📚 Independent Events and Tree Diagrams | 独立事件与树状图
Probability is the branch of mathematics that quantifies how likely events are to occur. In many real-life situations, we need to calculate the probability of two or more events happening together. Two powerful tools help us do this with clarity and precision: the concept of independent events and the visual aid known as a tree diagram. In this revision guide, we will define independence, apply the multiplication rule, and learn how to construct and interpret tree diagrams step by step, exactly as required by the Edexcel IGCSE Mathematics syllabus.
概率是数学中用于量化事件发生可能性的分支。在现实生活中的许多情境里，我们需要计算两个或多个事件同时发生的概率。两个强大的工具能帮助我们清晰而精确地完成这一任务：独立事件的概念和被称为”树状图”的可视化工具。在本复习指南中，我们将定义独立性，运用乘法法则，并逐步学习如何构建和解读树状图，完全遵循 Edexcel IGCSE 数学教学大纲的要求。
1. What Are Independent Events? | 什么是独立事件？
Two events are said to be independent if the occurrence of one event does not affect the probability of the other event occurring. In other words, knowing that one event has happened gives you no information about whether the other event will happen.
如果两个事件中一个事件的发生不影响另一个事件发生的概率，则称这两个事件为相互独立事件。换句话说，知道其中一个事件已经发生，并不能为你提供关于另一个事件是否会发生的信息。
Classic examples of independent events include:
独立事件的典型例子包括：
- Tossing a fair coin and rolling a fair die – the coin result has no effect on the die result.
- 掷一枚均匀硬币和掷一颗均匀骰子——硬币的结果对骰子的结果没有任何影响。
- Drawing a ball from a bag and then replacing it before drawing again – the composition of the bag is restored, so the second draw is unaffected by the first.
- 从袋中抽出一个球，在再次抽取之前将其放回——袋中的组成得以恢复，因此第二次抽取不受第一次的影响。
- The weather on two different days (in most practical models) – rain today generally does not change the probability of rain a month later.
- 连续两天的天气（在大多数实际模型中）——今天下雨通常不会改变一个月后下雨的概率。
By contrast, drawing a ball from a bag without replacement creates dependent events, because the composition of the bag changes after the first draw. We will examine this difference in detail later in this guide.
相比之下，不放回地从袋中抽球会产生相关事件，因为第一次抽取后袋中球的组成发生了变化。我们将在本指南后面详细讨论这一区别。
2. The Multiplication Rule | 乘法法则
For two independent events A and B, the probability that both A and B occur is found by multiplying their individual probabilities together. This is known as the multiplication rule.
对于两个独立事件 A 和 B，A 和 B 同时发生的概率等于它们各自概率的乘积，这被称为乘法法则。
P(A and B) = P(A) × P(B)
The rule extends naturally to three or more independent events:
该法则可以自然地推广到三个或更多独立事件：
P(A and B and C) = P(A) × P(B) × P(C)
For example, the probability of rolling a 4 on a fair die and getting tails on a fair coin is:
例如，掷一枚均匀骰子得到 4 点并且掷一枚均匀硬币得到反面的概率为：
P(4 and T) = 1/6 × 1/2 = 1/12
Notice that the multiplication rule only applies when the events are independent. If the events are dependent, we must adjust the second probability to reflect the changed situation — we will return to this in Section 8.
请注意，乘法法则仅适用于独立事件。如果事件是相关的，我们必须调整第二个概率以反映变化后的情况——我们将在第 8 节中回到这一点。
3. Introduction to Tree Diagrams | 树状图入门
A tree diagram is a visual method for listing all possible outcomes of two or more events and calculating their probabilities. It is especially useful for multi-stage probability problems where each stage has several possible results.
树状图是一种列出两个或多个事件所有可能结果并计算其概率的可视化方法。它对于每个阶段有多种可能结果的多阶段概率问题尤其有用。
Key features of a tree diagram:
树状图的关键特征：
- Each branch represents a possible outcome of one event, labelled with its probability.
- 每条分支代表一个事件的一种可能结果，并标注其概率。
- Branches emerging from the same point must have probabilities that sum to 1, since they represent all possible results of that event.
- 从同一点出发的分支，其概率之和必须等于 1，因为它们代表该事件的所有可能结果。
- To find the probability of a combined outcome, multiply along the branches that lead to it.
- 要找到组合结果的概率，需要沿着通向该结果的分支相乘。
- If a question asks for the probability of multiple paths (e.g., “at least one head”), add the probabilities of the relevant paths.
- 如果题目要求多条路径的概率（例如”至少一次正面”），则需要相加相关路径的概率。
A tree diagram is not just a drawing — it is a structured calculation tool. Every path from the starting point to a terminal endpoint corresponds to one complete combination of outcomes, and the product of the probabilities along that path gives the probability of that combination.
树状图不仅仅是一个图形——它是一个结构化的计算工具。从起点到终端端点的每一条路径对应一个完整的结果组合，沿该路径的概率乘积即为该组合的概率。
4. How to Draw a Tree Diagram | 如何绘制树状图
Follow these steps to construct a tree diagram correctly and efficiently:
按照以下步骤正确高效地绘制树状图：
- Step 1: Identify the number of stages (events) in the problem. Draw a starting point on the left.
- 第1步：确定问题中的阶段（事件）数量。在左侧画一个起点。
- Step 2: For the first event, draw one branch for each possible outcome. Write the probability on each branch.
- 第2步：对于第一个事件，为每个可能的结果画一条分支。在每条分支上写出概率。
- Step 3: From the end of every branch, draw branches for the outcomes of the second event. Write their probabilities on the branches. Repeat for further events.
- 第3步：从每一条分支的末端，画出第二个事件结果的各条分支。在分支上标出概率。对后续事件重复此操作。
- Step 4: Multiply along each complete path to obtain the probability of that final outcome. Write the product at the end of the path.
- 第4步：沿着每条完整路径相乘，得到该最终结果的概率。在路径末端写出乘积。
- Step 5: Check that the sum of all final probabilities equals 1. This verifies that no outcomes were missed.
- 第5步：检查所有最终概率之和是否等于 1。这样可以验证没有遗漏任何结果。
Let us now apply these steps to concrete examples, starting with the simplest case.
现在让我们将这些步骤应用于具体的例子，从最简单的案例开始。
5. Worked Example: Tossing a Coin Twice | 例题演练：掷两次硬币
Consider tossing a fair coin twice. Since the outcome of the first toss does not affect the second, the two tosses are independent events.
考虑掷两次均匀硬币。由于第一次掷的结果不会影响第二次，两次掷硬币是独立事件。
P(H) = 1/2, P(T) = 1/2
The tree diagram has two stages. From the start, two branches: H (1/2) and T (1/2). From the end of each, two more branches for the second toss, each with probability 1/2.
树状图包含两个阶段。从起点分出两条分支：H（正面，1/2）和 T（反面，1/2）。从每条分支末端再分出两条分支表示第二次掷的结果，每条概率均为 1/2。
Multiplying along each path gives the following results:
沿每条路径相乘得到以下结果：
| Outcome | Probability |
| HH | 1/2 × 1/2 = 1/4 |
| HT | 1/2 × 1/2 = 1/4 |
| TH | 1/2 × 1/2 = 1/4 |
| TT | 1/2 × 1/2 = 1/4 |
Notice that 1/4 + 1/4 + 1/4 + 1/4 = 1, confirming the diagram is complete. Each outcome is equally likely because the coin is fair and the tosses are independent.
注意 1/4 + 1/4 + 1/4 + 1/4 = 1，确认树状图是完整的。由于硬币均匀且各次掷相互独立，每个结果等可能发生。
Suppose the question asks: “What is the probability of getting exactly one head?” We identify the paths HT and TH and add their probabilities:
假设题目问：”恰好出现一次正面的概率是多少？”我们确定路径 HT 和 TH，并将它们的概率相加：
P(exactly one head) = 1/4 + 1/4 = 1/2
This demonstrates the two key operations on a tree diagram: multiply along branches, add across paths.
这展示了树状图上的两个关键操作：沿分支相乘，跨路径相加。
6. Worked Example: Drawing Balls with Replacement | 例题演练：有放回地抽球
A bag contains 5 red balls and 3 blue balls. A ball is drawn at random, its colour is recorded, and the ball is replaced before a second ball is drawn. Find the probability that both balls are red.
一个袋中装有 5 个红球和 3 个蓝球。随机抽取一个球，记录其颜色，在抽取第二个球之前放回该球。求两个球都是红色的概率。
Because the ball is replaced, the two draws are independent. The probabilities remain constant across both draws:
因为球被放回，两次抽取是独立的。两次抽取的概率保持不变：
P(R) = 5/8, P(B) = 3/8
The tree diagram has two stages, each with branches R (5/8) and B (3/8). Note that the probabilities on the second-stage branches are identical to the first-stage branches, precisely because of the replacement.
树状图包含两个阶段，每个阶段分支为 R（红，5/8）和 B（蓝，3/8）。请注意，第二阶段分支上的概率与第一阶段分支完全相同，这正是因为有放回。
The four possible outcomes and their probabilities are:
四种可能的结果及其概率为：
| Outcome | Calculation | Probability |
| RR | 5/8 × 5/8 | 25/64 |
| RB | 5/8 × 3/8 | 更多咨询请联系16621398022（同微信）
CommentsMore posts |
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导Cancel reply