Conditional Probability Worksheet | AQA International A Level Maths | 条件概率练习与考点解析

📚 Conditional Probability Worksheet | AQA International A Level Maths | 条件概率练习与考点解析

Conditional probability appears throughout the AQA International A Level Mathematics specification, especially in the statistics content. This worksheet-style revision guide covers the key definitions, formulas, diagram techniques, and exam-style questions you need to master conditional probability. Work through each section carefully, and use the paired Chinese notes to strengthen your understanding of notation and logic.

条件概率贯穿 AQA 国际 A Level 数学大纲，尤其在统计部分出现频率很高。这份练习式复习指南涵盖关键定义、公式、图表技巧以及考试风格题目，帮助你掌握条件概率。请认真完成每一节，并利用配对中文注释加强对符号和逻辑的理解。

1. Conditional Probability Formula | 条件概率公式

The conditional probability of event A given event B is written P(A | B). It measures how likely A is to occur when we already know that B has occurred. The formal definition is:

事件 B 已发生时事件 A 的条件概率记作 P(A | B)。它衡量在已知 B 发生的前提下 A 发生的可能性。正式定义为：

P(A | B) = P(A ∩ B) ÷ P(B), P(B) > 0

Always check that the condition has positive probability. If P(B) = 0, the conditional probability is not defined.

务必检查条件事件的概率为正。如果 P(B)=0，条件概率无定义。

For example, if P(A ∩ B) = 0.2 and P(B) = 0.5, then P(A | B) = 0.2 ÷ 0.5 = 0.4.

例如，若 P(A ∩ B)=0.2 且 P(B)=0.5，则 P(A | B)=0.2 ÷ 0.5 = 0.4。

Remember that P(A | B) is not the same as P(B | A). Reversing the condition changes the meaning and usually the numerical value.

请记住 P(A | B) 与 P(B | A) 不同。交换条件会改变含义，通常也会改变数值。

2. Tree Diagrams | 树状图

Tree diagrams are especially useful for multi-stage experiments, such as drawing counters without replacement. Label each branch with its probability, and multiply along branches to find the probability of a particular path. Add path probabilities to find the total probability of an event.

树状图特别适合多阶段试验，例如不放回抽取计数物。在每个分支上标注概率，沿分支相乘得到特定路径的概率；将各路径概率相加得到事件的总概率。

A bag contains 5 red and 3 blue counters. Two counters are drawn without replacement. The tree has first branches P(R) = 5/8 and P(B) = 3/8. If the first is red, the second branches are P(R | R) = 4/7 and P(B | R) = 3/7. If the first is blue, the second branches are P(R | B) = 5/7 and P(B | B) = 2/7.

一个袋子里有 5 个红色和 3 个蓝色计数物，不放回抽取两个。树状图第一层分支为 P(R)=5/8 和 P(B)=3/8。若第一个是红色，第二层分支为 P(R | R)=4/7 和 P(B | R)=3/7。若第一个是蓝色，第二层分支为 P(R | B)=5/7 和 P(B | B)=2/7。

To find P(second red), add the two relevant paths: (5/8 × 4/7) + (3/8 × 5/7) = 20/56 + 15/56 = 35/56 = 5/8.

求 P(第二个为红色) 时，将两条相关路径相加：(5/8 × 4/7) + (3/8 × 5/7) = 20/56 + 15/56 = 35/56 = 5/8。

This shows that without replacement, the second draw is not independent of the first, even if the unconditional probability happens to match the first draw in this symmetric example.

这表明在不放回情境下，第二次抽取与第一次不独立，即使在这个对称例子中无条件概率恰好与第一次相同。

3. Venn Diagrams and Two-Way Tables | 韦恩图与双向表

Venn diagrams and two-way tables help organise overlapping events. The intersection P(A ∩ B) is the overlap; the union P(A ∪ B) is the total probability in either A or B or both. Conditional probability focuses on the overlap relative to the given event.

韦恩图和双向表有助于整理重叠事件。交集 P(A ∩ B) 是重叠部分；并集 P(A ∪ B) 是 A 或 B 或两者发生的总概率。条件概率关注的是交集占已知事件的比例。

The table below classifies 100 students by whether they study French (F) and Spanish (S).

下表将 100 名学生按是否学习法语 (F) 和西班牙语 (S) 分类。