和实验概率。请始终使用清晰的语言：实验概率取决于实际结果，而理论概率在结果等可能时是一个固定的值。

7. Mutually Exclusive Events and Sum of Probabilities | 互斥事件及概率之和

Two events are mutually exclusive if they cannot happen at the same time. For example, when you roll a dice, the events ‘rolling a 2’ and ‘rolling an odd number’ are mutually exclusive – a single dice roll cannot be both 2 and odd. The sum of the probabilities of all mutually exclusive outcomes in a sample space is always equal to 1.

如果两个事件不可能同时发生，那么它们就是互斥的。例如，掷一枚骰子时，事件“掷出 2”和“掷出奇数”就是互斥的——单次掷骰子不可能既是 2 又是奇数。样本空间中所有互斥结果的概率之和总是等于 1。

An important rule follows from this: for any event A, the probability that A does not happen is 1 – P(A). If the probability that it rains tomorrow is 0.3, then the probability it does not rain is 1 – 0.3 = 0.7. This is often called the complement rule and can save you a lot of calculating.

由此可以得出一个重要的规则：对于任何事件 A，A 不发生的概率等于 1 – P(A)。如果明天下雨的概率是 0.3，那么不下雨的概率就是 1 – 0.3 = 0.7。这通常被称为补集规则，可以为你节省大量计算。

Always check your work by ensuring that the probabilities of all possible separate outcomes add up to exactly 1 (or 100%). If they do not, you may have missed an outcome or made a calculation mistake.

检查你的作业时，一定要确保所有可能独立结果的概率之和恰好等于 1（或 100%）。如果不等于 1，你很可能遗漏了某个结果或者出现了计算错误。

8. Probability as Fractions, Decimals and Percentages | 概率的分数、小数和百分比形式

Probabilities can be expressed as fractions, decimals or percentages. In the KS3 exam, you must be comfortable converting between these three forms and choosing the most appropriate one for the context.

概率可以用分数、小数或百分比来表示。在 KS3 考试中，你必须能够熟练地在这三种形式之间进行转换，并能根据情境选择最合适的一种形式。

Fraction	Decimal	Percentage
1/2	0.5	50%
1/4	0.25	25%
3/4	0.75	75%
1/5	0.2	20%
1/10	0.1	10%

To convert a fraction to a decimal, divide the numerator by the denominator. To convert a decimal to a percentage, multiply by 100. To convert a percentage to a fraction, write it over 100 and simplify. Always show your working clearly, as marks are often given for correct conversions.

要将分数转化为小数，用分子除以分母。要将小数转化为百分比，乘以 100。要将百分比转化为分数，写成百分之几并约简。务必清晰地展示解题步骤，因为正确的转换过程常常能得分。

9. Two-Step Experiments and Tree Diagrams | 两步试验与树形图

Some experiments involve doing two or more things one after the other, such as drawing two counters from a bag or flipping a coin and rolling a dice. To find the probabilities of combined outcomes, you can use a tree diagram.

有些实验涉及先后进行两个或多个操作，例如从一个袋子中抽取两个筹码，或者抛一枚硬币再掷一枚骰子。为了求出复合结果的概率，你可以使用树形图。

A tree diagram shows all the possible outcomes of the first step as branches, and then from each of those branches it shows the outcomes of the second step. Along each branch, you write the probability of that outcome happening at that stage. To find the probability of a whole path (e.g., red then red), you multiply the probabilities along the branches.

树形图用分支展示第一步所有可能的结果，然后从每个分支再分出第二步的结果。沿着每一条分支，你写出该阶段该结果发生的概率。要求出整条路径的概率（例如，先红后红），你就把路径上的概率相乘。

For example, a bag contains 3 red and 2 blue counters. You take one counter, note the colour, and put it back. You then take a second counter. The probability of picking two reds is: P(first red) × P(second red) = 3/5 × 3/5 = 9/25. This uses replacement. If you do not replace the first counter, the probabilities on the second set of branches change because the totals and numbers of colours both decrease by one.

例如，一个袋子里有 3 个红筹码和 2 个蓝筹码。你取出一个筹码，记下颜色，再放回去。然后取出第二个筹码。取出两个红筹码的概率为：P(第一个红) × P(第二个红) = 3/5 × 3/5 = 9/25。这使用了放回方式。如果你不放回第一个筹码，那么第二组分支上的概率会发生变化，因为总数和该颜色的个数都减少了一个。

Always read the question carefully to know whether the experiment is with or without replacement. Label your tree diagram neatly, and remember to multiply fractions correctly. Tree diagrams also help

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KS3 Maths: Probability Revision Guide | KS3 数学：概率 考点精讲