Venn Diagrams in Probability | 韦恩图在概率问题中的应用

📚 Venn Diagrams in Probability | 韦恩图在概率问题中的应用

The Venn diagram is one of the most useful visual tools in A-Level probability. It turns abstract set operations into a picture, so students can see exactly which outcomes belong to which events and how overlapping regions affect a probability.

韦恩图是 A-Level 概率部分最具实用价值的可视化工具之一。它将抽象的集合运算转化为直观图形，让我们一眼看出哪些结果属于什么事件，以及区域重叠会如何影响概率。

1. The Sample Space and Events | 样本空间与事件

A probability model begins with a sample space S, containing every possible outcome of a random experiment. An event is a subset of S; we represent S as a rectangle and each event as a circle inside it.

概率模型的起点是样本空间 S，它包含一次随机试验的所有可能结果。事件是 S 的子集；我们用矩形表示 S，用矩形内部的圆表示各个事件。

For a single roll of a fair die, S = {1, 2, 3, 4, 5, 6}. The event A = “the score is even” = {2, 4, 6} can be drawn as a circle inside the rectangle. The probability of A is the area of that circle divided by the total area of the rectangle.

例如，掷一颗均匀骰子时，S = {1, 2, 3, 4, 5, 6}。事件 A = “点数为偶数” = {2, 4, 6} 可以画成样本空间矩形内的一个圆。A 的概率就是圆面积占整个矩形面积的比例。

In more advanced questions we normally use numbers or probabilities written in each region. The rectangle then represents the total probability 1, and every region inside it represents a part of that total.

在高阶题目中，我们通常在每个区域中写出频数或概率。此时矩形代表总概率 1，矩形内的每个区域都代表总概率的一部分。

2. Set Notation: Union, Intersection and Complement | 集合记号：并、交与补

Three set operations are essential: A ∪ B means outcomes in A or B or both, A ∩ B means outcomes in both A and B, and A′ means outcomes not in A.

有三个基本运算必须掌握：A ∪ B 表示属于 A 或属于 B 或同时属于两者的结果；A ∩ B 表示同时属于 A 和 B 的结果；A′ 表示不属于 A 的结果。

If A ∩ B = ∅, we say A and B are mutually exclusive: they have no outcomes in common. If A ∪ B = S, then A and B are exhaustive: together they cover the whole sample space.

若 A ∩ B = ∅，称 A 与 B 互斥，即它们没有共同结果；若 A ∪ B = S，则 A 与 B 构成完备事件组，即它们共同覆盖整个样本空间。

Notation / 记号	Meaning / 含义
A ∪ B	A or B or both / 属于 A 或 B 或两者
A ∩ B	Both A and B / 同时属于 A 与 B
A′	Not A / 不属于 A 的结果
∅	Empty set / 空集
S	Sample space / 样本空间

3. The Addition Rule and Total Probability | 加法法则与总概率

From a Venn diagram, P(A ∪ B) can be found by adding P(A) and P(B) and then subtracting P(A ∩ B) once. In a diagram, the overlapping region would otherwise be counted twice.

利用韦恩图，P(A ∪ B) 等于 P(A) 加 P(B) 再减去一次 P(A ∩ B)。否则重叠区域的概率会被重复计算两次。

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

If A and B are mutually exclusive, then A ∩ B = ∅, so P(A ∩ B) = 0. The addition rule then simplifies to P(A ∪ B) = P(A) + P(B).

如果 A 与 B 互斥，则 A ∩ B = ∅，因此 P(A ∩ B) = 0。此时加法法则简化为 P(A ∪ B) = P(A) + P(B)。

Because the rectangle is the whole sample space, the sum of the probabilities of all non-overlapping regions inside it must equal 1. This also gives the complement rule P(A′) = 1 − P(A).

因为矩形代表整个样本空间，矩形内所有互不重叠区域概率之和必为 1。这同时给出补事件公式 P(A′) = 1 − P(A)。

4. Filling in a Venn Diagram and Working Backwards | 填写韦恩图并反求未知量

When a question gives the probabilities of individual events and of intersections, place every number in the correct region before doing any probability calculation.

题目若给出了各事件的概率以及交集概率，务必先将所有数填到正确的区域，再进行概率计算。

Start with the innermost intersection. For two events this is A ∩ B; for three events it is A ∩ B ∩ C.

先从最内部的交集开始填。两个事件时是 A ∩ B；三个事件时是 A ∩ B ∩ C。
Subtract these intersection values from each event total to find the “only” regions, such as A ∩ B′ or A ∩ B′ ∩ C′.

再用各事件总概率减去对应交集概率，得到“仅属于该事件”的区域，例如 A ∩ B′ 或 A ∩ B′ ∩ C′。
Finally subtract all filled regions from 1 to find the region outside all circles.

最后用 1 减去所有已填区域，得到所有圆之外的外部区域。

5. Conditional Probability in Venn Diagrams | 韦恩图中的条件概率

Conditional probability measures the probability of A given that B has already occurred. In a Venn diagram we restrict attention to the B circle, so the denominator becomes P(B), not 1.

条件概率衡量的是在 B 已发生的条件下 A 发生的概率。在韦恩图中，我们把注意力限制在 B 的圆圈之内，因此分母是 P(B)，而不是 1。

P(A | B) = P(A ∩ B) / P(B)

It is a common mistake to use the whole sample space as the denominator. When a condition is given, only the region inside the given event may be considered.

一个常见错误是仍然用整个样本空间作分母。当题目给出条件时，只能考虑给定事件内部的那部分区域。

For example, if a Venn diagram shows P(B) = 0.4 and P(A ∩ B) = 0.15, then P(A | B) = 0.15 ÷ 0.4 = 0.375.

例如，若韦恩图中 P(B) = 0.4，P(A ∩ B) = 0.15，则 P(A | B) = 0.15 ÷ 0.4 = 0.375。

6. Independence and Mutually Exclusive Events | 独立事件与互斥事件

Independent events satisfy the multiplicative condition P(A ∩ B) = P(A) × P(B). This is an algebraic condition, not a visual one: independent events may overlap in a Venn diagram.

独立事件满足乘法条件 P(A ∩ B) = P(A) × P(B)。这是代数条件，不是视觉条件：独立事件在韦恩图中完全可以有重叠区域。

Mutually exclusive events satisfy P(A ∩ B) = 0, so their circles do not overlap. Two non-empty mutually exclusive events cannot be independent in any meaningful case, because knowing B has occurred makes A impossible.

互斥事件满足 P(A ∩ B) = 0，因此两个圆不重叠。除退化情形外，两个非空互斥事件不可能独立，因为一旦已知 B 发生，A 的概率就变成 0。

Always check independence using the product rule rather than simply looking at the diagram. Two circles that overlap are not automatically independent, and two circles that do not overlap are not automatically dependent in the way many students expect.

判断独立事件时，必须使用乘法条件检查，而不是只看图形。两个圆相交并不自动代表独立；两个圆不相交也并不意味着就是“不独立”的直观解释。

7. Worked Example 1: Two Events | 例 1：两个事件

A class has 100 students. 45 take Mathematics (M), 35 take Physics (P), and 20 take both. Draw a Venn diagram and find: (a) P(M ∪ P); (b) P(M′ ∩ P); (c) P(P | M′).

某班有 100 名学生。45 人选数学（M），35 人选物理（P），20 人两门都选。画出韦恩图并求：(a) P(M ∪ P)；(b) P(M′ ∩ P)；(c) P(P | M′)。

First fill the intersection: n(M ∩ P) = 20. Mathematics only: 45 − 20 = 25. Physics only: 35 − 20 = 15. Neither subject: 100 − 25 − 20 − 15 = 40.

先填交集：n(M ∩ P) = 20。只选数学：45 − 20 = 25。只选物理：35 − 20 = 15。两门都不选：100 − 25 − 20 − 15 = 40。

(a) P(M ∪ P) = (25 + 20 + 15) ÷ 100 = 60 ÷ 100 = 0.6.

(a) P(M ∪ P) = (25 + 20 + 15) ÷ 100 = 60 ÷ 100 = 0.6。

(b) M′ ∩ P means Physics but not Mathematics, so P(M′ ∩ P) = 15 ÷ 100 = 0.15.

(b) M′ ∩ P 表示选物理但不选数学，所以 P(M′ ∩ P) = 15 ÷ 100 = 0.15。

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Venn Diagrams in Probability | 韦恩图在概率问题中的应用