Cambridge KS3 Maths – Page 179: Probability and Tree Diagrams | 剑桥初中数学 – 第179页：概率与树状图

📚 Cambridge KS3 Maths – Page 179: Probability and Tree Diagrams | 剑桥初中数学 – 第179页：概率与树状图

Probability helps us measure how likely an event is to happen. On page 179 of the Cambridge KS3 Mathematics course, you will explore how to calculate probabilities, use probability scales, and draw tree diagrams to represent multiple outcomes clearly. Tree diagrams are powerful tools for solving combined event problems, such as flipping two coins or pulling coloured socks from a drawer. Mastering these skills will build a strong foundation for IGCSE and beyond.

概率帮助我们衡量事件发生的可能性。在剑桥初中数学课程的第 179 页中，你将学习如何计算概率、使用概率尺度以及绘制树状图来清晰地表示多重结果。树状图是解决组合事件问题（例如抛两枚硬币或从抽屉里取袜子）的强大工具。掌握这些技能将为 IGCSE 及更高阶段的学习打下坚实基础。

1. What Is Probability? | 什么是概率？

Probability is a number between 0 and 1 that describes the chance of an event occurring. A probability of 0 means the event is impossible, while a probability of 1 means it is certain. Most events have probabilities somewhere in between.

概率是一个介于 0 和 1 之间的数字，描述某个事件发生的可能性。概率为 0 表示事件不可能发生，概率为 1 表示事件必然发生。大多数事件的概率介乎两者之间。

We can write probability as a fraction, decimal, or percentage. For example, when flipping a fair coin, the probability of getting heads is 1/2, 0.5, or 50%. The notation P(heads) = 1/2 is commonly used.

我们可以用分数、小数或百分数来表示概率。例如，抛一枚公平硬币时，得到正面的概率是 1/2、0.5 或 50%。常用记法为 P(正面) = 1/2。

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

The sample space is the set of all possible outcomes of an experiment. For a single dice roll, the sample space is {1, 2, 3, 4, 5, 6}. Each individual result is called an outcome. Understanding the sample space is the first step in calculating any probability.

样本空间是某个实验所有可能结果的集合。对于掷一个骰子，样本空间为 {1, 2, 3, 4, 5, 6}。每一个单独的结果称为一个“结果”。理解样本空间是计算任何概率的第一步。

For combined events, such as rolling two dice, you can create a sample space diagram (grid) to list all 36 possible pairs. This organized approach prevents missing outcomes and ensures accurate probability calculations.

对于组合事件，例如掷两个骰子，你可以绘制样本空间图（网格）列出全部 36 个可能的数对。这种有条理的方法可以防止遗漏结果，确保概率计算准确。

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3. The Probability Scale | 概率尺度

The probability scale is a visual line from 0 to 1. Marking events on this line helps you compare their likelihoods. Words like ‘impossible’, ‘unlikely’, ‘evens’, ‘likely’, and ‘certain’ correspond to ranges on the scale.

概率尺度是一条从 0 到 1 的视觉化线段。将事件标注在该线上有助于比较它们的可能性大小。“不可能”、“不太可能”、“等可能”、“很可能”、“必然”等词语对应尺度上的不同区间。

For instance, the chance of the sun rising tomorrow is almost 1 (certain), while the chance of rolling a 7 on a standard dice is 0 (impossible). A 50% chance lies exactly in the middle, often called an even chance.

例如，明天太阳升起的概率几乎为 1（必然），而掷标准骰子得到 7 的概率为 0（不可能）。50% 的概率恰好位于中间，常被称为等可能的概率。

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4. Calculating Basic Probabilities | 计算基本概率

The basic probability formula is: P(event) = number of favourable outcomes / total number of possible outcomes. This only applies when all outcomes are equally likely. It is essential to count outcomes systematically.

基本的概率公式为：P(事件) = 有利结果的数量 / 所有可能结果的总数。这仅在所有结果等可能时成立。系统性地清点结果是十分重要的。

In a bag with 3 red pens and 5 blue pens, the probability of picking a red pen at random is 3 / 8. The total outcomes are 8, and favourable ones are 3. Always simplify fractions if possible: 3/8 is already in simplest form.

在一个装有 3 支红笔和 5 支蓝笔的袋子里，随机抽到红笔的概率是 3/8。总结果数为 8，有利结果数为 3。如果可能，记得化简分数：3/8 已是最简形式。

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5. Complementary Events | 互补事件

The complement of an event A is the event that A does not happen, written as A’ or not A. The probabilities of an event and its complement always add up to 1: P(A) + P(not A) = 1. This rule saves time when calculating ‘at least one’ style problems.

事件 A 的补事件是指 A 不发生的那个事件，记作 A’ 或非 A。任一事件与其补事件的概率之和总是 1：P(A) + P(非 A) = 1。在计算“至少一个”这类问题时，这一规则能节省时间。

If the probability of rain tomorrow is 0.3, then the probability of no rain is 1 – 0.3 = 0.7. Using complements often avoids adding many separate probabilities.

如果明天下雨的概率是 0.3，那么不下雨的概率就是 1 – 0.3 = 0.7。利用补事件常常可以避免将多个单独概率相加。

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6. Introducing Tree Diagrams | 树状图入门

A tree diagram is a branching structure that shows all possible outcomes of a sequence of events. Each branch represents an outcome and is labelled with its probability. Tree diagrams are especially useful for two or more stages.

树状图是一种分支结构，展示一连串事件的所有可能结果。每一分支代表一个结果，并标有其概率。树状图对于涉及两个或两个以上阶段的问题尤其有用。

To draw a tree diagram, start with a single point, then draw branches for each possible outcome of the first event. From the end of each of those branches, draw branches for the second event, and continue if there are more stages.

绘制树状图时，从一个点出发，为第一个事件的每种可能结果画出分支。再从这些分支末端，为第二个事件画出分支；如有更多阶段则继续延伸。

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7. Tree Diagram for Independent Events | 独立事件的树状图

Two events are independent if the outcome of one does not affect the outcome of the other. Flipping a coin and rolling a dice are independent events. On a tree diagram, the probabilities on the second set of branches remain the same regardless of the first outcome.

如果一件事的结果不影响另一件事的结果，那么这两个事件就是独立的。抛硬币和掷骰子是独立事件。在树状图上，第二组分支的概率不会因第一组结果的不同而改变。

For a coin flip followed by a dice roll, the coin has branches H (1/2) and T (1/2). From each, draw six dice branches, each with probability 1/6. The probability of any combined outcome, like H and 5, is 1/2 × 1/6 = 1/12.

对于先抛硬币再掷骰子的情况，硬币有两支分支 H（1/2）和 T（1/2）。从每支再画出六支骰子分支，每支概率 1/6。任何组合结果（例如 H 和 5）的概率为 1/2 × 1/6 = 1/12。

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8. Tree Diagram for Dependent Events | 相关事件的树状图

Events are dependent when the outcome of the first event changes the probability of the second event. Picking items from a bag without replacing them is a classic example. The probabilities on the second branches must be updated according to what has already happened.

如果第一件事的结果改变了第二件事的概率，那么这些事件是相关的。不放回地从袋子中抽取物品就是一个典型的例子。第二组分支的概率必须根据已经发生的情况进行调整。

Imagine a bag with 2 red and 3 green marbles. If you pick a red first and do not replace it, the bag now has 1 red and 3 green left, so the probability of red on the second pick becomes 1/4, not 2/5.

假设一个袋子里有 2 个红色弹珠和 3 个绿色弹珠。如果你先抽到一个红色弹珠且不放回，袋子里就剩下 1 红 3 绿，因此第二次抽到红色弹珠的概率变为 1/4，而不是 2/5。

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9. Using Tree Diagrams to Find Probabilities | 利用树状图求概率

To find the probability of a sequence of outcomes, multiply the probabilities along the path. To find the probability of an event that can happen in more than one way, calculate each path probability and then add them together. This is the ‘multiply along, add across’ rule.

要找到一系列结果的概率，将路径上的各概率相乘。若一个事件可以通过多种路径发生，则分别计算每条路径的概率，然后将它们相加。这就是“沿路径相乘，路径间相加”的规则。

For example, if you flip two coins, the probability of getting exactly one head is found by looking at the paths HT and TH. Each path probability is 1/2 × 1/2 = 1/4, so total P(exactly one head) = 1/4 + 1/4 = 1/2.

例如，抛两枚硬币时，恰好出现一次正面的概率通过观察 HT 和 TH 两条路径求得。每条路径概率为 1/2 × 1/2 = 1/4，因此总 P(恰好一次正面) = 1/4 + 1/4 = 1/2。

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10. Common Mistakes and Tips | 常见错误与提示

Students often forget to update probabilities for dependent events, or they add all the endpoint probabilities instead of multiplying along branches. Always double‑check whether the problem involves replacement or no replacement.

学生常忘记为相关事件更新概率，或者一味地将所有端点概率全部相加，而不是先沿分支相乘。务必仔细检查题目中是否涉及“放回”还是“不放回”。

Another error is assuming outcomes are equally likely when they are not. If a spinner is biased, the sections do not have equal probabilities. Read the question text carefully to identify given probabilities. Drawing a tree diagram clearly and labelling every branch with a fraction or decimal is the best safeguard.

另一个常见错误是在结果并非等可能时却假定它们等可能。如果转盘是偏心的，各区域就不具有等概率。仔细阅读题目信息以识别给定的概率。清晰地画出树状图，并用分数或小数标注每一分支，是最佳的防错手段。

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11. Practice Question Walkthrough | 典型例题讲解

Question: A box contains 4 black pens and 2 green pens. Two pens are taken out at random without replacement. Draw a tree diagram and find the probability that at least one of the pens is black.

题目：一个盒子里有 4 支黑笔和 2 支绿笔。随机取出两支且不放回。画出树状图，并求至少有一支是黑笔的概率。

Step 1: First pick – P(black) = 4/6 = 2/3, P(green) = 2/6 = 1/3. Step 2: If black was taken first, remaining pens: 3 black, 2 green – so P(black second) = 3/5, P(green second) = 2/5. If green was taken first, remaining: 4 black, 1 green – so P(black second) = 4/5, P(green second) = 1/5.

步骤 1：第一次抽取 – P(黑) = 4/6 = 2/3，P(绿) = 2/6 = 1/3。步骤 2：若第一次抽到黑笔，剩余 3 黑 2 绿 – 所以 P(第二次黑) = 3/5，P(第二次绿) = 2/5。若第一次抽到绿笔，剩余 4 黑 1 绿 – 所以 P(第二次黑) = 4/5，P(第二次绿) = 1/5。

Now calculate probability of at least one black: this is the complement of getting no black (i.e., both green). Path for both green: 1/3 × 1/5 = 1/15. Therefore P(at least one black) = 1 – 1/15 = 14/15.

现在计算至少一支黑笔的概率：这等价于未抽到黑笔（即两支全绿）的补事件。两支全绿的路径：1/3 × 1/5 = 1/15。因此 P(至少一支黑) = 1 – 1/15 = 14/15。

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12. Summary and Beyond | 总结与拓展

Probability and tree diagrams are essential tools for organising outcomes logically. Always remember to set up your sample space, label branches with correct probabilities, and apply the multiply‑and‑add rules carefully. These techniques are directly tested in Cambridge Checkpoint and will be expanded in IGCSE topics such as conditional probability and Venn diagrams.

概率与树状图是逻辑组织结果的重要工具。务必记住建立样本空间，用正确的概率标注分支，并谨慎应用相乘与相加规则。这些技巧会在剑桥 Checkpoint 考试中直接考查，并将在 IGCSE 中有条件概率、文氏图等主题中进一步拓展。

Keep practising with different structures – three branches, biased dice, or picking sweets from a bag. The more you draw and label, the more intuitive these diagrams become. Page 179 is just the beginning; the logic you learn here will support statistical reasoning for years to come.

请用不同的结构多加练习——三支分支、偏心骰子、或从袋中取糖果等。画得越多、标注得越多，这些图就会越直观。第 179 页仅仅是个开始；你在这里学到的逻辑思维将支持你未来多年的统计推理。

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