Venn Diagrams & Conditional Probability | 韦恩图与条件概率

📚 Venn Diagrams & Conditional Probability | 韦恩图与条件概率

In Edexcel IGCSE Mathematics, the most powerful tools for solving probability problems are Venn diagrams and probability tree diagrams. Conditional probability — the probability that event A happens given that event B has already happened — is a core exam topic that appears regularly on both Paper 1 and Paper 2. This revision article explains the set notation you need, the key formulas, and typical exam questions step by step.

在Edexcel IGCSE数学中，解决概率问题最强大的两个工具是韦恩图（文氏图）和概率树形图。条件概率——即在事件B已经发生的条件下事件A发生的概率——是核心考点，在Paper 1和Paper 2中经常出现。本文将系统讲解所需的集合符号、关键公式以及典型考试题的解题步骤。

1. Sets and Set Notation | 集合与集合符号

A set is a collection of objects, called elements. The universal set, written as ξ (xi), contains every element we are considering. We use capital letters to name sets, for example A and B.

集合是一组对象的总体，其中的对象称为元素。全集用符号 ξ 表示，它包含我们所考虑的所有元素。我们用大写字母来命名集合，例如A和B。

The following symbols are essential in exam questions:

以下符号在考试题目中至关重要：

A ∪ B — the union of A and B: all elements in A or B (or both)
A ∩ B — the intersection of A and B: elements in both A and B
A′ (A complement) — elements not in A
n(A) — the number of elements in set A
ξ — the universal set
∅ — the empty set

A ∪ B — A与B的并集：属于A或属于B（或同时属于两者）的所有元素
A ∩ B — A与B的交集：既属于A又属于B的元素
A′（A的补集）— 不属于A的元素
n(A) — 集合A中元素的个数
ξ — 全集
∅ — 空集

You must be able to translate between these symbols, words, and the shaded regions of a diagram. A common exam instruction is “write down the set shown by the shaded region”, so practise describing each region of a two-circle Venn diagram.

你必须能够在符号、文字和图中阴影区域之间进行转换。考试中常见的指令是”写出阴影部分所表示的集合”，因此要练习描述双圆韦恩图中的每一个区域。

2. Reading Venn Diagrams | 读懂韦恩图

A Venn diagram shows sets as circles inside a rectangle. The rectangle represents the universal set ξ, and every element must be placed in exactly one region of the diagram.

韦恩图用矩形内的圆形表示集合。矩形代表全集 ξ，每一个元素都必须放在图中唯一的一个区域里。

For two sets A and B, the diagram is divided into four main regions:

对于两个集合A和B，韦恩图分为四个主要区域：

A only (inside A, outside B)
B only (inside B, outside A)
both A and B (the intersection, A ∩ B)
neither A nor B (outside both circles, inside the rectangle)

仅属于A（在A内、B外）
仅属于B（在B内、A外）
同时属于A和B（交集 A ∩ B）
既不属于A也不属于B（两个圆之外、矩形之内）

When numbers are written inside the regions, always read them carefully. A number written inside region A ∩ B belongs to both sets, so do not count it twice.

当区域中写有数字时，一定要仔细阅读。写在A ∩ B区域内的数同时属于两个集合，因此不要重复计算。

3. The Addition Rule | 加法法则

To find the number of elements in A ∪ B, we add n(A) and n(B), but the intersection has been counted twice. We therefore subtract it once:

要求A ∪ B中元素的个数，我们把n(A)与n(B)相加，但交集被重复计算了一次，因此要减去一次：

n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

Dividing every term by n(ξ) gives the probability version of the addition rule:

将每一项都除以n(ξ)，就得到概率形式的加法法则：

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

If A and B are mutually exclusive, they cannot happen at the same time, so P(A ∩ B) = 0 and the rule simplifies to P(A ∪ B) = P(A) + P(B).

如果A与B互斥，则它们不可能同时发生，因此P(A ∩ B) = 0，公式简化为P(A ∪ B) = P(A) + P(B)。

4. The Conditional Probability Formula | 条件概率公式

Conditional probability is written as P(A|B), read as “the probability of A given B”. It means the probability that A occurs, knowing that B has already occurred.

条件概率记作P(A|B)，读作”在B发生的条件下A发生的概率”。它表示在已知B已经发生的情况下，A发生的概率。

The formula is given on the Edexcel IGCSE formula sheet and is the key to nearly every conditional probability question:

该公式出现在Edexcel IGCSE的公式表中，是解决几乎所有条件概率问题的关键：

P(A|B) = P(A ∩ B) ÷ P(B)

An equivalent statement is P(A ∩ B) = P(A|B) × P(B). This multiplication version is exactly what we use when multiplying along the branches of a probability tree.

等价形式为P(A ∩ B) = P(A|B) × P(B)。这个乘法形式正是我们在概率树中沿分支相乘所使用的公式。

5. Conditional Probability from a Venn Diagram | 从韦恩图求条件概率

When a Venn diagram contains numbers, you can find conditional probabilities by restricting your attention to the “given” set.

当韦恩图中含有数字时，你可以通过把注意力限制在”已知条件”的集合上，来求出条件概率。

For example, P(A|B) means “A happens given B”. Because we know B has happened, the sample space is now only the set B, not ξ. The numerator is the probability of being in A ∩ B, and the denominator is P(B):

例如，P(A|B)表示”在B发生的条件下A发生”。因为已知B发生，样本空间现在是集合B，而不是ξ。分子是落在A ∩ B中的概率，分母是P(B)：

P(A|B) = n(A ∩ B) ÷ n(B)

This is much faster than converting to fractions of ξ. Simply read the two relevant numbers from the diagram and divide.

这种方法比先转换成ξ的分式要快得多。只需从图中读出两个相关数字并相除即可。

6. Probability Tree Diagrams | 概率树形图

A probability tree shows every possible combination of events. Each branch segment has a probability, and the probabilities on branches from the same point must add to 1.

概率树形图展示所有可能的事件组合。每一段分支都有一个概率，从同一点出发的各分支概率之和必须等于1。

Two rules are essential:

两条规则至关重要：

Multiply along branches to find the probability of a sequence of events.
Add the results from separate branches when the question asks for “either … or …”.

求多个事件依次发生的概率时，沿分支相乘。
当题目问”或”时，将不同分支的结果相加。

For questions without replacement, the denominator changes after the first draw. For example, if a red ball is removed from a bag, both the number of red balls and the total number of balls decrease. This makes the second set of branches conditional probabilities.

对于不放回抽取的题目，第一次抽取后分母会改变。例如，从袋中取出一个红球后，红球数和总数都会减少。这使得第二组分支上的概率成为条件概率。

7. Worked Example: Venn Diagram | 例题精讲：韦恩图

A school has 120 students. 70 students play football (F), 50 students play tennis (T), and 30 students play both. A student is chosen at random.

某学校有120名学生。70名学生踢足球（F），50名学生打网球（T），其中30名学生两项都参加。现随机选择一名学生。

Step 1: Fill the Venn diagram regions.

第一步：填写韦恩图的各个区域。

Only football: 70 − 30 = 40
Only tennis: 50 − 30 = 20
Both: 30
Neither: 120 − 40 − 20 − 30 = 30

只踢足球：70 − 30 = 40
只打网球：50 − 30 = 20
两项都参加：30
两项都不参加：120 − 40 − 20 − 30 = 30

Step 2: Find the probability that the student plays neither sport.

第二步：求该学生两项运动都不参加的概率。

P(neither) = 30 ÷ 120 = 1 ÷ 4

Step 3: Find P(F|T), the probability that the student plays football, given that they play tennis.

第三步：求P(F|T)，即在已知该学生打网球的条件下，他踢足球的概率。

Given tennis means the sample space is T only. Within T, the number who also play football is 30, and the total in T is 50:

“已知打网球”意味着样本空间仅为T。在T内部，同时踢足球的人数为30，而T的总人数为50：

P(F|T) = 30 ÷ 50 = 3 ÷ 5

8. Worked Example: Tree Diagram | 例题精

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Venn Diagrams & Conditional Probability | 韦恩图与条件概率