📚 Conditional Probability | 条件概率
Conditional probability is one of the most important ideas in Edexcel A-Level Statistics. It measures how the chance of one event changes when we know that another event has happened. Understanding P(A | B) is essential for tree diagrams, two-way tables and independence tests, and it drives many real-world models such as disease testing, quality control and insurance risk.
条件概率是 Edexcel A-Level 统计部分最重要的概念之一。它衡量当我们已知另一个事件发生时，某个事件发生的概率如何变化。理解 P(A | B) 对树状图、双向表和独立性检验都至关重要，同时也是疾病检测、质量控制、保险定价等现实模型的核心。
1. What Is Conditional Probability? | 什么是条件概率？
Suppose A and B are two events with P(B) > 0. The conditional probability of A given B is written P(A | B) and read as "the probability of A given B". It tells us how likely A is when we already know that B has occurred.
设 A 和 B 为两个事件，且 P(B) > 0。已知 B 发生条件下 A 的概率记作 P(A | B)，读作 "在 B 发生的条件下 A 的概率"。它表示在已经知道 B 发生的情况下，A 发生的可能性有多大。
The vertical bar means "given that we already know", not division, and it does not mean that B causes A. It only updates the information available to us: knowing B changes the sample space we are working in, so the same event A can carry a completely different probability under the new information.
竖线表示 "在已知……的条件下"，它既不是除号，也不表示 B 导致了 A。它只是更新了我们掌握的信息：知道 B 发生会改变我们所考虑的样本空间，因此同一个事件 A 在新信息下可能拥有完全不同的概率。
2. The Formal Definition | 正式定义
For any two events A and B with P(B) > 0, the formal definition of conditional probability is:
对任意两个事件 A 和 B，只要 P(B) > 0，条件概率的正式定义是：
P(A | B) = P(A ∩ B) / P(B)
The numerator is the probability that A and B happen together, and the denominator is the probability of the event we are told has happened. The condition P(B) > 0 is essential: dividing by zero is undefined, so P(A | B) only exists when B is a possible event.
分子是 A 与 B 同时发生的概率，分母是已知已经发生的那个事件 B 的概率。P(B) > 0 这个条件必不可少：除以零没有意义，所以只有当 B 是可能事件时，P(A | B) 才有定义。
Swapping the roles of A and B gives the other conditional probability:
把 A 和 B 的角色互换，就得到另一个条件概率：
P(B | A) = P(A ∩ B) / P(A)
Both formulas share the same numerator P(A ∩ B), which is exactly why one conditional probability can always be converted into the other.
两个公式的分子都是 P(A ∩ B)，这正是两种条件概率可以互相换算的原因。
3. Why B Becomes the New Sample Space | 为什么 B 成为新的样本空间
Intuitively, once we know that B has happened, the only outcomes still possible are those inside B. The event B becomes the new "universe", and we then ask what fraction of B also lies inside A. In a Venn diagram this is exactly the ratio of the shaded overlap to the whole of B.
直观地说，一旦知道 B 已经发生，仍然可能的结果就只剩下 B 内部的那些结果。事件 B 成为新的 "全集"，我们接着问 B 中有多大比例同时也落在 A 里面。在韦恩图中，这正是重叠部分与 B 整体的面积之比。
A fair six-sided die is rolled. Let B be "the score is even" and A be "the score is at least 4". Then B = {2, 4, 6} and A ∩ B = {4, 6}, so P(B) = 3/6 and P(A ∩ B) = 2/6. Hence P(A | 3/6) = 2/3, which is larger than the unconditional P(A) = 3/6 = 1/2. The extra information "the score is even" has increased the chance of A.
掷一枚均匀的六面骰子。设 B 为 "点数是偶数"，A 为 "点数至少为 4"。则 B = {2, 4, 6}，A ∩ B = {4, 6}，所以 P(B) = 3/6，P(A ∩ B) = 2/6。因此 P(A | B) = (2/6) ÷ (3/6) = 2/3，大于无条件的 P(A) = 3/6 = 1/2。多出的 "点数是偶数" 这一信息提高了 A 发生的可能性。
4. The Multiplication Rule | 乘法法则
Rearranging the definition of conditional probability gives the multiplication rule, the workhorse of almost every A-Level probability question:
把条件概率的定义稍作变形，就得到乘法法则，它是几乎所有 A-Level 概率题的解题主力：
P(A ∩ B) = P(A | B) × P(B) = P(B | A) × P(A)
In words: the probability that both events happen equals the probability of the first event multiplied by the conditional probability of the second one given the first.
用文字表述：两个事件同时发生的概率，等于第一个事件的概率乘以在第一个事件发生条件下第二个事件的条件概率。
For three or more events the same idea extends into the chain rule:
对于三个或更多事件，同样的思路可以推广为链式法则：
P(A ∩ B ∩ C) = P(A) × P(B | A) × P(C | A ∩ B)
This is exactly the calculation we carry out when we multiply along the branches of a tree diagram.
这正好就是我们在树状图上沿着分支相乘时所做的计算。
5. Conditional Probability from Two-Way Tables | 用双向表求条件概率
Two-way tables are the fastest way to handle conditional probability in an exam, because the condition simply tells you which row or column total to use as the denominator.
双向表是考试中处理条件概率最快的方法，因为条件只是告诉你该用哪一行或哪一列的合计作为分母。
| Group 组别 | Passed 通过 | Failed 未通过 | Total 合计 |
|---|---|---|---|
| Boys 男生 | 42 | 18 | 60 |
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