Conditional Probabilities in Venn Diagrams | 维恩图中的条件概率

📚 Conditional Probabilities in Venn Diagrams | 维恩图中的条件概率

Conditional probability is one of the most examined ideas in Edexcel A-Level Mathematics. When a question says ‘given that’ or ‘if you know’, you are being asked to restrict the sample space, and a Venn diagram makes this restriction visual and easier to handle.

条件概率是 Edexcel A-Level 数学中最常考的概念之一。当题目出现 given that 或 if you know 时，实际上是要你缩小样本空间，而维恩图正好能把这种限制直观地表现出来。

1. Conditional Probability: The Core Idea | 条件概率：核心思想

The probability of A given B, written P(A | B), measures how likely A is when we already know that B has occurred.

已知 B 发生时 A 的条件概率，记作 P(A | B)，衡量在已经知道 B 发生的前提下 A 发生的可能性。

You are no longer considering the whole sample space; you are only looking inside the set B. The total probability inside B becomes your new denominator.

此时你不再考虑整个样本空间，而是只看集合 B 的内部。B 内部的总概率成为新的分母。

P(A | B) = P(A ∩ B) / P(B), P(B) > 0

The denominator must be positive because we cannot condition on an impossible event. If P(B) = 0, the conditional probability is undefined.

分母必须为正，因为不能以不可能事件作为条件。如果 P(B) = 0，条件概率没有定义。

2. Venn Diagrams as Probability Maps | 维恩图作为概率地图

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