📚 Combined Events and Conditional Probability for Edexcel A-Level Maths | Edexcel A-Level 数学组合事件与条件概率
In Edexcel A-Level Mathematics, probability questions often combine several events using tree diagrams, Venn diagrams and two-way tables. This article explains the key rules for mutually exclusive, independent and conditional events, and shows how to avoid common errors in exam responses.
在 Edexcel A-Level 数学中，概率题常常通过树形图、维恩图和双向表将多个事件结合。本文讲解互斥事件、独立事件与条件概率的关键法则，并说明如何避免考试作答中的常见错误。
1. Key Definitions and Notation | 关键定义与符号
Use P(A) for the probability of event A, P(A ∩ B) for A and B both happening, and P(A ∪ B) for A or B happening. The sample space S contains all possible outcomes, and every probability must lie between 0 and 1 inclusive.
用 P(A) 表示事件 A 的概率，P(A ∩ B) 表示 A 和 B 同时发生，P(A ∪ B) 表示 A 或 B 发生。样本空间 S 包含所有可能结果，每个概率都必须落在 0 到 1 之间并包含端点。
P(A ∩ B) means 'A and B' or the intersection of the two events.
P(A ∩ B) 表示 A 与 B 的交集，即两个事件同时发生。
P(A ∪ B) means 'A or B' or the union of the two events.
P(A ∪ B) 表示 A 与 B 的并集，即 A 或 B 发生。
P(A' ) = 1 - P(A) is the complement of A, meaning A does not happen.
P(A' ) = 1 - P(A) 是 A 的补集，表示 A 不发生。
Clear notation earns method marks in Edexcel exams, even if the final answer is wrong.
在 Edexcel 考试中，清晰的符号表示即使最终答案有误也能获得方法分。
2. Mutually Exclusive Events | 互斥事件
Two events are mutually exclusive if they cannot occur together, so P(A ∩ B) = 0. For mutually exclusive events, the addition rule simplifies to P(A ∪ B) = P(A) + P(B).
若两个事件不能同时发生，则它们互斥，因此 P(A ∩ B) = 0。对于互斥事件，加法法则简化为 P(A ∪ B) = P(A) + P(B)。
This rule is used when selecting a single card that cannot be both a King and a Queen, or when a score cannot be both odd and even.
当抽出一张牌不可能同时是 K 和 Q，或一个得分不可能同为奇数和偶数时，使用该法则。
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
注意通用加法法则中要减去交集，避免重复计算。互斥时交集为 0，所以减去 0 后就是简化形式。
3. Independent Events | 独立事件
t if the occurrence of one does not affect the probability of the other. In this case, P(A ∩ B) = P(A) × P(B).
如果事件 A 和 B 独立，即一个发生不影响另一个的概率，则 P(A ∩ B) = P(A) × P(B)。
Check that the events in a question are independent before multiplying probabilities. Independent is different from mutually exclusive: mutually exclusive events cannot both happen, but independent events can both happen and do not
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