📚 Understanding Probability | 理解概率
Probability is a branch of mathematics that helps us measure how likely an event is to happen. It is used everywhere – from weather forecasts and games of chance to sports predictions and medical research. At Key Stage 3, you learn to describe probability using numbers between 0 and 1, and you explore how to calculate it for simple, everyday situations.
概率是数学的一个分支，帮助我们衡量事件发生的可能性有多大。它无处不在——从天气预报和彩票游戏到体育预测和医学研究。在 Key Stage 3 阶段，你将学习用 0 到 1 之间的数字来描述概率，并探索如何计算日常生活中简单事件的概率。
1. What Is Probability? | 什么是概率？
Probability is a number that tells us the chance that a particular outcome will occur. It always lies between 0 and 1, where 0 means the event is impossible and 1 means it is certain to happen. The larger the probability, the more likely the event.
概率是一个数字，告诉我们某个特定结果发生的可能性。这个数字总是在 0 到 1 之间，0 表示事件不可能发生，1 表示事件必然发生。概率越大，事件发生的可能性就越高。
Probability can be written as a fraction, a decimal or a percentage. For example, an even chance can be expressed as ½, 0.5 or 50%. Understanding how to move between these forms is an important skill.
概率可以用分数、小数或百分数表示。例如，均等的机会可以表示为 ½、0.5 或 50%。学会在这些形式之间进行转换是一项重要的技能。
2. The Probability Scale | 概率尺度
The probability scale is a line that runs from 0 (impossible) to 1 (certain). Words such as 'unlikely', 'even chance' and 'likely' help us place events on this scale without using numbers.
概率尺度是从 0（不可能）延伸到 1（必然）的一条线。“不太可能”、“均等机会”、“很可能”等词语帮助我们在不使用数字的情况下将事件放置在尺度上。
A probability of 0 means the event can never happen, like rolling a 7 on a fair six-sided dice. A probability of 1 means the event will definitely happen, like the fact that tomorrow's weather will be either wet or dry (when we consider only these two simple outcomes).
概率为 0 意味着事件绝不会发生，比如在公平的六面骰子上掷出 7。概率为 1 意味着事件肯定发生，比如明天的天气要么下雨要么不下雨（当我们只考虑这两种简单结果时）。
A probability of 0.5 or ½ describes an even chance – the event is just as likely to happen as not to happen. Tossing a fair coin and getting 'heads' is a classic example of an even chance.
概率为 0.5 或 ½ 表示均等的机会——事件发生与不发生的可能性完全相同。抛掷一枚公平硬币得到正面就是均等机会的一个典型例子。
3. Key Terms in Probability | 概率的关键术语
To work with probability, you need to understand a few important terms. An outcome is a possible result of an experiment or event. When you roll a dice, the possible outcomes are 1, 2, 3, 4, 5 and 6.
要处理概率问题，你需要理解几个重要的术语。结果是试验或事件中一个可能出现的结果。当你掷一个骰子时，可能的结果是 1、2、3、4、5 和 6。
An event is a set of one or more outcomes. For instance, 'rolling an even number' includes the outcomes 2, 4 and 6. The sample space is the list of all possible outcomes. In a coin toss, the sample space is {heads, tails}.
事件由一个或多个结果组成的集合。例如，“掷出偶数”包含 2、4 和 6 这些结果。样本空间是所有可能结果的列表。在抛硬币中，样本空间是{正面, 反面}。
We say that outcomes are equally likely when each one has the same chance of happening. A fair dice and a fair coin both produce equally likely outcomes. If outcomes are not equally likely, we cannot use the simple counting rule for probability.
当每个结果发生的可能性相同时，我们称这些结果是等可能的。公平的骰子和公平的硬币都会产生等可能的结果。如果结果不是等可能的，我们就不能使用简单的计数法则来计算概率。
4. Calculating Probability | 计算概率
For a single event where all outcomes are equally likely, the probability can be found using this formula:
对于所有结果等可能的单一事件，概率可以用以下公式计算：
robability of an event = Number of favourable outcomes ÷ Total number of possible outcomes
事件的概率 = 有利结果的数量 ÷ 所有可能结果的总数
We call the outcomes that match the event 'favourable outcomes'. The total number of possible outcomes is the size of the sample space. Both numbers must be positive integers, and the favourable outcomes cannot exceed the total number of outcomes.
我们把与事件匹配的结果称为“有利结果”。可能结果的总数就是样本空间的大小。这两个数都必须是正整数，且有利结果的数量不能超过可能结果的总数。
For example, when rolling a fair six-sided dice, the probability of getting a 5 is 1/6, because there is one favourable outcome (5) and six possible outcomes altogether. Since all numbers are equally likely, the calculation is straightforward.
例如，掷一个公平六面骰子时，得到 5 的概率是 1/6，因为有利结果只有一个（5），而可能结果总共有六个。由于所有数字等可能，计算就很简单。
5. Examples of Simple Events | 简单事件的例子
Let's practise with a few common situations. When you flip a fair coin, the sample space has two equally likely outcomes: heads and tails. The probability of getting heads is ½, and so is the probability of getting tails.
让我们用几个常见的情形来练习。当你抛掷一枚公平硬币时，样本空间有两个等可能的结果：正面和反面。得到正面的概率是 ½，得到反面的概率也是 ½。
If a bag contains 4 red counters, 3 blue counters and 1 green counter, there are 8 counters in total. The probability of picking a red counter at random is 4/8, which simplifies to ½. The probability of picking a blue is 3/8, and picking a green is 1/8.
如果一个袋子里有 4 个红色筹码、3 个蓝色筹码和 1 个绿色筹码，总共有 8 个筹码。随机取出一个红色筹码的概率是 4/8，化简为 ½。取出蓝色的概率是 3/8，取出绿色的概率是 1/8。
A spinner is divided into five equal sections labelled A, B, C, D and E. The probability of landing on C is 1/5. The probability of landing on a vowel (A or E) is 2/5, since there are two favourable outcomes out of five.
一个转盘被分成五个相等的扇区，分别标有 A、B、C、D 和 E。 指针停在 C 上的概率是 1/5。停在元音字母（A 或 E）上的概率是 2/5，因为五个结果中有两个有利结果。
6. Fractions, Decimals and Percentages | 分数、小数和百分比
Probability can be written in three forms, and it is useful to be able to convert between them. A probability of ½ can also be written as 0.5 or 50%. In the same way, ¼ becomes 0.25 or 25%, and ¾ becomes 0.75 or 75%.
概率可以用三种形式书写，能够在这几种形式之间转换是很有用的。概率 ½ 也可以写成 0.5 或 50%。类似地，¼ 变成 0.25 或 25%，¾ 变成 0.75 或 75%。
nstance, the probability 3/8 as a decimal is 3 ÷ 8 = 0.375, and as a percentage it is 37.5%.
要将分数转换为小数，用分子除以分母。要将小数转换为百分数，乘以 100。例如，概率 3/8 转换为小数是 3 ÷ 8 = 0.375，转换为百分数是 37.5%。
When comparing probabilities, it is often helpful to write them all in the same form. For example, which is more likely: an event with probability 2/5, one with probability 0.45, or one with probability 38%? Converting 2/5 to 0.40 and 38% to 0.38 shows that 0.45 is the greatest.
在比较概率时，把它们都写成同一种形式通常很有帮助。例如，哪个事件更可能发生：概率为 2/5 的事件，概率为 0.45 的事件，还是概率为 38% 的事件？将 2/5 转换为 0.40，38% 转换为 0.38，可以看出 0.45 最大。
7. Experimental Probability | 实验概率
Sometimes we cannot find a probability just by listing equally likely outcomes. We might need to conduct an experiment or look at past data to estimate how likely an event is. This is called experimental probability or relative frequency.
有时我们无法仅仅通过列出等可能的结果来求出概率。我们可能需要通过实验或查看过去的数据来估计事件发生的可能性。这叫做实验概率或相对频数。
and lands point up 162 times, the experimental probability of point up is 162/200 = 0.81.
实验概率的计算方法是：事件发生的次数 ÷ 试验总次数。例如，如果一枚图钉被抛下 200 次，有 162 次钉尖朝上落下，则钉尖朝上的实验概率是 162/200 = 0.81。
The more trials you carry out, the closer the experimental probability usually gets to the true theoretical probability. However, they are often not exactly the same, especially with only a small number of trials.
你进行的试验次数越多，实验概率通常越接近真实的理论概率。但是，它们往往并不完全相等，特别是在试验次数较少时。
8. Expected Outcomes | 预期结果
Once you know the probability of an event, you can predict how many times you would expect that event to occur in a certain number of trials. This is called the expected frequency or expected number of outcomes.
一旦你知道了某个事件的概率，你就可以预测在若干次试验中该事件预计会发生多少次。这被称为期望频数或预期结果数。
is 0.3, and you pass through 50 times, you expect to get a red light 0.3 × 50 = 15 times.
期望频数的公式是：预期次数 = 概率 × 试验次数。例如，如果在某个路口遇到红灯的概率是 0.3，而你经过 50 次，那么你预期会遇到红灯 0.3 × 50 = 15 次。
expect to roll a 6 about 1/6 × 600 = 100 times, but the actual number might be 98 or 102.
这并不保证事件恰好发生那么多次。这是一个长期平均值。如果你掷一枚公平骰子 600 次，你预期大约掷出 6 的次数是 1/6 × 600 = 100 次，但实际数字可能是 98 或 102。
9. Complementary Events | 互补事件
probability of its complement is always 1. If and only if we know P(A), then P(not A) = 1 – P(A).
每个事件都有一个互补事件，它包含了不在该事件中的所有结果。一个事件的概率与其互补事件的概率之和总是等于 1。如果我们知道 P(A)，那么 P(非 A) = 1 – P(A)。
obability that it will rain tomorrow is 0.2, so the probability that it will not rain is 1 – ts are complementary and cover all possible weather outcomes (assuming we only consider rain or no rain).
假设明天下雨的概率是 0.2，那么不下雨的概率就是 1 – 0.2 = 0.8。这是因为这两个事件互补，并且覆盖了所有可能的天气结果（假设我们只考虑下雨或不下雨）。
When rolling a dice, the event 'getting a 3' has probability 1/6. The complementary event 'not getting a 3' has probability 5/6. Using complements can often make calculations quicker, especially when it is easier to find the probability of an event NOT happening.
掷骰子时，事件“得到 3”的概率是 1/6。互补事件“没有得到 3”的概率是 5/6。使用互补事件常常能使计算更快捷，尤其是当更容易求出事件不发生的概率时。
10. Using Frequency Trees | 使用频率树
A frequency tree is a simple way to record and organise outcomes from two or more trials or characteristics. It helps us work out probabilities when we know the totals and how the data splits at each stage.
频率树是一种记录和组织两次或多次试验或特征结果的简单方法。当我们知道总数以及数据在每个阶段如何拆分时，它可以帮我们求出概率。
Imagine there are 30 students in a class: 18 girls and 12 boys. Of the girls, 10 wear glasses and 8 do not. Of the boys, 4 wear glasses and 8 do not. A frequency tree splits the total into branches, making it easy to see how many students belong to each combined group.
假设班上有 30 名学生：18 名女生和 12 名男生。女生中有 10 人戴眼镜，8 人不戴。男生中有 4 人戴眼镜，8 人不戴。频率树将总数拆分为分支，让我们很容易看出每个组合组中有多少学生。
= 8/30 = 4/15. Frequency trees work well when outcomes are combined in a step-by-step way.
根据频率树，我们可以求出如下概率：P(女生且戴眼镜) = 10/30 = 1/3，或者 P(男生且不戴眼镜) = 8/30 = 4/15。当结果以分步方式组合时，频率树的效率很高。
11. Key Points to Remember | 需要记住的关键点
Always check that the total number of possible outcomes is counted correctly. Write probabilities in their simplest form unless the question asks otherwise, and remember that probabilities must be between 0 and 1 inclusive.
始终要确保正确计数了可能结果的总数。除非题目另有要求，应将概率写成最简形式，并记住概率必须介于 0 和 1 之间（含 0 和 1）。
A probability scale can help you judge whether an answer makes sense. If someone says the probability of winning is 1.5, you know it is impossible because a probability cannot be larger than 1. Similarly, a negative probability has no meaning in our context.
概率尺度可以帮助你判断答案是否合理。如果有人说获胜的概率是 1.5，你就知道那是不可能的，因为概率不能大于 1。同样，在我们上下文里，负概率也是没有意义的。
For complementary events, using '1 – P(event)' can save time. Practise converting between fractions, decimals and percentages so that you can handle word problems efficiently and compare different probabilities quickly.
对于互补事件，使用“1 – P(事件)”可以节省时间。练习在分数、小数和百分数之间进行转换，这样你就能高效地处理应用题，也能快速比较不同的概率。
12. Summary | 总结
Probability gives us a way to measure uncertainty. By identifying the sample space and counting favourable outcomes, you can calculate the theoretical probability of a simple event. When outcomes are not equally likely, experiments and past data help you find relative frequencies.
概率为我们提供了一种度量不确定性的方法。通过识别样本空间并计数有利结果，你可以计算简单事件的理论概率。当结果不是等可能时，实验和过去数据可以帮助你找出相对频数。
Whether you are spinning a spinner, drawing a card or predicting the weather, the same basic rules apply. Move confidently between numbers, words and symbols, and always check that your final probability lies between 0 and 1. With these tools, you are well on your way to mastering probability at Key Stage 3.
无论你是旋转转盘、抽牌还是预测天气，都适用相同的基本规则。自信地在数字、词语和符号之间转换，并始终检查你的最终概率是否在 0 和 1 之间。有了这些工具，你就在 Key Stage 3 阶段顺利掌握了概率。
Published by TutorHao | KS3 Mathematics Revision Series | aleveler.com
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