Probability Basics: Event Types and Probability Calculations | 概率基础：事件类型与概率计算

📚 Probability Basics: Event Types and Probability Calculations | 概率基础：事件类型与概率计算

Probability is the mathematical language of uncertainty. It measures how likely an event is to occur, and it underpins everything from gambling odds to weather forecasting and quality control in manufacturing. In the IGCSE Edexcel Mathematics syllabus, probability is a core topic that tests both your conceptual understanding and your computational fluency.

概率是描述不确定性的数学语言。它衡量某一事件发生的可能性大小，从博彩赔率到天气预报，再到制造业中的质量控制，处处都有概率的应用。在 Edexcel IGCSE 数学大纲中，概率是一个核心专题，既考查你的概念理解，又考查你的计算能力。

1. What Is Probability? | 什么是概率？

Probability is a number between 0 and 1 that describes the likelihood of an event happening. A probability of 0 means the event is impossible; a probability of 1 means the event is certain. All other probabilities lie somewhere in between.

概率是介于 0 和 1 之间的一个数，用于描述事件发生的可能性大小。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。其余所有概率都介于这两者之间。

0 ≤ P(A) ≤ 1

For example, the probability of rolling a 7 on a standard six-sided die is 0 (impossible), while the probability of rolling a number less than 7 is 1 (certain).

例如，掷一枚标准六面骰子得到 7 点的概率为 0（不可能），而掷出小于 7 的数的概率为 1（必然）。

2. Sample Space and Events | 样本空间与事件

The sample space, denoted S, is the set of all possible outcomes of an experiment. An event is any subset of the sample space — that is, a collection of outcomes that we are interested in.

样本空间（记作 S）是某个试验所有可能结果组成的集合。事件是样本空间的任意子集，即我们感兴趣的一组结果。

Example: When rolling a die, the sample space is S = {1, 2, 3, 4, 5, 6}. The event ‘rolling an even number’ is E = {2, 4, 6}.

例如：掷一枚骰子时，样本空间为 S = {1, 2, 3, 4, 5, 6}。事件“掷出偶数”为 E = {2, 4, 6}。

When every outcome in the sample space is equally likely, the probability of an event E is:

当样本空间中每个结果等可能发生时，事件 E 的概率为：

P(E) = n(E) / n(S)

where n(E) is the number of outcomes in the event and n(S) is the total number of outcomes in the sample space.

其中 n(E) 是事件 E 中包含的结果个数，n(S) 是样本空间中的结果总数。

3. Types of Events | 事件的类型

In IGCSE probability, you must be able to identify and distinguish the following types of events:

在 IGCSE 概率中，你必须能够识别并区分以下类型的事件：

Simple Event: A single outcome, e.g., rolling a 5 on a die.
简单事件：单一结果，例如掷骰子得到 5 点。
Compound Event: An event that combines two or more outcomes, e.g., rolling an even number.
复合事件：由两个或更多结果组成的事件，例如掷出偶数。
Complementary Events: Two events A and A’ where A’ contains all outcomes not in A.
互补事件：事件 A 与 A’，其中 A’ 包含所有不在 A 中的结果。
Mutually Exclusive Events: Events that cannot happen at the same time, e.g., rolling a 2 and rolling a 5 in a single throw.
互斥事件：不能同时发生的事件，例如一次掷骰子中掷出 2 点又掷出 5 点。
Independent Events: Events where the occurrence of one does not affect the probability of the other.
独立事件：一个事件的发生不影响另一个事件概率的事件。
Dependent Events: Events where the occurrence of one affects the probability of the other.
相依事件：一个事件的发生会影响另一个事件概率的事件。

4. The Probability Scale | 概率标度

The probability scale runs from 0 to 1. In exams, you may be asked to place events on a probability scale, or to convert between fractions, decimals and percentages.

概率标度从 0 到 1。在考试中，你可能会被要求在概率标度上标示事件，或在分数、小数和百分数之间进行转换。

0	0.25	0.5	0.75	1
Impossible 不可能	Unlikely 不太可能	Even chance 五五开	Likely 很可能	Certain 必然

Remember: 0% = 0, 50% = 0.5 = ½, and 100% = 1. Be comfortable converting between these representations.

记住：0% = 0，50% = 0.5 = ½，100% = 1。要熟练掌握这些表示形式之间的转换。

5. The Basic Probability Formula | 基本概率公式

For equally likely outcomes, the probability of an event is calculated as:

对于等可能结果，事件概率的计算公式为：

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

Worked example: A bag contains 3 red balls, 2 blue balls and 5 green balls (10 in total). What is the probability of picking a blue ball at random?

例题：一个袋子里有 3 个红球、2 个蓝球和 5 个绿球（共 10 个）。随机取出一个球，取到蓝球的概率是多少？

P(blue) = 2/10 = 1/5 = 0.2 = 20%

P(蓝球) = 2/10 = 1/5 = 0.2 = 20%

Always simplify your fractions where possible, and show your working clearly — you often get method marks.

尽量化简分数，并清晰地写出解题步骤——步骤分通常能拿到手。

6. The Complement Rule | 互补事件法则

For any event A, its complement A’ is the event that A does not happen. Since either A or A’ must occur:

对于任意事件 A，其互补事件 A’ 表示“A 不发生”这一事件。由于 A 与 A’ 必有其一发生：

P(A) + P(A’) = 1

P(A’) = 1 – P(A)

Example: If the probability that it rains tomorrow is 0.35, then the probability that it does not rain is 1 – 0.35 = 0.65.

例：如果明天下雨的概率为 0.35，那么不下雨的概率为 1 – 0.35 = 0.65。

Tip: The complement rule is extremely useful when ‘at least one’ appears in a question, because ‘at least one

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Probability Basics: Event Types and Probability Calculations | 概率基础：事件类型与概率计算