📚 Combined Events in Probability | 概率中的组合事件
In Edexcel A-Level Mathematics, combined events bring together the core ideas of probability, set notation, tree diagrams and conditional reasoning. This article covers the key rules and exam techniques you need for AS and A-Level questions.
在 Edexcel A-Level 数学中，组合事件将概率、集合符号、树状图和条件推理的核心思想结合在一起。本文涵盖 AS 和 A-Level 考试中你需要掌握的关键规则与解题技巧。
1. Probability Foundations | 概率基础
ility of 0 means the event is impossible, while a probability of 1 means the event is certain.
在 Edexcel A-Level 数学中，事件 A 的概率记作 P(A)，始终满足 0 ≤ P(A) ≤ 1。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。
The sample space S is the set of all possible outcomes. For combined events, you often need to count outcomes using lists, two-way tables, Venn diagrams or tree diagrams before applying probability rules.
样本空间 S 是所有可能结果的集合。对于组合事件，你通常需要先使用列表、双向表、维恩图或树状图来计数结果，再应用概率法则。
Probability can be understood as a long-run relative frequency, but at A-Level you mainly work with theoretical probabilities based on equally likely outcomes, such as rolling fair dice or drawing cards from a well-shuffled pack.
概率可以理解为长期相对频率，但在 A-Level 阶段，你主要使用基于等可能结果的理论概率，例如掷公平骰子或从洗好的牌中抽牌。
0 ≤ P(A) ≤ 1
2. Set Notation and Venn Diagrams | 集合符号与维恩图
Combined events are often described using set notation: A ∩ B is the intersection, meaning both A and B occur; A ∪ B is the union, meaning A or B or both occur; A′ is the complement, meaning A does not occur.
组合事件通常用集合符号描述：A ∩ B 表示交集，即 A 和 B 同时发生；A ∪ B 表示并集，即 A 或 B 或两者都发生；A′ 表示补集，即 A 不发生。
Venn diagrams are useful for visualising overlaps and gaps. In A-Level exam questions, you often need to place probabilities or frequencies in regions of a Venn diagram and then calculate missing values using the fact that the total probability sums to 1.
维恩图有助于直观展示重叠与空白区域。在 A-Level 考试题中，你经常需要将概率或频数填入维恩图的区域，然后利用总概率之和为 1 来计算缺失值。
The region inside the rectangle represents the sample space S. Each circle represents an event, and the overlap shows outcomes that belong to both events. The area outside the circles represents the complement of the union, written (A ∪ B)′.
矩形内部区域代表样本空间 S。每个圆代表一个事件，重叠部分表示同时属于两个事件的结果。圆外区域表示并集的补集，写作 (A ∪ B)′。
3. Mutually Exclusive Events | 互斥事件
re mutually exclusive if they cannot happen at the same time. In set notation this means A ∩ B = ∅, and in a Venn diagram the two circles do not overlap.
如果两个事件 A 和 B 不能同时发生，则称它们互斥。用集合符号表示即 A ∩ B = ∅，在维恩图中两个圆不相交。
For example, when a fair die is rolled once, the events 'roll an odd number' and 'roll an even number' are mutually exclusive because no outcome belongs to both sets. However, 'roll a number less than 4' and 'roll an even number' are not mutually exclusive because the number 2 satisfies both conditions.
例如，掷一个公平骰子一次，事件 '掷出奇数' 和 '掷出偶数' 是互斥的，因为没有任何结果同时属于两个集合。然而，'掷出小于 4 的数' 和 '掷出偶数' 不是互斥的，因为数字 2 同时满足两个条件。
For mutually exclusive events, the addition rule simplifies to P(A ∪ B) = P(A) + P(B). This is because there is no intersection to subtract.
对于互斥事件，加法法则简化为 P(A ∪ B) = P(A) + P(B)。这是因为没有交集需要减去。
P(A ∪ B) = P(A) + P(B) when A ∩ B = ∅
4. Independent Events | 独立事件
if the occurrence of one does not affect the probability of the other. Mathematically, P(A ∩ B) = P(A) × P(B).
如果两个事件 A 和
Published by TutorHao | A-Level Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图