Combined Events in Probability | 概率中的复合事件

📚 Combined Events in Probability | 概率中的复合事件

In Edexcel A-Level Mathematics, combined events are central to probability and statistics. A combined event occurs when two or more simple events are considered together, such as rolling a die and flipping a coin at the same time. Understanding how to combine probabilities correctly is essential for solving exam questions on chance, risk and decision-making.

在爱德思A-Level数学中，复合事件是概率与统计的核心内容。当两个或多个简单事件被同时考虑时，例如同时掷骰子和抛硬币，就构成复合事件。正确掌握概率的合并方法，是解答考试中关于机会、风险和决策问题的关键。

1. What Are Combined Events? | 什么是复合事件？

A simple event has a single outcome. A combined event involves two or more simple events. For example, ‘rolling a 6 on a die’ is simple, while ‘rolling a 6 and getting heads on a coin’ is combined.

简单事件只有一个结果。复合事件涉及两个或多个简单事件。例如，掷骰子得到6是简单事件，而掷骰子得到6且抛硬币得到正面是复合事件。

In Edexcel exams, combined events often appear in the Statistics sections of the A-Level specification. You need to identify whether events are mutually exclusive, independent or conditional before applying a rule.

在爱德思考试中，复合事件常出现在A-Level大纲的统计部分。你需要先判断事件是互斥、独立还是条件关系，再运用相应法则。

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

The sample space S is the set of all possible outcomes. For two combined events, the sample space can be shown as a list, a table or a grid. For instance, rolling a fair die and flipping a fair coin gives 6 × 2 = 12 equally likely outcomes.

样本空间S是所有可能结果的集合。对于两个复合事件，样本空间可以用列表、表格或网格表示。例如，掷一枚均匀骰子并抛一枚均匀硬币，共有6 × 2 = 12个等可能结果。

A two-way table is often the fastest way to display combined outcomes. Each cell represents one intersection of events, such as (die = 4, coin = tails).

双向表通常是展示复合结果最快的方法。每个单元格代表一个事件交集，例如（骰子=4，硬币=反面）。

The probability of any event A is P(A) = number of favourable outcomes / total number of outcomes in S, provided all outcomes are equally likely.

任何事件A的概率为P(A) = 有利结果数 / 样本空间S中的总结果数，前提是所有结果等可能。

P(A) = n(A) / n(S)

3. Mutually Exclusive Events | 互斥事件

Two events are mutually exclusive if they cannot happen at the same time. For example, when rolling a die, the events ‘rolling an odd number’ and ‘rolling a 6’ are mutually exclusive.

如果两个事件不能同时发生，则它们互斥。例如，掷骰子时，掷出奇数和掷出6是互斥事件。

For mutually exclusive events A and B, the probability of A or B occurring is simply the sum: P(A or B) = P(A) + P(B).

对于互斥事件A和B，A或B发生的概率就是两者之和：P(A或B) = P(A) + P(B)。

P(A ∪ B) = P(A) + P(B) for mutually exclusive A, B

A common exam error is using the addition rule for non-mutually exclusive events without subtracting the overlap.

常见的考试错误是：对非互斥事件使用加法法则时，没有减去重叠部分。

4. Independent Events | 独立事件

Two events are independent if the outcome of one does not affect the probability of the other. For example, flipping a coin and rolling a die are independent because the coin result does not change the die probabilities.

如果两个事件中一个的结果不影响另一个的概率，则它们独立。例如，抛硬币和掷骰子是独立的，因为硬币结果不改变骰子的概率。

For independent events A and B, the probability that both A and B happen is the product: P(A and B) = P(A) × P(B).

对于独立事件A和B，A和B同时发生的概率为乘积：P(A且B) = P(A) × P(B)。

P(A ∩ B) = P(A) × P(B) for independent A, B

Do not confuse independence with mutual exclusivity. Mutually exclusive events cannot both happen, so if one occurs, the other has zero probability. Independent events can both happen.

不要混淆独立与互斥。互斥事件不能同时发生，因此如果其中一个发生，另一个概率为零；而独立事件可以同时发生。

5. The Addition Rule | 加法法则

The general addition rule works for any two events A and B: P(A or B) = P(A) + P(B) − P(A and B).

一般加法法则适用于任意两个事件A和B：P(A或B) = P(A) + P(B) − P(A且B)。

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

The subtraction removes the double-counted intersection. This formula is especially useful for combined events that are not mutually exclusive, such as drawing a card that is either a heart or a king.

减去交集的目的是消除重复计数。这个公式对于非互斥的复合事件特别有用，例如抽到一张红桃或K的牌。

If A and B are mutually exclusive, P(A ∩ B) = 0, so the general rule reduces to the simple sum.

如果A和B互斥，则P(A ∩ B) = 0，因此一般法则退化为简单求和。

6. The Multiplication Rule | 乘法法则

The general multiplication rule for any two events is P(A and B) = P(A) × P(B | A), where P(B | A) is the conditional probability of B given A.

任意两个事件的一般乘法法则是P(A且B) = P(A) × P(B | A)，其中P(B | A)是给定A发生时B的条件概率。

P(A ∩ B) = P(A) × P(B | A)

When events are independent, P(B | A) = P(B), so the multiplication rule simplifies to P(A) × P(B). If events are dependent, you must use the conditional form.

当事件独立时，P(B | A) = P(B)，因此乘法法则简化为P(A) × P(B)。如果事件不独立，则必须使用条件形式。

This rule is essential for solving combined probability questions where events occur sequentially, such as selecting two balls from a bag without replacement.

该法则对于解决顺序发生的复合概率问题至关重要，例如从袋子中不放回地抽取两个球。

7. Tree Diagrams | 树状图

Tree diagrams are a powerful tool for combined events. Each branch shows a possible outcome and its probability. To find the probability of a complete path, multiply the probabilities along the branches.

树状图是处理复合事件的强大工具。每个分支显示一个可能结果及其概率。要求出一条完整路径的概率，将路径上的概率相乘。

For example, a bag contains 3 red and 2 blue balls. Two balls are drawn without replacement. The tree diagram has red and blue branches for each draw, with probabilities changing after the first draw.

例如，一个袋子装有3个红球和2个蓝球。不放回地抽取两个球。树状图在每次抽取时都有红球和蓝球分支，且第一次抽取后概率发生变化。

Tree diagrams are especially useful for dependent events, because they show how probabilities update at each stage.

树状图对于不独立事件特别有用，因为它显示了概率在每一阶段如何更新。

Always label the branches with probabilities that sum to 1 at each node. Check your tree diagram before answering.

始终在每个节点处标记分支概率，并使它们总和为1。在作答前检查树状图。

8. Venn Diagrams | 维恩图

Venn diagrams show the overlap between events. The circles represent events, and the overlapping region represents their intersection P(A ∩ B).

维恩图显示事件之间的重叠。圆圈表示事件，重叠区域表示它们的交集P(A ∩ B)。

Venn diagrams help you visualise the addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). They are also useful for finding conditional probabilities from given frequencies.

维恩图有助于直观理解加法法则：P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。它们也有助于从给定频数中求条件概率。

In exam questions, you may be given a Venn diagram with counts or probabilities inside each region. The total of all disjoint regions is 1.

在考试题中，可能会给出一张维恩图，各区域内标有频数或概率。所有互斥区域的总和为1。

Be careful to distinguish between ‘A and B’ (intersection) and ‘A or B’ (union). Venn diagrams make this clear.

注意区分A且B（交集）和A或B（并集）。维恩图能使这一点更清晰。

9. Conditional Probability | 条件概率

Conditional probability measures the chance of an event given that another event has already occurred. The formula is P(A | B) = P(A ∩ B) / P(B), provided P(B) > 0.

条件概率衡量在另一个事件已经发生的情况下某事件发生的可能性。公式为P(A | B) = P(A ∩ B) / P(B)，前提是P(B) > 0。

P(A | B) = P(A ∩ B) ÷ P(B)

Rearranging gives the multiplication rule: P(A ∩ B) = P(B) × P(A | B). This is often used in exam problems involving two or more stages.

移项得到乘法法则：P(A ∩ B) = P(B) × P(A | B)。这常用于涉及两个或多个阶段的考试问题。

Conditional probability is closely linked to tree diagrams and two-way tables. When a question says ‘given that’, it is asking for a conditional probability.

条件概率与树状图和双向表密切相关。当题目中出现 ‘given that’ 时，通常是在求条件概率。

10. Combined Events in Practice: Exam Technique | 复合事件的解题技巧

When solving combined events questions, always identify the events, state whether they are independent or mutually exclusive, choose the correct rule, and calculate step by step.

解答复合事件问题时，务必先识别事件，判断它们是独立还是互斥，选择正确的法则，并逐步计算。

A simple table can organise given probabilities and help you decide between addition and multiplication rules.

一个简单的表格可以整理已知概率，并帮助你决定使用加法还是乘法法则。

Let us look at a typical Edexcel-style question: A bag contains 5 green and 3 yellow sweets. A child takes two sweets at random without replacement. Find the probability that both sweets are green.

让我们看一道典型爱德思风格的问题：一个袋子装有5颗绿色和3颗黄色糖果。一个孩子随机取出两颗糖果且不放回。求两颗都是绿色的概率。

First, P(first green) = 5/8. After removing one green, 4 green and 3 yellow remain, so P(second green | first green) = 4/7. Multiply: P(both green) = (5/8) × (4/7) = 20/56 = 5/14.

首先，P(第一颗绿色) = 5/8。取走一颗绿色后，剩下4绿3黄，因此P(第二颗绿色 | 第一颗绿色) = 4/7。相乘：P(两颗绿色) = (5/8) × (4/7) = 20/56 = 5/14。

P(both green) = (5/8) × (4/7) = 5/14

Always check that your final probability is between 0 and 1, and simplify fractions where possible.

始终检查最终概率是否在0到1之间，并尽可能化简分数。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Combined Events in Probability | 概率中的复合事件