Probability Basics | 概率基础

📚 Probability Basics | 概率基础

In the Cambridge KS3 Mathematics course, probability is a key topic that allows you to quantify uncertainty. Page 112 of the coursebook introduces the fundamental rules and notations you need to calculate and interpret the likelihood of events. This bilingual revision note explains each concept step by step, with paired English and Chinese paragraphs to reinforce your understanding and prepare you for Checkpoint assessments.

在剑桥 KS3 数学课程中，概率是一个重要主题，让你能够把不确定性的程度量化。教材第 112 页介绍了计算和解释事件可能性所需的基本规则和符号。这篇双语复习笔记逐步解释每个概念，通过配对的英文和中文段落加深理解，帮助你为 Checkpoint 测评做好准备。

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1. What is Probability? | 什么是概率？

Probability is a branch of mathematics that deals with the chance that a particular event will happen. It is expressed as a number between 0 and 1, where 0 means the event is impossible and 1 means it is certain to happen.

概率是数学的一个分支，研究某个特定事件发生的可能性。它用 0 到 1 之间的一个数来表示，其中 0 表示事件不可能发生，1 表示事件一定发生。

Everyday language uses words like ‘likely’, ‘unlikely’, ‘certain’, or ‘fifty-fifty’, but probability assigns numerical values to these words. This precision helps us compare risks and make informed decisions in games, science, and real life.

日常语言使用“很可能”、“不太可能”、“一定”或“五五开”这样的词语，而概率则给这些词语赋予数值。这种精确性帮助我们在游戏、科学和现实生活中比较风险并做出明智的决策。

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2. The Probability Scale | 概率标度

The probability scale is a line from 0 to 1. You can mark events along this line to show how likely they are. An impossible event, such as rolling a 7 on a standard six‑sided die, sits at 0. A certain event, like the sun rising tomorrow, sits at 1.

概率标度是一条从 0 到 1 的线。你可以把事件标记在这条线上，以显示它们的可能性大小。不可能事件，比如在标准六面骰子上掷出 7，位于 0。必然事件，比如太阳明天升起，位于 1。

The midpoint, 0.5 or 1/2, represents an even chance – something that is equally likely to happen or not happen. Events closer to 1 are more likely; events closer to 0 are less likely.

中点 0.5 或 1/2 表示均等机会 —— 发生和不发生的可能性相等。越接近 1 的事件越可能发生；越接近 0 的事件越不可能发生。

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3. Sample Space and Outcomes | 样本空间与结果

The sample space is the set of all possible outcomes of an experiment. For a fair coin toss, the sample space is {Heads, Tails}. For a six‑sided die, it is {1, 2, 3, 4, 5, 6}. Listing the sample space systematically helps you count outcomes correctly.

样本空间是实验所有可能结果的集合。抛一次公平硬币，样本空间为 {正面, 反面}。对于六面骰子，样本空间为 {1, 2, 3, 4, 5, 6}。系统列出样本空间有助于正确计数结果。

An outcome is a single result from the sample space. If you flip a coin and it lands on Heads, that is one outcome. When all outcomes are equally likely, you can calculate probabilities using a simple ratio.

一个结果就是样本空间中的一个单项。如果你掷硬币落地为正面，那是一个结果。当所有结果等可能时，你可以用一个简单比值来计算概率。

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4. Calculating Probability of a Single Event | 计算单一事件的概率

The probability of an event E is found using the formula:

事件 E 的概率可以用以下公式求得：

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

P(E) = 有利结果的数量 ÷ 总等可能结果的数量

For example, the probability of rolling a 4 on a fair die is 1/6, because there is one favourable outcome and six possible outcomes. Always check that your outcomes are equally likely before using this formula.

例如，掷一粒公平骰子得到 4 的概率是 1/6，因为有一个有利结果和六种可能结果。在使用此公式之前，务必确认你的结果是等可能的。

Probabilities can be simplified just like fractions. The probability 2/8 should be simplified to 1/4. Write your final answer in its simplest form unless the question asks for a decimal or percentage.

概率可以像分数一样进行化简。概率 2/8 应该化简为 1/4。除非题目要求用小数或百分比，否则请把最终答案写成最简形式。

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5. Mutually Exclusive Events | 互斥事件

Two events are mutually exclusive if they cannot happen at the same time. On a single die roll, getting a 2 and getting a 5 are mutually exclusive – you cannot roll both numbers together.

如果两个事件不能同时发生，它们就是互斥的。单次掷骰子时，得到 2 和得到 5 就是互斥的 —— 你不可能同时掷出两个数字。

When events are mutually exclusive, the probability that either one or the other occurs is the sum of their individual probabilities: P(A or B) = P(A) + P(B). This is known as the addition rule for mutually exclusive events.

当事件互斥时，任何一个事件发生的概率等于各自概率之和：P(A 或 B) = P(A) + P(B)。这称为互斥事件的加法法则。

Always verify that events have no overlap before adding their probabilities. If they can occur together, the addition rule must be adjusted to avoid double counting.

在加总概率之前，一定要验证事件没有重叠。如果它们能同时发生，就必须调整加法法则以避免重复计数。

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6. Complementary Events | 互补事件

The complement of an event A is the event ‘A does not happen’, written as P(not A) or P(A’). Since an event either happens or does not happen, the sum of their probabilities is always 1: P(A) + P(not A) = 1.

事件 A 的互补事件是“A 不发生”，写作 P(非 A) 或 P(A′)。由于一个事件要么发生要么不发生，它们的概率之和总是 1：P(A) + P(非 A) = 1。

This relationship is very useful for finding probabilities of complicated events. If calculating P(A) directly is difficult, you can find P(not A) first and subtract from 1.

这种关系在求复杂事件的概率时非常有用。如果直接计算 P(A) 有困难，你可以先求 P(非 A)，再用 1 减去它。

Example: The probability that it will rain tomorrow is 0.3, so the probability that it will not rain is 1 − 0.3 = 0.7. Complement events are always mutually exclusive and exhaustive.

示例：明天下雨的概率是 0.3，那么明天不下雨的概率就是 1 − 0.3 = 0.7。互补事件总是互斥且穷尽的。

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7. Experimental Probability | 实验概率

Experimental probability is based on actual trials or observations. It is calculated as the number of times an event occurs divided by the total number of trials. For instance, if you toss a coin 100 times and get 47 heads, the experimental probability of heads is 47/100 = 0.47.

实验概率基于实际试验或观察。它通过事件发生的次数除以试验总次数来计算。例如，如果你掷硬币 100 次得到 47 次正面，那么正面的实验概率就是 47/100 = 0.47。

Experimental probability may differ from theoretical probability, especially with small sample sizes. As the number of trials increases, the experimental probability usually gets closer to the theoretical value – this is known as the law of large numbers.

实验概率可能与理论概率不同，特别是在小样本量的情况下。随着试验次数的增加，实验概率通常会接近理论值 —— 这被称为大数定律。

In KS3, you will often be asked to compare experimental results with theoretical expectations and discuss why results might vary.

在 KS3 阶段，你经常会被要求比较实验结果与理论期望值，并讨论结果可能存在差异的原因。

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8. Relative Frequency | 相对频数

Relative frequency is another name for experimental probability. It is found using the same method: Relative frequency = Frequency of the event ÷ Total number of trials. It represents the proportion of trials in which the event occurred.

相对频数是实验概率的另一个名称，求法相同：相对频数 = 事件发生的频数 ÷ 试验总次数。它表示该事件在试验中发生的比例。

For example, a spinner with three colours is spun 50 times. If it lands on red 18 times, the relative frequency of red is 18/50 = 9/25 or 0.36.

例如，一个有三种颜色的转盘被转动 50 次。如果它停在红色上 18 次，那么红色的相对频数就是 18/50 = 9/25 或 0.36。

Remember that relative frequency can only be determined after conducting an experiment. It is an estimate of the true probability and becomes more reliable as more trials are performed.

请记住，相对频数只能在实验完成后确定。它是真实概率的估计值，并且随着试验次数的增加会变得更加可靠。

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9. Expressing Probabilities | 概率的表示形式

Probability can be written as a fraction, a decimal, or a percentage. You should be comfortable converting between these three forms. For instance, a probability of 1/4 is equal to 0.25 and 25%. The table below shows common equivalents:

概率可以用分数、小数或百分比表示。你应该能熟练地在这三种形式之间转换。例如，概率 1/4 等于 0.25 和 25%。下表列出了一些常见的等价形式：

Fraction (分数)	Decimal (小数)	Percentage (百分比)
1/2	0.5	50%
1/3	0.333… (or 0.3̅)	33.3̅%
2/5	0.4	40%
3/4	0.75	75%
1/1	1	100%

When a question asks for an answer in a specific form, make sure you convert your answer accordingly. Leaving a probability as an unsimplified fraction may lose marks.

当题目要求用特定形式作答时，确保你进行了相应的转换。把概率留作未化简的分数可能会丢分。

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10. Common Errors and How to Avoid Them | 常见错误及如何避免

One common mistake is forgetting that probabilities must be between 0 and 1 inclusive. A probability of 1.2 or −0.3 makes no sense. Always check that your final answer lies in this range.

一个常见错误是忘记概率必须在 0 到 1 之间，含 0 和 1。概率为 1.2 或 −0.3 毫无意义。始终检查你的最终答案是否在此范围内。

Another error is adding probabilities of events that are not mutually exclusive without adjusting for the overlap. Before using the addition rule, list the sample space to confirm there is no common outcome.

另一个错误是将非互斥事件的概率相加而没有对重叠部分进行调整。使用加法法则之前，列出样本空间以确认没有共同结果。

Finally, avoid confusing experimental probability with theoretical probability. Experimental probability comes from data; theoretical probability is what you expect from equally likely outcomes. Use the correct term in explanations.

最后，要避免混淆实验概率和理论概率。实验概率来自数据；理论概率是基于等可能结果的期望值。在解释时使用正确的术语。

Practice drawing sample space diagrams for combined events. A systematic list or a table helps prevent miscounting outcomes, a key skill for more advanced probability work.

多多练习为组合事件画样本空间图。系统性列表或表格有助于防止结果数错，这是处理更复杂概率问题的关键技能。

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11. Worked Example | 详细解析示例

Question: A bag contains 3 red marbles, 2 blue marbles, and 5 green marbles. One marble is picked at random. Find the probability that it is (a) red, (b) not green, (c) red or blue. Give your answers as fractions in simplest form.

题目：一个袋子装有 3 颗红色弹珠、2 颗蓝色弹珠和 5 颗绿色弹珠。随机抽取一颗。求它是 (a) 红色，(b) 非绿色，(c) 红色或蓝色的概率。用最简分数给出答案。

Solution:

Total marbles = 3 + 2 + 5 = 10. All outcomes are equally likely.

总弹珠数 = 3 + 2 + 5 = 10。所有结果等可能。

• Probability of red = 3 ÷ 10 = 3/10.

  红色概率 = 3 ÷ 10 = 3/10。

• Probability of not green = probability of red or blue = (3+2) ÷ 10 = 5/10 = 1/2. Alternatively, P(green) = 5/10 = 1/2, so P(not green) = 1 − 1/2 = 1/2.

  非绿色概率 = 红色或蓝色的概率 = (3+2) ÷ 10 = 5/10 = 1/2。或者，P(绿色) = 5/10 = 1/2，所以 P(非绿色) = 1 − 1/2 = 1/2。

• Probability of red or blue: these events are mutually exclusive, so add their probabilities: 3/10 + 2/10 = 5/10 = 1/2.

  红色或蓝色的概率：这些事件互斥，所以概率相加：3/10 + 2/10 = 5/10 = 1/2。

Always re-read the question: part (c) asks for the probability of red or blue, which is the same as ‘not green’ in this context, confirming the consistency of our answers.

答题后务必重读题目：第 (c) 问要求红色或蓝色的概率，这与本题中“非绿色”一致，从而验证了我们答案的连贯性。

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12. Key Takeaways | 核心要点

Probability is a measure of how likely something is to happen and always falls between 0 and 1. You now have tools to find probabilities for single events, use the addition rule for mutually exclusive events, and work with complementary events.

概率是衡量某事发生可能性的指标，总是介于 0 和 1 之间。你现在拥有求单个事件概率、对互斥事件使用加法法则以及处理互补事件的工具。

The concepts of sample space, experimental probability, and relative frequency form the foundation for all future work in statistics and data handling. Regular practice with converting between fractions, decimals, and percentages will build your confidence and speed.

样本空间、实验概率和相对频数的概念是所有统计与数据处理后续内容的基础。经常练习分数、小数和百分比的转换将增强你的信心并提高速度。

Remember that probability appears not only in Checkpoint tests but also in everyday decision making. Approach each problem methodically: list possible outcomes, identify favourable ones, and apply the correct rule.

请记住，概率不仅出现在 Checkpoint 考试中，也出现在日常决策中。有条理地处理每一个问题：列出所有可能的结果，找出有利结果，再应用正确的法则。

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