Combined Events (098) in Probability | 概率中的组合事件 (098)

📚 Combined Events (098) in Probability | 概率中的组合事件 (098)

In Edexcel A‑Level Mathematics, the study of probability extends far beyond single events. Combined events occur when we consider two or more outcomes happening together, such as tossing a coin and rolling a die, or drawing cards from a deck without replacement. Topic reference 098 focuses on how to model, calculate and interpret such situations using clear rules and diagrams. A strong grasp of combined events is essential for success in the statistics components of the exam, as well as for understanding real‑world risk, decision‑making and data analysis.

在 Edexcel A‑Level 数学中，概率的学习远不止单个事件。当我们考虑两个或更多结果同时发生时，就产生了组合事件，例如同时抛硬币和掷骰子，或者从一副牌中不放回地连续抽牌。考点 098 聚焦于如何使用清晰的法则和图形来建模、计算和解释这类情境。牢固掌握组合事件对于在考试统计部分取得成功，以及理解现实世界中的风险、决策和数据分析都至关重要。

1. Understanding Combined Events | 理解组合事件

A combined event is any outcome that involves more than one characteristic or trial. For example, 'rolling an even number and getting a head' when a die and a coin are used together is a combined event. To work with these events systematically, we first define a sample space that lists every possible paired outcome. This space can be represented as a list, a table or a grid, ensuring no outcome is overlooked.

组合事件是指任何涉及多个特征或多次试验的结果。例如，当同时使用一个骰子和一枚硬币时，“掷出偶数 且 得到正面”就是一个组合事件。为了系统地处理这些事件，我们首先定义一个样本空间，列出所有可能的成对结果。这个空间可以用列表、表格或网格表示，确保没有遗漏任何结果。

The size of the sample space is found by multiplying the number of outcomes for each individual event, provided the trials are independent. This fundamental counting principle is the starting point for all probability calculations involving combined events.

如果各次试验相互独立，样本空间的大小可以通过将每个单独事件的结果数相乘来求得。这个基本的计数原理是所有涉及组合事件的概率计算的起点。

2. Sample Spaces and Outcomes | 样本空间与结果

Let us take a simple case: tossing a fair coin and rolling a fair six‑ s 2 × 6 = 12 equally likely outcomes, which can be written as ordered pairs: (H,1), (H,2), … we can directly count the number of outcomes that satisfy a given combined event, such as 'head and a number greater than 4'.

让我们看一个简单的例子：抛一枚均匀的硬币并掷一个均匀的六面骰子。硬币的结果是 {H, T}，骰子的结果是 {1, 2, 3, 4, 5, 6}。组合样本空间包含 2 × 6 = 12 个等可能的结果，可以写成有序对：(H,1), (H,2), …, (T,6)。从这个空间中，我们可以直接数出满足某个给定组合事件的结果数目，例如“正面且点数大于 4”。

When events are not equally likely, or when the number of outcomes is large, we rely on rules of probability rather than simple enumeration. The sample space concept, however, remains the theoretical foundation for all subsequent rules.

当结果不是等可能时，或者结果数量很大时，我们依靠概率法则而不是简单的列举。然而，样本空间的概念仍然是所有后续法则的理论基础。

3. Mutually Exclusive Events | 互斥事件

Two events are mutually exclusive if they cannot occur at the same time. In a single roll of a die, the events 'rolling a 2' and 'rolling a 5' are mutually exclusive. For combined events, we must be careful: if we draw one card from a deck, the events 'drawing a heart' and 'drawing a spade' are mutually exclusive, but 'drawing a heart' and 'drawing a king' are not, because the king of hearts belongs to both.

如果两个事件不能同时发生，则它们互斥。在掷一次骰子时，“掷出 2 点”和“掷出 5 点”就是互斥事件。对于组合事件，我们必须小心：从一副牌中抽一张牌，“抽到红心”和“抽到黑桃”是互斥的，但“抽到红心”和“抽到 K”不是互斥的，因为红心 K 同时属于两者。

The key probability rule for mutually exclusive events is the addition rule in its simplest form: the probability of A or B occurring is the sum of their individual probabilities. Symbolically,

P(A ∪ B) = P(A) + P(B)

This holds only when A and B are mutually exclusive. In exam questions, you will often need to check this condition before applying the formula.

互斥事件的关键概率法则是加法法则的最简形式：A 或 B 发生的概率等于它们各自概率之和。用符号表示为 P(A ∪ B) = P(A) + P(B)。这一定律仅当 A 与 B 互斥时才成立。在考试题目中，你通常需要先检查这一条件，然后再应用该公式。

4. The Addition Rule | 加法法则

When events are not mutually exclusive, simply adding their probabilities would count the intersection twice. The general addition rule corrects this by subtracting the probability of the intersection:

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

This formula is universally valid, whether the events are mutually exclusive (where P(A ∩ B) = 0) or not. For combined events such as 'it rains on Monday' and 'it rains on Tuesday', you need historical data to estimate P(A ∩ B) before you can find the probability of at least one rainy day.

当事件不互斥时，简单地将概率相加会重复计算交集部分。一般加法法则通过减去交集的概率来修正这一点：P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。该公式无论事件互斥与否均成立，当互斥时 P(A ∩ B) = 0。对于“星期一下雨”和“星期二下雨”这样的组合事件，你需要历史数据来估计 P(A ∩ B)，然后才能求出至少一天下雨的概率。

Always identify whether the word 'or' in a question means the union of events, and never assume mutual exclusivity unless the context explicitly forbids overlap. Drawing a Venn diagram is an excellent way to visualise the relationships.

务必识别问题中的“或”是否意味着事件的并集，并且除非上下文明确禁止重叠，否则绝不要假设互斥。绘制文氏图是可视化这些关系的绝佳方法。

5. Independent Events | 独立事件

Two events are independent if the occurrence of one does not affect the probability of the other. For instance, the outcome of a coin toss does not influence the result of a die roll. In contrast, drawing two cards from a deck without replacement makes the second draw dependent on the first, because the composition of the deck changes.

如果一件事件的发生不影响另一事件的概率，则这两个事件独立。例如，抛硬币的结果不影响掷骰子的结果。相反，从一副牌中不放回地连续抽两张牌时，第二次抽牌的结果依赖于第一次，因为牌的构成发生了变化。

Formally, events A and B are independent if and only if

P(A ∩ B) = P(A) × P(B)

This multiplication condition is both a definition and a test for independence. In many exam problems, you may be given P(A), P(B) and P(A ∩ B) and asked to decide whether A and B are independent.

形式上，事件 A 和 B 独立当且仅当 P(A ∩ B) = P(A) × P(B)。这个乘法条件既是定义，也是独立性检验。在许多考题中，你可能会得到 P(A)、P(B) 和 P(A ∩ B)，并被要求判断 A 和 B 是否独立。

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6. The Multiplication Rule | 乘法法则

For any two events, the probability of both occurring can be found by multiplying the probability of one event by the conditional probability of the second given the first:

P(A ∩ B) = P(A) × P(B | A)

If A and B are independent, then P(B | roduct of their individual probabilities. This chain multiplication can be extended to three or more events, which is particularly useful when working with tree diagrams for sequential trials.

对于任意两个事件，两者同时发生的概率可以通过一个事件的概率乘上给定第一个事件后第二个事件的条件概率来求得：P(A ∩ B) = P(A) × P(B | A)。如果 A 与 B 独立，那么 P(B | A) = P(B)，该法则便退化为各自概率的乘积。这种链式乘法可以推广到三个或更多事件，在用树形图处理连续试验时尤其有用。

When a question involves sampling without replacement, remember that the probabilities change at each stage, so the multiplication rule must be applied with care, updating the denominator and numerator each time.

当题目涉及不放回抽样时，请记住每一步的概率都会变化，因此必须谨慎应用乘法法则，每次都要更新分母和分子。

7. Tree Diagrams for Combined Events | 组合事件的树形图

Tree diagrams provide a structured visual method for calculating probabilities of sequences of events. Each branch represents a possible outcome at a given stage, labelled with its probability. To find the probability of a complete path, multiply the probabilities along the branches. If there is more than one path leading to the same final outcome, add the path probabilities together.

树形图为计算事件序列的概率提供了一种结构化的可视化方法。每一分支代表给定阶段的一个可能结果，并标注其概率。要找到一条完整路径的概率，只需将沿途各分支的概率相乘。如果有多条路径通向同一个最终结果，则将各路径的概率相加。

Tree diagrams are indispensable for 'at least one' problems and for conditional probability questions. Always label the ends of branches clearly and check that the probabilities on branches emanating from the same point sum to 1. If conditional probabilities change after the first outcome, make sure the second‑stage probabilities reflect that change.

树形图在“至少一次”类问题和条件概率题目中不可或缺。务必清楚标注各分支的末端，并检查从同一点伸出的分支上的概率之和是否为 1。如果条件概率在第一个结果之后发生改变，一定要确保第二阶段概率反映了这种变化。

8. Conditional Probability | 条件概率

Conditional probability measures the likelihood of an event occurring given that another event has already happened. The standard formula is

P(A | B) = P(A ∩ B) / P(B)

provided P(B) > 0. This concept is central to many Edexcel A‑Level questions, including those set in medical testing, weather forecasting and quality control contexts.

条件概率衡量在已知另一事件已经发生的条件下某事件发生的可能性。标准公式为 P(A | B) = P(A ∩ B) / P(B)，其中 P(B) > 0。这个概念是许多 Edexcel A‑Level 考题的核心，包括设置在医学检测、天气预测和质量控制情境下的题目。

When working with tables of data, you can compute conditional probabilities directly by restricting the sample space to the given condition. For example, if a two‑way table shows the number of left‑handed and right‑handed students by gender, the probability that a student is left‑handed given that they are female is simply the number of left‑handed females divided by the total number of females.

在处理数据表格时，你可以通过将样本空间限制在给定条件下来直接计算条件概率。例如，如果一个双向表显示了按性别分类的左撇子与右撇子学生人数，则在已知为女生的条件下该学生是左撇子的概率，就等于左撇子女生人数除以女生总人数。

9. Using Venn Diagrams | 使用文氏图

Venn diagrams are powerful tools for visualising relationships among combined events, especially when dealing with unions, intersections and complements. A typical diagram for two events shows a rectangle representing the whole sample space, with overlapping circles for A and B. Numbers or probabilities can be placed in each region to represent P(A ∩ B), P(A ∩ B'), P(A' ∩ B) and P(A' ∩ B').

文氏图是可视化组合事件之间关系的有力工具，尤其是在处理并集、交集和补集时。一张典型的两个事件文氏图显示一个代表整个样本空间的矩形，以及两个重叠的圆分别代表 A 和 B。每个区域可以填入数字或概率，以表示 P(A ∩ B)、P(A ∩ B')、P(A' ∩ B) 和 P(A' ∩ B')。

From a completed Venn diagram you can quickly read off probabilities required for addition rule calculations, test for independence and answer 'given that' questions by focusing on the appropriate restricted space. Many exam questions ask you to fill in a Venn diagram from a worded description, so practice translating 'at least one', 'both' and 'neither' into the correct regions.

从一张填写完整的文氏图中，你可以快速读取加法法则计算所需的概率，检验独立性，并通过聚焦于适当的受限空间来回答“已知…的条件下”问题。许多考试题目要求你根据文字描述填制文氏图，因此要练习将“至少一个”、“两者都”和“两者都不”转化为正确的区域。

10. Common Pitfalls and Exam Tips | 常见错误与应试技巧

ith independent events. Remember: mutually exclusive events cannot happen together, so P(A ∩ B) = 0, whereas independent events have P(A ∩ nd can certainly happen together. Another pitfall is forgetting to update probabilities in 'without replacement' scenarios, leading to incorrect second‑stage branch probabilities on tree diagrams.

一个常见错误是混淆互斥事件与独立事件。请记住：互斥事件不能同时发生，因此 P(A ∩ B) = 0，而独立事件满足 P(A ∩ B) = P(A)P(B)，且完全可以同时发生。另一个常见错误是在“不放回”情况下忘记更新概率，导致树形图上第二阶段分支的概率出错。

When using the general addition rule, ensure you subtract the intersection to avoid double‑counting. In conditional probability questions, write down the formula first, then compute each part carefully. Always keep your working logical and tidy, drawing diagrams whenever possible. Finally, in 'at least one' problems, it is often simpler to calculate 1 − P(none) rather than adding many individual path probabilities.

使用一般加法法则时，请确保减去交集部分以避免重复计算。在条件概率问题中，先写出公式，然后仔细计算每个部分。始终保持解题步骤逻辑清晰、书写整洁，并尽可能绘制图表。最后，在“至少一次”类问题中，计算 1 − P(一个都没有)往往比将许多单个路径概率相加更简单。

Edexcel Mark schemes reward clear notation and the correct use of probability rules. Even if you find a correct numerical answer, showing the formula and intermediate steps will earn method marks if a slip occurs later. Practice with past paper 098‑style questions to become fluent in switching between tree diagrams, Venn diagrams and algebraic rules.

Edexcel 的评分标准奖励清晰的符号表示和概率法则的正确运用。即使你得出了正确的数值答案，写出公式和中间步骤也能在后头出现小失误时为你争取到方法分。多练习类似 098 号真题风格的题目，熟练地在树形图、文氏图和代数法则之间自如转换。

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Published by TutorHao | Probability Revision Series | aleveler.com

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