📚 Tree Diagrams for A-Level Edexcel Maths | A-Level Edexcel 数学：树状图概率核心方法
Tree diagrams are a visual tool used in A-Level Edexcel Mathematics to organise the outcomes of multi-stage probability experiments. They allow you to multiply probabilities along branches and add probabilities between mutually exclusive paths, linking directly to the multiplication and addition rules required by the Edexcel specification.
树状图是 A-Level Edexcel 数学中用于整理多阶段概率试验结果的可视化工具。它们允许你沿分支相乘概率，并在互斥路径之间相加概率，这与 Edexcel 考试大纲要求的乘法和加法规则直接对应。
1. What is a Tree Diagram? | 什么是树状图？
A tree diagram is a branching diagram that shows every possible outcome of a sequence of events. Each branch represents one outcome, and the branch label is usually its probability.
树状图是一种分支图，展示一系列事件的每一种可能结果。每个分支代表一个结果，分支标签通常为其概率。
In Edexcel exam questions, tree diagrams are most common when two or more items are selected from a bag, when a coin and die are thrown together, or when a test has dependent stages.
在 Edexcel 考题中，树状图最常用于从袋子中选取两个或多个物品、同时抛硬币和掷骰子，或某个试验具有依赖阶段的情况。
2. Key Components: Branches, Nodes, Outcomes | 关键组成部分：分支、节点、结果
A node is a point where the tree splits. Branches leaving a node must be mutually exclusive and their probabilities must sum to 1.
节点是树的分叉点。离开节点的分支必须互斥，其概率之和必须为 1。
For example, if a bag contains 3 red and 7 blue counters, the first node has two branches: red with probability 3/10 and blue with probability 7/10.
例如，若一个袋子中有 3 个红色和 7 个蓝色计数器，则第一个节点有两条分支：红色概率为 3/10，蓝色概率为 7/10。
Each final outcome is found at the end of a complete path from the start to the right-hand side of the diagram.
每个最终结果都位于从起点到图右侧完整路径的末端。
3. The Multiplication Rule on Branches | 分支上的乘法规则
To find the probability of a specific path from the start to a final outcome, multiply the probabilities written on the branches along that path.
要找到从起点到最终结果的某条路径的概率，将沿该路径的分支上写出的概率相乘。
For two events A and B, the probability of both A and B occurring is given by the general multiplication rule:
对于两个事件 A 和 B，A 和 B 同时发生的概率由一般乘法规则给出：
P(A ∩ B) = P(A) × P(B | A)
For independent events this simplifies to P(A ∩ B) = P(A) × P(B).
对于独立事件，这简化为 P(A ∩ B) = P(A) × P(B)。
On a tree diagram, this means you simply multiply the probabilities written on the branches of a single path.
在树状图上，这意味着你只需将单一路径上分支所写的概率相乘即可。
4. The Addition Rule for Multiple Paths | 多条路径的加法规则
If a final outcome can occur by two or more different paths, add the probabilities of those separate paths. This works because the paths are mutually exclusive.
如果某个最终结果可以通过两个或多个不同路径发生，则将这些单独路径的概率相加。这是可行的，因为这些路径是互斥的。
For example, the probability of getting exactly one head in two coin flips is P(H then T) + P(T then H) = 1/4 + 1/4 = 1/2.
例如，两次抛硬币恰好得到一次正面的概率为 P(先正后反) + P(先反后正) = 1/4 + 1/4 = 1/2。
This is the addition rule applied to mutually exclusive final outcomes that share a common description, such as exactly one yellow sweet.
这是将加法规则应用于具有共同描述的互斥最终结果，例如恰好一个黄色糖果。
5. Replacement and Independence | 放回与独立性
When an item is replaced after selection, the probabilities on the second set of branches stay the same as the first. This indicates independence.
当一个物品在选择后被放回时，第二组分支上的概率与第一组保持相同。这表明事件是独立的。
When items are not replaced, the second-stage probabilities depend on the first outcome, so the tree labels change on different second branches.
当物品没有被放回时，第二阶段概率取决于第一阶段结果，因此树状图在不同第二阶段分支上的标签会发生变化。
The table below shows the branch probabilities for a bag containing 4 red and 5 blue counters when two counters are drawn without replacement.
下表显示了一个装有 4 个红色和 5 个蓝色计数器的袋子，在不放回地抽取两个计数器时的分支概率。
| First draw | Second draw branch | Probability |
| Red (4/9) | Red after red | 3/8 |
| Red (4/9) | Blue after red | 5/8 |
| Blue (5/9) | Red after blue | 4/8 = 1/2 |
| Blue (5/9) | Blue after blue | 4/8 = 1/2 |
6. Conditional Probability from a Tree | 树状图中的条件概率
Tree diagrams are especially useful for conditional probability. If you are given that a later event has occurred, you focus only on the paths where that event happens.
树状图对条件概率尤其有用。如果已知后一个事件已经发生，你只需关注该事件发生的那些路径。
The probability of an earlier event A given a later event B is:
已知后一事件 B 发生时前一事件 A 发生的概率为：
P(A | B) = P(A ∩ B) / P(B)
On a tree diagram, P(B) is the sum of all path probabilities ending in B, and the numerator is the relevant single path probability.
在树状图上，P(B) 是所有以 B 结尾的路径概率之和，分子是相关单一路径的概率。
In sampling without replacement, this method automatically accounts for the changing denominators at later stages.
在不放回抽样中，这种方法会自动处理后续阶段中分母的变化。
7. Bayes’ Theorem and Reverse Probability | 贝叶斯定理与逆概率
Many A-Level Edexcel exam questions ask a reverse probability: you know the outcome, but you need the probability that it came from a particular earlier branch. This is an application of Bayes’ theorem.
许多 A-Level Edexcel 考题会问逆概率：你已知结果，但需要求它来自某个特定早期分支的概率。这是贝叶斯定理的应用。
You can answer these questions directly from the tree without memorising a separate formula: identify the target path and divide its probability by the total probability of reaching the known outcome.
你可以直接从树状图回答这些问题，而无需记忆单独公式：确定目标路径，并将其概率除以到达已知结果的总概率。
Example: A factory has machine A producing 60% of items and machine B producing 40%. Machine A has a 2% defect rate, B has 5%. If a randomly selected item is defective, what is the probability it came from machine A?
示例：某工厂机器 A 生产 60% 的产品，机器 B 生产 40%。机器 A 的次品率为 2%，机器 B 为 5%。如果随机抽到一个次品，它来自机器 A 的概率是多少？
The tree gives P(A and defective) = 0.6 × 0.02 = 0.012, and P(B and defective) = 0.4 × 0.05 = 0.02. So P(A | defective) = 0.012 / (0.012 + 0.02) = 0.012 / 0.032 = 3/8.
树状图给出 P(A 且次品) = 0.6 × 0.02 = 0.012，P(B 且次品) = 0.4 × 0.05 = 0.02。因此 P(A | 次品) = 0.012 / (0.012 + 0.02) = 0.012 / 0.032 = 3/8。
8. Tree Diagrams with Three or More Stages | 三个或更多阶段的树状图
A tree diagram can be extended to three or more events, but the number of final branches grows quickly, so you should only draw branches that are relevant to the question.
树状图可以扩展到三个或更多事件，但最终分支的数量会迅速增长，因此你应该只画与问题相关的分支。
For three events A, B and C, the probability of one path is P(A) × P(B | A) × P(C | A ∩ B).
对于三个事件 A、B 和 C，一条路径的概率为 P(A) × P(B | A) × P(C | A ∩ B)。
Edexcel questions rarely require a full three-stage tree; more often they ask you to extend one branch or use a labelled partial tree.
Edexcel 考题很少要求完整的三阶段树；更常见的是要求你延伸一个分支或使用已标记的部分树。
For longer sequences, a clear partial tree with only the needed paths is faster and reduces the chance of arithmetic errors.
对于更长的序列，只画出所需路径的部分树会更快速，并减少算术错误的机会。
9. Common Mistakes and How to Avoid Them | 常见错误及如何避免
One common mistake is forgetting that branch probabilities at a node must add up to 1. Always check this before calculating.
一个常见错误是忘记节点上的分支概率必须加起来为 1。在计算之前一定要检查这一点。
Another common mistake is mixing up the multiplication and addition rules: multiply along a path, add across alternative paths.
另一个常见错误是混淆乘法和加法规则：沿路径相乘，跨备选路径相加。
In conditional probability, students often write P(A | B) when they actually computed P(A ∩ B). Make sure to divide by the probability of the given event.
在条件概率中，学生经常在实际上计算的是 P(A ∩ B) 时写成 P(A | B)。务必除以给定事件的概率。
When sampling without replacement, do not recycle first-stage probabilities on the second-stage branches. Update the denominator and numerator after each selection.
在不放回抽样时，不要在第二阶段分支上重复使用第一阶段的概率。每次选择后都要更新分母和分子。
10. Worked
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