📚 Combined Events | 组合事件
When we carry out more than one simple experiment at a time — such as flipping two coins, rolling two dice, or picking two cards from a deck — we are dealing with combined events. In IGCSE Mathematics, combined events help us extend basic probability ideas to situations with multiple stages or multiple characteristics. Understanding how to list outcomes systematically, when to add probabilities and when to multiply them, and how to use diagrams like tree diagrams and Venn diagrams is essential for solving exam problems accurately.
当我们同时进行多个简单试验时——例如同时抛两枚硬币、掷两个骰子，或者从一副牌中抽取两张牌——我们处理的就是组合事件。在IGCSE数学中，组合事件将基本的概率概念扩展到多阶段或多特征的情境中。系统地列出结果、知道何时将概率相加、何时相乘，以及如何使用树形图、文氏图等工具，对于准确解决考试题目至关重要。
1. What Are Combined Events? | 什么是组合事件？
A combined event is an event that involves two or more individual outcomes happening together or in succession. Examples include getting a head on the first coin and a tail on the second coin, rolling a sum of 7 on two dice, or selecting a red ball first and a blue ball second from a bag. The focus is not on a single trial, but on how multiple trials interact.
组合事件是指涉及两个或多个单独结果同时发生或相继发生的事件。例子包括第一枚硬币正面朝上且第二枚硬币反面朝上、两个骰子的点数之和为7，或从袋子里先取出红球再取出蓝球。其重点不在于单一试验，而在于多个试验如何相互作用。
In IGCSE, we learn to calculate probabilities of combined events using rules for 'OR' and 'AND', as well as visual representations such as sample space diagrams, tree diagrams and Venn diagrams. Tackling these events often starts with identifying whether the individual outcomes affect each other — that is, whether they are independent or dependent.
在IGCSE中，我们学习使用“或”和“与”的规则以及样本空间图、树形图和文氏图等可视化表示来计算组合事件的概率。处理这些事件通常首先要判断各个单独结果是否互相影响，即它们是独立的还是相依的。
2. Sample Spaces and Systematic Listing | 样本空间与系统列举
pairs. A systematic list ensures no outcome is missed: start with the first die showing 1 and list all possibilities for the second die, then move to 2, and so on.
样本空间是一个组合事件所有可能结果的集合。对于两枚公平硬币，样本空间是{正正，正反，反正，反反}。对于两个普通骰子，共有 6 × 6 = 36 个可能的有序数对。系统列举可确保不漏掉任何结果：从第一个骰子显示1开始，列出第二个骰子的所有可能，然后移到2，依此类推。
ered pair or the sum in each cell. This visual layout makes it easy to count favourable outcomes for events like ‘sum = 8’ or ‘ as n outcomes, the combined trial has m × n outcomes, provided the trials are independent.
一个非常有用的工具是双向表。对于掷两个骰子的情况，可以将第一个骰子的结果写在顶部，第二个骰子的结果写在侧边，在每个单元格中填入有序数对或和。这种可视化布局可以让人轻松地计算出诸如“和为8”或“点数相同”等事件的有利结果。计数乘法原理指出，如果一次试验有 m 种结果，另一次试验有 n 种结果，且两次试验相互独立，则组合试验有 m × n 种结果。
3. The OR Rule for Mutually Exclusive Events | 互斥事件的“或”法则
Two events are mutually exclusive if they cannot happen at the same time. For example, when rolling a fair six‑sided die, getting a 2 and getting a 5 are mutually exclusive — a single roll cannot give both. For mutually exclusive events A and B, the probability that A or B occurs is simply the sum of their individual probabilities.
如果两个事件不可能同时发生，则它们是互斥的。例如，掷一个公平的六面骰子时，得到2点和得到5点是互斥的——单次掷骰不可能同时出现两个结果。对于互斥事件 A 和 B，事件 A 或 B 发生的概率就是它们各自概率的和。
P(A or B) = P(A) + P(B)
more than two mutually exclusive events: simply add all their probabilities. Exam questions often ask for ‘the probability that the score is a 4 or a 5’ — always check whether the events can overlap before using this simple addition.
这有时被称为互斥事件的加法法则。在掷骰例子中，P(2或5) = 1/6 + 1/6 = 2/6 = 1/3。同样的规则也适用于两个以上的互斥事件：只需将它们所有的概率相加。考试题常问“得分是4或5的概率”——在使用这种简单加法之前，务必检查事件是否可能重叠。
4. The OR Rule for Non‑Mutually Exclusive Events | 非互斥事件的“或”法则
When events are not mutually exclusive, they share some outcomes. For instance, consider drawing one card at random from a standard deck of 52 playing cards. Event A is ‘the card is a heart’ and Event B is ‘the card is a king’. These events can occur together because the king of hearts is both a heart and a king.
当事件不是互斥时，它们会共享一些结果。例如，考虑从标准的52张扑克牌中随机抽取一张牌。事件A是“这张牌是红心”，事件B是“这张牌是K”。这些事件可能同时发生，因为红心K既是红心又是K。
ng of hearts twice. To correct this, we subtract the probability of the intersection once. The general addition rule is:
如果我们简单地将 P(红心) = 13/52 和 P(K) = 4/52 相加，就会把红心K计算了两次。为了纠正这一点，我们减去一次交集的概率。通用的加法法则是：
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
For the card example, P(heart ∪ king) = 13/52 + 4/52 – ing. This formula is crucial when working with Venn diagrams and combined events in IGCSE.
在扑克牌的例子中，P(红心 ∪ K) = 13/52 + 4/52 – 1/52 = 16/52 = 4/13。务必识别重叠部分并减去一次，以避免重复计算。在使用文氏图和IGCSE中的组合事件时，这个公式至关重要。
5. The AND Rule for Independent Events | 独立事件的“与”法则
Two events are independent if the outcome of one does not affect the probability of the other. Classic examples are flipping a fair coin and rolling a fair die — the coin result does not change the chance of rolling a 6. For independent events A and B, the probability that both A and B happen is the product of their individual probabilities.
如果一个事件的结果不影响另一个事件的概率，则这两个事件是独立的。经典的例子是抛一枚公平硬币和掷一个公平骰子——硬币的结果不会改变掷出6点的概率。对于独立事件 A 和 B，两者同时发生的概率是它们各自概率的乘积。
P(A ∩ B) = P(A) × P(B)
lso applies to more than two independent events. It is important to check that events really are independent: flipping the same coin twice produces independent results; rolling two separate dice are independent; but drawing two cards without replacement makes the events dependent, so you cannot simply multiply the original probabilities.
例如，如果你抛一枚公平硬币并掷一个公平骰子，P(正面且5点) = 1/2 × 1/6 = 1/12。此乘法规则同样适用于两个以上的独立事件。重要的是检查事件是否真正独立：抛同一枚硬币两次会产生独立的结果；掷两个独立的骰子是独立的；但无放回地抽取两张牌会使事件相依，此时就不能简单地将原始概率相乘。
6. Dependent Events and Conditional Probability | 相依事件与条件概率
When the outcome of the first event changes the probability of the second event, we have dependent events. Selecting two balls from a bag without replacement is the most common IGCSE example. If a bag contains 4 red and 6 blue balls, the probability that the first ball is red is 4/10. If the first ball is red and not replaced, 9 balls remain — 3 red and 6 blue — so the probability the second ball is red given the first was red is now 3/9.
当第一个事件的结果改变了第二个事件的概率时，我们面对的是相依事件。无放回地从袋子中取两个球是IGCSE中最常见的例子。如果一个袋子里有4个红球和6个蓝球，第一个球是红色的概率为4/10。如果第一个球是红色且不放回，则剩下9个球——3个红球和6个蓝球——因此在第一个球是红色的条件下第二个球是红色的概率变为3/9。
The probability of both events happening is found by multiplying along the branch of a tree diagram or using the conditional probability formula:
两个事件同时发生的概率可通过沿树形图分支相乘或使用条件概率公式求得：
P(A ∩ B) = P(A) × P(B | A)
Here P(B | A) is read as ‘ dependent, always adjust the probability for the second event based on what has already happened. Conditional probability questions often require you to identify this ‘given that’ scenario clearly.
这里 P(B | A) 读作“在A发生的条件下B发生的概率”。在袋子的例子中，P(第一个红且第二个红) = 4/10 × 3/9 = 12/90 = 2/15。当事件相依时，务必根据已发生的情况调整第二个事件的概率。条件概率问题通常要求你清晰地识别这种“已知…条件下”的情境。
7. Tree Diagrams for Combined Events | 组合事件的树形图
Tree diagrams are one of the most powerful tools for handling combined events, especially multi‑stage experiments. Each branch represents a possible outcome at a given stage, labelled with its probability. For two stages, you draw two sets of branches from left to right. To find the probability of a particular sequence of outcomes, multiply the probabilities along the relevant branches.
树形图是处理组合事件最强大的工具之一，尤其适用于多阶段试验。每个分支代表给定阶段的一个可能结果，并标注其概率。对于两个阶段，你从左到右绘制两组分叉。要找到特定结果序列的概率，将相关分支上的概率相乘。
For independent events, the probabilities on the second set of branches are the same regardless of the first outcome. For dependent events, the second‑stage probabilities change depending on the first outcome. After drawing the tree, you can calculate the probability of any combined outcome by multiplying along the branches. If a question asks for ‘at least one’ or ‘either this or that’, you may need to add the probabilities of several final outcomes.
对于独立事件，无论第一个结果如何，第二组分叉上的概率保持不变。对于相依事件，第二阶段概率会根据第一个结果而改变。画好树形图后，你可以通过沿分支相乘来计算任何组合结果的概率。如果题目问“至少有一个”或“非此即彼”，你可能需要将几个最终结果的概率相加。
n and 5 yellow sweets. You take two sweets without replacement. Draw the tree: first sweet — green (3/8) or yellow (5/8). If green taken, second sweet — green (2/7) or yellow (5/7); if yellow first, second sweet — = 15/28. Tree diagrams keep your working organised and are strongly recommended in exams.
例如，一个袋子里有3颗绿色糖果和5颗黄色糖果。你不放回地取两颗糖果。画树形图：第一颗——绿色(3/8)或黄色(5/8)。如果第一颗取走绿色，第二颗——绿色(2/7)或黄色(5/7)；如果第一颗是黄色，第二颗——绿色(3/7)或黄色(4/7)。拿到两颗不同颜色的概率为(3/8 × 5/7) + (5/8 × 3/7) = 15/56 + 15/56 = 30/56 = 15/28。树形图能保持解题过程有序，强烈建议在考试中使用。
8. Venn Diagrams and Combined Probability | 文氏图与组合概率
Venn diagrams show the relationships between events in a visual way. Each event is represented by a circle inside a rectangle that represents the sample space. Overlapping regions show outcomes common to both events. Numbers written in each region can be frequencies or probabilities. From a Venn diagram, you can easily read off P(A), P(B), P(A ∩ B), and P(A ∪ B).
文氏图以可视化的方式展示事件之间的关系。每个事件用一个圆圈表示，矩形代表样本空间。重叠区域显示两个事件共有的结果。每个区域中填写的数字可以是频数或概率。从文氏图中，你可以轻松读出 P(A)、P(B)、P(A ∩ B) 和 P(A ∪ B)。
30 = 4/5. Venn diagrams are particularly helpful when you need the probability of combinations like ‘A but not B’ or ‘exactly one of A and B’.
例题：班上有30名学生，18人学习法语，14人学习西班牙语，8人两种语言都学。在交集区域填入8，然后只学法语的为18−8=10，只学西班牙语的为14−8=6，两种都不学的为30−10−8−6=6。随机选取一名学生至少学习一门外语的概率为(10+8+6)/30 = 24/30 = 4/5。当需要计算诸如“A 但非 B”或“恰好只发生 A 和 B 中的一个”的组合概率时，文氏图特别有用。
9. Frequency Trees and Two‑Way Tables | 频率树与双向表
Before moving directly to probabilities, you might organise data using frequency trees and two‑way tables. A frequency tree records the number of outcomes along branches, while a two‑way table arranges frequencies by two categories. Both tools make it easier to calculate experimental probabilities or to tackle grouped data questions.
在直接处理概率之前，你可以使用频率树和双向表整理数据。频率树沿分支记录结果的数量，而双向表则按两个类别排列频数。这两种工具都为计算试验概率或处理分组数据问题提供了方便。
For example, a sports survey of 50 children shows: 30 boys and 20 girls; 18 boys prefer football, 12 prefer basketball; 10 girls prefer football, 10 prefer basketball. You can display this in a two‑way table:
例如，对50名儿童的体育调查显示：男孩30人，女孩20人；18名男孩偏爱足球，12名偏爱篮球；10名女孩偏爱足球，10名偏爱篮球。你可以将其展示在一个双向表中：
| Football | Basketball | Total | |
|---|---|---|---|
| Boys | 18 | 12 | 30 |
| Girls | 10 | 10 | 20 |
| Total | 28 | 22 | 50 |
can quickly find probabilities: P(choosing a girl who prefers football) = 10/50 = 1/5. Two‑ ties, for example, the probability that a randomly chosen football fan is a girl is 10/28.
利用这个表格，你可以快速求出概率：P(选到一个偏爱足球的女孩) = 10/50 = 1/5。双向表也有助于将条件概率形象化，例如，随机选出的一个足球迷是女孩的概率为10/28。
10. Multiplication Principle for Counting Outcomes | 计数结果的乘法原理
× n. This extends to any number of stages: multiply the numbers of choices at each stage.
当列出所有可能结果变得不切实际时，乘法原理（或称计数乘法规则）可快速求得结果的总数。如果在第一步有 m 种选择，第二步有 n 种选择，那么组合选择的总数为 m × n。这可以推广到任意多个阶段：将每个阶段的选择数相乘即可。
r instance, a restaurant offers 4 starters and 6 main courses. The number of different two‑course meals (starter then main) is 4 × 6 = 24. If they also offer 3 desserts, a three‑ tions and simplifies probability problems where you only need the total possible outcomes.
例如，一家餐厅提供4种开胃菜和6种主菜。两菜套餐（开胃菜加主菜）的不同组合总数为 4 × 6 = 24。如果他们还提供3种甜点，那么三菜套餐的组合数就是 4 × 6 × 3 = 72。该原理为样本空间计算提供了基础，也简化了那些仅需知道可能结果总数的概率问题。
In an exam, you might be asked to find the probability of a specific combination
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