Probability | 概率入门

📚 Probability | 概率入门

Probability is the branch of mathematics that deals with how likely an event is to happen. From flipping a coin to predicting the weather, we use probability every day to make sense of uncertainty. In KS3 Mathematics, you will learn how to describe chance using numbers, words, and diagrams. This article covers the key ideas you need, often found in exercises like those on page 238 of your workbook, and helps you build a solid foundation for future study.

概率是数学中处理事件发生可能性的分支。从抛硬币到预测天气，我们每天都在运用概率来理解不确定性。在 KS3 数学中，你将学会如何用数字、词语和图形来描述可能性。本文涵盖了你需要掌握的核心概念，这些内容常见于练习册第 238 页之类的题目，会帮助你为今后的学习打下坚实基础。

1. What is Probability? | 什么是概率？

Probability measures the chance that a particular event will occur. It is always expressed as a number between 0 and 1, where 0 means the event is impossible and 1 means the event is certain. For example, the probability of the sun rising tomorrow is almost 1, while the probability of finding a unicorn in your garden is 0.

概率衡量某个特定事件发生的可能性。它总是一个介于 0 和 1 之间的数字，0 表示事件不可能发生，1 表示事件必然发生。例如，明天太阳升起的概率几乎为 1，而在花园里发现独角兽的概率为 0。

We can also write probabilities as fractions, decimals, or percentages. A probability of ½ is the same as 0.5 or 50%. On page 238, you might see questions asking you to match these different forms.

我们也可以将概率写成分数、小数或百分数。概率 ½ 等价于 0.5 或 50%。在第 238 页的练习中，你可能会碰到要求匹配这些不同形式的题目。

2. Probability Scale | 概率尺度

A probability scale is a number line from 0 to 1 that helps us visualise how likely events are. We place events along this line using their numerical probabilities. Words such as ‘impossible’, ‘unlikely’, ‘even chance’, ‘likely’, and ‘certain’ are also used to describe positions on the scale.

概率尺度是一条从 0 到 1 的数轴，帮助我们直观地看出事件发生的可能性大小。我们可以根据事件的概率值把它标在数轴上，同时用“不可能”“不太可能”“机会均等”“很可能”“必然”等词语来描述其位置。

0 – Impossible (e.g., rolling a 7 on a standard six‑sided die)
¼ – Unlikely (e.g., picking a red ball when only 1 out of 5 balls is red)
½ – Even chance (e.g., getting heads on a coin toss)
¾ – Likely (e.g., drawing a picture card from a standard deck – roughly 77%)
1 – Certain (e.g., the next day being Monday after Sunday)

0 – 不可能（例如掷一枚标准六面骰子得到 7 点）
¼ – 不太可能（例如从 5 个球中挑出唯一的红球）
½ – 机会均等（例如抛硬币得到正面）
¾ – 很可能（例如从一副扑克牌中抽到人头牌——概率约为 77%）
1 – 必然（例如星期天之后是星期一）

When filling in a probability scale like the one on p238, always check that your numbers are between 0 and 1 and that the positions make sense relative to each other.

在填写像第 238 页那样的概率尺度时，一定要确保所有数字都在 0 到 1 之间，并且各个位置之间的相互关系合理。

3. Calculating Basic Probability | 计算基本概率

For equally likely outcomes, the probability of an event A is calculated using the formula:

对于等可能的结果，事件 A 的概率用以下公式计算：

P(A) = Number of favourable outcomes ÷ Total number of equally likely outcomes

Example: When rolling a fair six‑sided die, the probability of rolling a 4 is 1/6 because there is one favourable outcome (4) and six possible outcomes in total.

例如：掷一枚公平的六面骰子，得到 4 点的概率是 1/6，因为有利结果（4 点）有 1 个，而所有可能结果共有 6 个。

Always simplify the fraction if possible. If a bag contains 2 blue, 3 yellow and 5 green marbles, the probability of picking a blue marble is 2/10 = 1/5. The answer can also be written as 0.2 or 20%.

如果可能，要把分数化简。如果一个袋子里有 2 个蓝弹珠、3 个黄弹珠和 5 个绿弹珠，那么抽出蓝弹珠的概率是 2/10 = 1/5。答案也可以写为 0.2 或 20%。

Event	Favourable outcomes	Total outcomes	Probability
Rolling a number > 2 on a die	4 (3,4,5,6)	6	4/6 = 2/3
Choosing a vowel from ‘MATHS’	1 (A)	5	1/5

Questions on page 238 often ask you to use this formula with spinners, dice, cards or coloured counters. Make sure you count the outcomes carefully.

第 238 页的题目经常让你将这个公式用于转盘、骰子、扑克牌或彩色计数片。一定要仔细数清楚所有可能结果。

4. Experimental vs Theoretical Probability | 实验概率与理论概率

Theoretical probability is what we expect to happen based on equally likely outcomes, while experimental probability (or relative frequency) is based on actual experiments or past data.

理论概率是根据等可能的结果，我们预期会发生的情况；而实验概率（或相对频率）则是基于实际实验或历史数据得到的结果。

The formula for experimental probability is:

实验概率的公式为：

Experimental probability = Number of times the event occurs ÷ Total number of trials

If you flip a coin 50 times and get 28 heads, the experimental probability of heads is 28/50 = 0.56, even though the theoretical probability is 0.5. The more trials you do, the closer the experimental probability usually gets to the theoretical value—this is known as the Law of Large Numbers.

如果你抛硬币 50 次，得到 28 次正面，那么正面的实验概率就是 28/50 = 0.56，尽管理论概率是 0.5。试验次数越多，实验概率通常会越接近理论值——这就是大数定律。

In KS3, you may be asked to compare these two types of probability and explain why they might differ, especially when the number of trials is small.

在 KS3 阶段，你可能会被要求比较这两种概率，并解释它们为何可能不同，尤其是在试验次数较少时。

5. 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 both outcomes cannot occur on a single roll.

如果两个事件不能同时发生，它们就是互斥的。例如，掷一枚骰子时，“得到 2 点”和“得到 5 点”是互斥的，因为一次投掷不可能同时出现两个结果。

For mutually exclusive events A and B, the probability of either A or B happening is simply the sum of their individual probabilities:

对于互斥事件 A 和 B，A 或 B 发生的概率就是它们各自概率之和：

P(A or B) = P(A) + P(B)

What is the probability of rolling a 2 or a 5 on a fair die? P(2) = 1/6, P(5) = 1/6, so P(2 or 5) = 1/6 + 1/6 = 2/6 = 1/3.

掷一枚公平骰子得到 2 点或 5 点的概率是多少？P(2) = 1/6，P(5) = 1/6，所以 P(2 或 5) = 1/6 + 1/6 = 2/6 = 1/3。

Non‑mutually exclusive events can happen at the same time (e.g., drawing a red card and a king from a deck), and their combined probability requires a different approach, but at KS3 the focus is normally on mutually exclusive cases.

非互斥事件可以同时发生（例如从一副牌中抽到红色牌和抽到 K），它们合并的概率需要用另一种方法计算，不过 KS3 通常只关注互斥事件。

6. Sample Space Diagrams | 样本空间图

A sample space is a list, table, or diagram that shows all possible outcomes of an experiment. For two events combined (like rolling two dice or spinning a two‑section spinner), a sample space diagram helps you count outcomes systematically.

样本空间是一个列出所有可能实验结果的清单、表格或图形。对于两个事件的组合（例如掷两枚骰子或转动一个双区转盘），样本空间图可以帮助你系统地数出所有结果。

For rolling two dice, a common sample space is a 6 × 6 grid with numbers 1–6 on each axis. Each cell shows the sum or the ordered pair. For example, there are 36 equally likely outcomes. This makes it easy to calculate probabilities like P(total = 7) = 6/36 = 1/6, because the sums of 7 occur in 6 different cells.

对于掷两枚骰子的情况，常用的样本空间是一个 6×6 的网格，横纵坐标分别标有 1 到 6。每个格子显示两枚骰子的点数之和或有序数对。如此一来，共有 36 个等可能结果。这就很容易计算诸如 P(总和为 7) = 6/36 = 1/6 的概率，因为和为 7 的情况出现在 6 个不同的格子里。

On page 238, you may see questions that ask you to complete a partially filled sample space diagram and then use it to find probabilities. The key is to stay organised and check that all possible combinations are included.

在第 238 页的练习中，你可能会看到要求补全部分样本空间图并利用它求概率的题目。关键是要有条理，并检查是否已包含所有可能的组合。

7. Probability of an Event Not Happening | 事件不发生的概率

If you know the probability of an event occurring, it is very easy to find the probability that it does not occur. Since something must either happen or not happen, the two probabilities add up to 1.

如果你知道一个事件发生的概率，就很容易求出它不发生的概率。因为事件要么发生要么不发生，两者的概率之和为 1。

P(not A) = 1 – P(A)

If the probability of rain tomorrow is 0.3, then the probability that it will not rain is 1 – 0.3 = 0.7. This is particularly useful in multi‑part questions where you need to find the chance of the ‘opposite’ or complementary event.

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。这在需要你求出“对立”事件或互补事件概率的多步骤问题中特别有用。

In KS3 examinations, you might be asked, ‘The probability that a train is late is 0.15. What is the probability it is on time?’ You simply calculate 1 – 0.15 = 0.85.

在 KS3 考试中，你可能会遇到这样的问题：“一列火车晚点的概率是 0.15。它准时的概率是多少？”你只需计算 1 – 0.15 = 0.85 即可。

8. Expected Frequency | 期望频数

Expected frequency tells us how many times we would expect an event to occur if we repeated an experiment a large number of times. It is based on the theoretical probability.

期望频数告诉我们，如果将一个实验重复很多次，我们预期某个事件会发生多少次。它基于理论概率计算。

Expected frequency = Probability × Number of trials

For example, if you spin a fair spinner with the letters A, B, B, C, D a total of 200 times, the probability of landing on B is 2/5 = 0.4, so the expected number of times the spinner lands on B is 0.4 × 200 = 80.

例如，某个公平转盘标有字母 A、B、B、C、D，转动 200 次，转到 B 的概率是 2/5 = 0.4，因此转到 B 的期望次数是 0.4 × 200 = 80。

This does not mean the event will happen exactly 80 times—it is only what we predict based on probability. The actual result may differ, and you can be asked to comment on whether the observed frequency matches the expectation.

这并不意味着该事件一定会恰好发生 80 次——它只是我们根据概率做出的预测。实际结果可能不同，你也可能需要评论观察频数是否与期望相符合。

9. Independent Events | 独立事件

Two events are independent if the outcome of one does not affect the outcome of the other. Tossing a coin and rolling a die are independent: the coin result does not change the die result. For independent events, you can multiply their probabilities to find the chance that both occur.

如果两个事件的结果互不影响，它们就是独立事件。抛硬币和掷骰子是独立的：硬币的结果不会改变骰子的结果。对于独立事件，你可以将它们的概率相乘来求出两者同时发生的概率。

P(A and B) = P(A) × P(B)

Example: The probability of getting heads on a coin and rolling a 3 on a die is P(heads) × P(3) = ½ × ⅙ = 1/12. You can often list all outcomes to verify this (2 × 6 = 12 total outcomes, 1 is heads‑and‑3).

例子：抛硬币得到正面并且掷骰子得到 3 点的概率是 P(正面) × P(3) = ½ × ⅙ = 1/12。你通常可以通过列出所有结果来验证这一点（总共有 2 × 6 = 12 种结果，其中 1 种是正面且 3 点）。

In KS3, questions might also involve tree diagrams to represent sequences of independent events. A simple two‑branch tree for a coin toss and a spinner is a common sight on page 238.

在 KS3 阶段，题目也可能会用树状图来表示一连串独立事件。在第 238 页经常能看到表示抛硬币和转盘的简单两分支树状图。

10. Key Terms and Common Pitfalls | 关键术语与常见误区

Here is a summary of important vocabulary and mistakes to avoid when working on probability exercises:

以下是做概率练习时需要掌握的重要词汇以及应避免的常见错误：

Outcome – a possible result of a trial (e.g., getting a 3 on a die)
Event – a set of one or more outcomes (e.g., rolling an even number)
Biased – when outcomes are not equally likely; probabilities are not equal
Fair – all outcomes have the same probability

结果 (Outcome) — 一次试验的可能结果（如掷骰子得到 3 点）
事件 (Event) — 一个或多个结果的集合（如掷出偶数）
有偏 (Biased) — 指结果不是等可能的，概率不相等
公平 (Fair) — 所有结果具有相同的概率

Common pitfalls: forgetting that probabilities must be between 0 and 1, adding probabilities for mutually exclusive events incorrectly (e.g., thinking P(A and B) = P(A) + P(B)), or confusing experimental and theoretical probability. Always read the question carefully to check whether events are independent or mutually exclusive.

常见误区：忘记概率必须介于 0 到 1 之间；错误地计算互斥事件（如以为 P(A 且 B) = P(A) + P(B)）；混淆实验概率和理论概率。务必仔细读题，判断事件是独立还是互斥。

Probability problems at this level often combine several of the ideas in this article. If you see a page reference like p238, it might be a mixed exercise. Take your time, show all your working, and check your answers with the total‑probability‑is‑1 rule when appropriate.

这个阶段的概率题目常常会综合运用本文中的多个知识点。如果你看到的页码引用是 p238，它可能是一份综合练习。请稳步进行，写出完整过程，并在适当的时候用“总概率为 1”来检验答案。

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Probability | 概率入门