📚 Probability: Tree Diagrams for Independent Events | 概率：独立事件的树形图
Tree diagrams are a powerful visual tool in probability that help you map out all possible outcomes of multi–stage experiments. When events are independent – meaning the outcome of one event does not affect the outcome of another – a tree diagram makes it straightforward to calculate combined probabilities. This revision guide, aligned with the Cambridge KS3 Mathematics syllabus, will walk you through the construction and interpretation of tree diagrams for independent events, using clear examples and step–by–step reasoning.
树形图是概率中一种强大的可视工具，能帮助你列出多阶段实验的所有可能结果。当事件相互独立时——即一个事件的结果不影响另一个事件的结果——树形图让计算组合概率变得简单明了。本复习指南与剑桥 KS3 数学教学大纲紧密衔接，将通过清晰的示例和分步推理，带你掌握独立事件树形图的绘制与解读。
1. What is a Tree Diagram? | 什么是树形图？
A tree diagram is a branching chart that shows every possible outcome of a sequence of events. Each branch represents a possible outcome at a certain stage, and probabilities are written along the branches. The diagram starts from a single point on the left and branches out to the right, making it easy to see all pathways.
树形图是一种分支图表，展示了一系列事件的所有可能结果。每条分支代表某一阶段的可能结果，概率标注在分支上。图表从左端一个点开始，向右分支展开，使所有路径一目了然。
2. Probability of a Single Event | 单一事件的概率
Before combining events, recall that the probability of a single event is a number between 0 and 1 that measures how likely it is to happen. It can be written as a fraction, decimal, or percentage. For example, when flipping a fair coin, the probability of getting heads is 1/2.
在合并事件之前，请回顾单一事件的概率是一个介于 0 和 1 之间的数，衡量事件发生的可能性。它可以用分数、小数或百分数表示。例如，抛一枚公平硬币时，得到正面的概率为 1/2。
3. Independent Events and the Multiplication Rule | 独立事件与乘法法则
P(A) × P(B). Tree diagrams graphically apply this multiplication along each complete path.
如果一个事件的发生不改变另一个事件的概率，则这两个事件相互独立。对于独立事件，两个事件同时发生的概率等于各自概率的乘积：P(A 与 B) = P(A) × P(B)。树形图以图形方式将这一乘法应用于每一条完整路径。
4. Drawing the First Branch | 画出第一个分支
Begin with a point on the left. Draw one branch for each outcome of the first event. Label each branch with the outcome and write its probability along the branch. The sum of probabilities on all first–level branches must equal 1. For example, when spinning a fair 3–colour spinner (red, blue, green), draw three branches with probabilities 1/3 each.
从左端一个点开始。为第一个事件的每个结果画一条分支。在每条分支上标注结果，并沿分支写下概率。第一层级所有分支的概率之和必须等于 1。例如，转动一个公平的三种颜色转盘（红、蓝、绿）时，画出三条分支，每条概率为 1/3。
5. Adding the Second Branch | 添加第二个分支
At the end of each first–stage branch, draw a new set of branches for the second event. Because the events are independent, the second–stage probabilities remain the same as they would be for that event on its own. If the second event is flipping a coin, each endpoint will have two branches: heads and tails, each with probability 1/2.
在每个第一阶段分支的末端，为第二个事件画出一组新分支。由于事件独立，第二阶段概率保持与该事件单独发生时相同。如果第二个事件是抛一枚硬币，每个端点将有两个分支：正面和反面，各自的概率为 1/2。
6. Calculating Probabilities Along the Branches | 沿分支计算概率
(1/3) and the second is coin heads (1/2), the combined probability is (1/3) × (1/2) = 1/6.
要找出某一特定结果组合的概率，将通往该最终结果的各分支概率相乘。将计算出的概率写在路径末端。例如，若第一个事件是转盘红色 (1/3)，第二个是硬币正面 (1/2)，则组合概率为 (1/3) × (1/2) = 1/6。
7. Sum of Probabilities on Each Level | 每个层级概率之和
Always check your tree diagram by verifying that the probabilities of all final outcomes sum to 1. Additionally, the probabilities on branches emanating from a single point should sum to 1. This consistency check helps catch errors early.
务必通过验证所有最终结果的概率之和是否为 1 来检查你的树形图。此外，从同一点发出的各分支概率之和应为 1。这种一致性检查有助于及早发现错误。
8. Worked Example: Flipping a Coin and Rolling a Die | 例题：抛硬币和掷骰子
Consider flipping a fair coin and then rolling a fair six– es 1,2,3,4,5,6, each with probability 1/6. The final outcomes are pairs like (H,1), (H,2), …, (T,6). The probability of, say, heads and a 4 is (1/2)×(1/6)=1/12.
考虑先抛一枚公平硬币，然后掷一个公平六面骰子。第一阶段结果有正面 H 和反面 T，概率各为 1/2。第二阶段有六个结果 1,2,3,4,5,6，概率各为 1/6。最终结果为 (H,1)、(H,2)、……、(T,6) 这样的组合。例如，正面且点数为 4 的概率是 (1/2)×(1/6)=1/12。
9. Worked Example: Coloured Spinners | 例题：彩色转盘
e branches from each. The probability of both spinners landing on red is (1/4)×(1/3)=1/12.
想象转动两个独立的转盘。转盘 A 有 4 个相等扇形：红、黄、绿、蓝（各 1/4）。转盘 B 有 3 个相等扇形：红、绿、紫（各 1/3）。画出树形图：第一阶段四条分支，第二阶段每条末端分出三条分支。两个转盘都停在红色的概率是 (1/4)×(1/3)=1/12。
10. Using Tree Diagrams for 'And' and 'Or' Rules | 使用树形图的“且”与“或”规则
Tree diagrams naturally handle 'and' situations through multiplication along a path. To solve 'or' situations, add the probabilities of the separate final outcomes that satisfy the condition. For example, the probability of getting exactly one head in two coin flips is P(HT) + P(TH), which can be read from the final column of a two–stage tree.
树形图通过路径上的乘法自然处理“且”的情况。对于“或”的情况，将满足条件的各个最终结果的概率相加。例如，两次抛硬币恰好得到一次正面的概率是 P(HT) + P(TH)，这可以从两阶段树形图的最后一列读出。
11. Common Mistakes to Avoid | 常见错误避免
One common mistake is forgetting that the second–stage probabilities may change if events are not independent. However, for independent events this does not happen. Another error is writing probabilities that do not sum to 1 on branches from the same point. Always label branches clearly and double–check your multiplications.
一个常见错误是忘记如果事件不独立，第二阶段概率可能会改变。但对于独立事件则不会发生。另一个错误是在同一点发出的分支上写下概率之和不等于 1。务必清晰标注分支，并仔细核对乘法计算。
12. Practice and Summary | 练习与小结
Tree diagrams for independent events provide a systematic way to organise outcomes and calculate combined probabilities. The key steps are: list all outcomes for each stage, draw branches, label with probabilities, multiply along paths for 'and', and add across paths for 'or'. Regular practice with spinners, dice, and coin problems will build confidence for assessments.
独立事件的树形图为组织结果和计算组合概率提供了一种系统方法。关键步骤是：列出每个阶段的所有结果，画出分支，标注概率，沿路径相乘求“且”，跨路径相加求“或”。通过转盘、骰子和硬币问题的经常练习，可以为测评建立信心。
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