Edexcel A-Level Maths: Combined Events and Probability | 爱德思A-Level数学：组合事件与概率

📚 Edexcel A-Level Maths: Combined Events and Probability | 爱德思A-Level数学：组合事件与概率

In Edexcel A-Level Mathematics, combined events bring together two or more simple events so you can calculate probabilities for ‘A and B’, ‘A or B’, and ‘given that’ situations. Mastering the addition rule, multiplication rule and conditional probability gives you a coherent toolkit for solving exam questions on Statistics and Mechanics.

在爱德思A-Level数学中，组合事件把两个或多个简单事件放在一起，使你能够计算“A 与 B”“A 或 B”以及“已知 B 发生时 A 发生”的概率。掌握加法法则、乘法法则和条件概率，就等于拥有了一套连贯的工具来解答统计与力学部分的考试题。

1. What Are Combined Events? | 什么是组合事件？

A combined event is formed by using words such as AND, OR and NOT to link two or more simple events. For example, roll a fair six-sided die and flip a coin: the event ‘score an even number and get heads’ combines the die event E = {2, 4, 6} with the coin event H = {heads}. When you list all possible combined outcomes, you form the sample space.

组合事件是用 AND、OR、NOT 等词连接两个或多个简单事件而形成的事件。例如，掷一枚均匀六面骰并抛一枚硬币：“掷出偶数且抛出正面”就是把骰子事件 E = {2, 4, 6} 与硬币事件 H = {正面} 组合起来。列出所有可能的组合结果，就得到样本空间。

In probability, a combined event is simply a subset of the sample space that is defined by more than one condition. You may be asked to find P(A and B), P(A or B), or P(A given B), and each expression has a distinct formula and meaning.

在概率论中，组合事件就是由多个条件定义的样本空间的子集。考试可能会要求你求出 P(A and B)、P(A or B) 或 P(A given B)，每一种表达方式都有不同的公式和含义。

2. Key Terminology: Outcomes, Sample Spaces and Events | 关键术语：结果、样本空间与事件

The sample space S is the set of all possible outcomes. An outcome is a single possible result, and an event is a subset of S. For combined experiments, you can use a table or grid to list all equally likely outcomes systematically.

样本空间 S 是所有可能结果组成的集合，结果是单个可能情形，事件是样本空间的子集。对于组合试验，你可以用表格或网格系统地列出所有等可能结果。

Outcome – one possible result, such as getting a 2 and heads.
Sample space – the complete set of outcomes, such as 6 × 2 = 12 pairs for a die and a coin.
Event – any collection of outcomes, such as E and H = {(2, H), (4, H), (6, H)}.

结果——一种可能的情形，例如掷出 2 且抛出正面。
样本空间——所有结果的完整集合，例如掷骰与抛硬币共有 6 × 2 = 12 对结果。
事件——任意结果的集合，例如 E 与 H = {(2, H), (4, H), (6, H)}。

3. The Addition Rule and Mutually Exclusive Events | 加法法则与互斥事件

To find the probability of A or B, use the addition rule. In general, if A and B can both happen, the formula is:

要求出 A 或 B 的概率，需要使用加法法则。一般来说，如果 A 和 B 可能同时发生，公式为：

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

This formula subtracts the overlap once so that outcomes in both A and B are not double counted. If A and B are mutually exclusive, they cannot happen together, so P(A ∩ B) = 0 and the rule simplifies to P(A ∪ B) = P(A) + P(B).

该公式减去一次重叠部分，避免同时属于 A 和 B 的结果被重复计算。如果 A 与 B 互斥，它们不能同时发生，因此 P(A ∩ B) = 0，公式简化为 P(A ∪ B) = P(A) + P(B)。

Example: A bag contains 5 red, 3 blue and 2 green balls. Find P(red or blue). Since red and blue are mutually exclusive, P(red or blue) = 5/10 + 3/10 = 8/10 = 4/5.

例题：一个袋子里有 5 个红球、3 个蓝球和 2 个绿球。求 P(红球或蓝球)。由于红球与蓝球互斥，P(红球或蓝球) = 5/10 + 3/10 = 8/10 = 4/5。

4. The Multiplication Rule and Independent Events | 乘法法则与独立事件

For independent events, the occurrence of one event does not affect the probability of the other. The multiplication rule for independent events is:

对于独立事件，一个事件的发生不会影响另一个事件发生的概率。独立事件的乘法法则为：

P(A ∩ B) = P(A) × P(B)

For example, the probability of rolling a 6 on a fair die and getting heads on a fair coin is 1/6 × 1/2 = 1/12. Always check the question for the word ‘replaced’, ‘fair’, or a clear statement of independence.

例如，均匀骰子掷出 6 且均匀硬币抛出正面的概率为 1/6 × 1/2 = 1/12。解题时务必留意题目中“放回”“均匀”或明确说明独立的字眼。

If events are not independent, you must use the general multiplication rule linked to conditional probability:

如果事件不独立，则必须使用与条件概率相关的一般乘法法则：

P(A ∩ B) = P(A) × P(B | A) = P(B) × P(A | B)

5. Conditional Probability and Combined Events | 条件概率与组合事件

Conditional probability measures the chance of A given that B has already occurred. The formula is:

条件概率衡量在事件 B 已经发生的条件下，事件 A 发生的概率。公式为：

P(A | B) = P(A ∩ B) ÷ P(B)

This formula applies only when P(B) ≠ 0. It is particularly useful when you know the joint probability and the probability of the condition. You can rearrange it as P(A ∩ B) = P(B) × P(A | B).

该公式仅在 P(B) ≠ 0 时适用。当你已知联合概率和条件事件的概率时，它特别有用。你也可以将其变形为 P(A ∩ B) = P(B) × P(A | B)。

Example: If P(A ∩ B) = 0.2 and P(B) = 0.5, then P(A | B) = 0.2 ÷ 0.5 = 0.4. This means that if B has happened, there is a 40% chance that A also happened.

例题：若 P(A ∩ B) = 0.2，P(B) = 0.5，则 P(A | B) = 0.2 ÷ 0.5 = 0.4。这意味着如果 B 已发生，A 也发生的概率为 40%。

6. Tree Diagrams for Combined Events | 组合事件的树状图

Tree diagrams are a powerful visual tool for combined events, especially when there are two or more stages. Along each branch write the probability for that stage; to find the probability of a complete path, multiply along the branches.

树状图是解决组合事件的强大可视化工具，尤其适用于有两个或多个阶段的试验。在每条分支上写出该阶段的概率；要求出某条完整路径的概率，将路径上各分支概率相乘。

Draw a branch for each possible outcome at every stage.
Label the branch probabilities clearly, using decimals or fractions.
Multiply along the path for joint probabilities.
Add the probabilities of mutually exclusive paths for ‘or’ events.

为每一阶段的每个可能结果画出分支。
用小数或分数清晰标注分支概率。
沿路径相乘得到联合概率。
将互斥路径的概率相加得到“或”事件的概率。

Tree diagrams are especially helpful for conditional probability because the probabilities on the second set of branches are written as P(B | A) and P(B | A’).

树状图尤其适合条件概率，因为第二组分支上的概率通常写作 P(B | A) 和 P(B | A’)。

7. Venn Diagrams and Combined Events | 维恩图与组合事件

Venn diagrams show overlaps between events and help you apply the addition rule correctly. The rectangle represents the sample space S, circles represent events, and the overlap represents A ∩ B.

维恩图显示事件之间的重叠，帮助你正确使用加法法则。矩形表示样本空间 S，圆表示事件，重叠部分表示 A ∩ B。

P(A) is the total probability inside circle A.
P(B) is the total probability inside circle B.
P(A ∩ B) is the overlap between A and B.
P(A ∪ B) is every region inside at least one circle.

P(A) 是圆 A 内部的总概率。
P(B) 是圆 B 内部的总概率。
P(A ∩ B) 是 A 与 B 的重叠部分。
P(A ∪ B) 是至少在一个圆内的所有区域。

Using a Venn diagram can make it easier to see why the addition rule subtracts P(A ∩ B). Fill in the known regions, then subtract to find missing probabilities.

使用维恩图可以更直观地理解为什么加法法则要减去 P(A ∩ B)。先填入已知区域，再用减法求出缺失的概率。

8. Two-Way Tables and Combined Events | 双向表与组合事件

Two-way tables organise data by two categories and are extremely useful for identifying joint frequencies and conditional probabilities. Each row and column gives a marginal total, and the grand total is the total number of observations.

双向表按两个类别整理数据，对于确定联合频数和条件概率非常有用。每一行和每一列给出边际总数，总计是所有观测值的总数。

Category	B	B’	Total
A	a	b	a + b
A’	c	d	c + d
Total	a + c	b + d	a + b + c + d

From a two-way table, P(A ∩ B) = a ÷ total, P(A) = (a + b) ÷ total, and P(B | A) = a ÷ (a + b). This is often a more accurate method than using a tree diagram when the table gives actual frequencies.

从双向表中可得 P(A ∩ B) = a ÷ 总计，P(A) = (a + b) ÷ 总计，P(B | A) = a ÷ (a + b)。当表格给出实际频数时，这通常比树状图更准确。

9. Complementary Events and ‘At Least One’ Problems | 对立事件与“至少一个”问题

The complement of event A is A’, meaning A does not happen. For any event A, P(A’) = 1 − P(A). This is especially useful in ‘at least one’ questions, which often can be solved faster by finding the probability of none and subtracting from 1.

事件 A 的对立事件是 A’，表示 A 不发生。对于任意事件 A，P(A’) = 1 − P(A)。这在“至少一个”问题中特别有用，通常可以通过先求“一个都没有”的概率再从 1 中减去来更快地解决。

P(at least one success) = 1 − P(no success)

Example: A fair coin is flipped 3 times. Find P(at least one head). The probability of no heads is P(three tails) = 1/2 × 1/2 × 1/2 = 1/8. Therefore P(at least one head) = 1 − 1/8 = 7/8.

例题：将一枚均匀硬币抛掷 3 次，求 P(至少一次正面)。没有正面的概率为 P(三次反面) = 1/2 × 1/2 × 1/2 = 1/8。因此 P(至少一次正面) = 1 − 1/8 = 7/8。

10. Common Mistakes and Exam Tips | 常见错误与考试技巧

Students often confuse P(A and B) with P(A or B), or assume independence when events affect each other. Always read the wording carefully: ‘or’ suggests addition, ‘and’ suggests multiplication, and ‘given that’ suggests conditional probability.

学生们经常混淆 P(A and B) 与 P(A or B)，或在事件相互影响时错误地假设它们独立。务必仔细阅读题意：“或”通常用加法，“且”通常用乘法，“已知”通常表示条件概率。

Do not add probabilities for independent events when you should multiply.
Do not double count the overlap in P(A ∪ B).
Check whether items are replaced or not, as this affects independence.
Use fractions in exam answers unless the question asks for decimals or percentages.
Always show the formula before substituting values.

不要把本应相乘的独立事件概率相加。
计算 P(A ∪ B) 时不要重复计算重叠部分。
检查物品是否放回，因为这会影响事件的独立性。
除非题目要求用小数或百分数作答，考试答案应使用分数。
在代入数值前务必写出公式。

11. Worked Example – Combined Probability | 典型例题精选

A box contains 4 white beads and 6 black beads. Two beads are drawn at random without replacement. Find the probability that the first bead is white and the second bead is black.

一个盒子里有 4 个白珠和 6 个黑珠。随机不放回地抽取两个珠子。求第一个珠子是白色且第二个珠子是黑色的概率。

Let W₁ be the event that the first bead is white, and B₂ be the event that the second bead is black. We need P(W₁ ∩ B₂). Since the first bead is not replaced, the events are not independent, so use the general multiplication rule:

设 W₁ 表示第一个珠子是白色，B₂ 表示第二个珠子是黑色。要求 P(W₁ ∩ B₂)。由于第一个珠子不放回，事件不独立，因此使用一般乘法法则：

P(W₁ ∩ B₂) = P(W₁) × P(B₂ | W₁) = 4/10 × 6/9 = 24/90 = 4/15

After taking one white bead, there are 9 beads left, of which 6 are black. Always adjust the denominator and numerator when sampling without replacement.

取出一个白珠后，还剩下 9 个珠子，其中 6 个是黑珠。在不放回抽样时，务必调整分母和分子。

12. Summary and Key Formulae | 总结与关键公式

Combined events are at the heart of Edexcel A-Level probability questions. Learn the rules below and practise applying them to tree diagrams, Venn diagrams and two-way tables.

组合事件是爱德思A-Level概率题的核心。掌握以下法则，并在树状图、维恩图和双向表中反复练习。

Rule	Formula	Use when
Addition rule	P(A ∪ B) = P(A) + P(B) − P(A ∩ B)	Finding ‘A or B’
Independent events	P(A ∩ B) = P(A) × P(B)	A and B do not affect each other
General multiplication	P(A ∩ B) = P(A) × P(B \| A)	Dependent or conditional events
Complement	P(A’) = 1 − P(A)	‘At least one’ or ‘not’ questions

Memorise these expressions and always define your events clearly. With consistent practice, combined event questions become an accessible source of marks in the exam.

熟记这些表达式，并始终清晰地定义事件。通过持续练习，组合事件题目将成为考试中容易得分的重要部分。

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Edexcel A-Level Maths: Combined Events and Probability | 爱德思A-Level数学：组合事件与概率