Construction and Application of Probability Tree Diagrams | 概率树形图的构建与应用

📚 Construction and Application of Probability Tree Diagrams | 概率树形图的构建与应用

A probability tree diagram is a visual method for breaking down a multi-stage experiment into a sequence of branches. Each branch carries a probability, and following the branches from the root to the leaves allows us to compute the probability of combined outcomes with clarity and precision.

概率树形图是一种将多阶段试验分解为连续分支的可视化方法。每条分支都带有概率，从根节点沿着分支走向叶节点，我们就可以清晰、精确地计算复合结果的概率。

1. What Is a Probability Tree Diagram? | 什么是概率树形图？

A probability tree diagram consists of nodes (branch points) and branches (paths). The first node represents the start of the experiment, and each set of branches from a node shows all possible outcomes of one stage.

概率树形图由节点（分支点）和分支（路径）组成。起始节点代表试验的开始，而从同一节点出发的每组分支显示某一阶段的所有可能结果。

Each branch is labelled with its probability. The final value at the end of a path is the probability of that particular sequence of outcomes, and these end values are sometimes called leaf probabilities.

每条分支都标注其概率。一条路径末端的数值就是该特定结果序列的概率，这些末端值有时被称为“叶概率”。

Tree diagrams are especially useful because they organise information clearly, making it easier to apply the multiplication and addition rules of probability.

树形图特别实用，因为它能清晰地组织信息，使我们更容易应用概率的乘法法则和加法法则。

2. Basic Rules on a Probability Tree | 树形图上的基本概率规则

There are three core rules that govern every probability tree diagram. Knowing these rules helps you both build and check your tree.

每一棵概率树形图都遵循三条核心法则。掌握这些法则有助于你构建和检查树形图。

Rule 1: The probabilities on the branches leaving any single node must sum to 1.

规则一：从任意一个节点出发的所有分支上的概率之和必须等于 1。
Rule 2: To find the probability of a particular path, multiply the probabilities along that path.

规则二：要求某一条特定路径的概率，就把该路径上各分支的概率相乘。
Rule 3: To find the probability of a combined event, add the probabilities of the mutually exclusive paths that satisfy the event.

规则三：要求某个复合事件的概率，就把所有满足该事件的互斥路径的概率相加。

These three rules mirror the fundamental laws of probability: complementarity, the multiplication rule, and the addition rule for mutually exclusive events.

这三条法则分别对应概率的基本定律：互补性、乘法法则，以及互斥事件的加法法则。

3. Constructing a Tree Diagram Step by Step | 分步构建树形图

Building a tree diagram is a systematic process. Follow these steps whenever you encounter a multi-stage probability question.

构建树形图是一个系统化的过程。当你遇到多阶段概率问题时，只需按以下步骤操作即可。

Step 1: Identify the stages of the experiment. Count how many times the action is performed; each stage becomes a new column of branches.

第一步：确定试验的阶段数。数一数该操作进行了多少次；每个阶段对应一层新的分支。
Step 2: At the first node, write all possible outcomes of the first stage on separate branches, with their probabilities.

第二步：在起始节点处，将第一阶段所有可能的结果分别写在不同的分支上，并标注各自的概率。
Step 3: From the end of every branch, draw the outcomes of the second stage. If probabilities change after the first event, use conditional probabilities.

第三步：从每条分支的末端继续画出第二阶段的结果。如果第一阶段之后概率发生了变化，就要使用条件概率。
Step 4: Multiply along each complete path to find the probability of each final outcome. Check that all leaf probabilities sum to 1.

第四步：沿每一条完整路径相乘，得到每个最终结果的概率。检查所有叶概率之和是否为 1。

Always label every branch clearly. A well-drawn tree is half the solution.

务必清楚标注每条分支。一棵画得好的树形图，等于已经解决了一半问题。

4. Independent Events | 独立事件

When two events are independent, the probability of the second event is unaffected by the outcome of the first. In this case, the same probabilities are used at every stage.

当两个事件相互独立时，第二个事件的概率不受第一个事件结果的影响。在这种情况下，每一阶段都使用相同的概率。

For example, tossing a fair coin twice: at the first node, P(Head) = ½ and P(Tail) = ½; at the second stage, the branches again carry P(Head) = ½ and P(Tail) = ½.

例如，抛一枚均匀硬币两次：在起始节点处，P(正面) = ½，P(反面) = ½；在第二阶段，分支上依然是 P(正面) = ½，P(反面) = ½。

P(Head then Tail) = ½ × ½ = ¼

P(正面然后反面) = ½ × ½ = ¼
P(Tail then Tail) = ½ × ½ = ¼

P(反面然后反面) = ½ × ½ = ¼

Independent events keep the tree symmetrical, and each path probability is simply the product of the stage probabilities.

独立事件使树形图保持对称，每条路径的概率就是各阶段概率的简单乘积。

5. Conditional Probability and Dependent Events | 条件概率与相依事件

When events are dependent, the probability of a later stage depends on what happened earlier. This is written as P(B|A), read as “the probability of B given A”.

当事件相依时，后一阶段的概率取决于先前发生的事件。这记作 P(B|A)，读作“在 A 发生的条件下 B 发生的概率”。

For example, drawing two balls from a bag containing 3 red and 2 blue balls, without replacement. If the first ball is red, only 2 red balls remain, so the probability of a red ball second is 2/4 = ½.

例如，从一个装有 3 个红球和 2 个蓝球的袋子中不放回地抽取两个球。若第一个球是红球，则袋中只剩 2 个红球，因此第二次抽到红球的概率为 2/4 = ½。

On the tree, the second-stage branches are labelled with conditional probabilities that change according to which first-stage branch was taken.

在树形图中，第二阶段的分支所标注的条件概率，会随第一阶段所走分支的不同而改变。

The probability of a path is then P(A) × P(B|A), which is exactly the definition of the multiplication rule for dependent events.

此时路径的概率为 P(A) × P(B|A)，这正是相依事件乘法法则的定义。

6. “Without Replacement” Problems | “不放回”问题

Drawing without replacement is the classic example of a dependent-event situation. The total number of items decreases after each draw, so probabilities adjust.

不放回抽取是相依事件情形的经典例子。每次抽取后物品总数减少，因此概率也随之调整。

Consider a bag with 5 red and 3 green marbles. Two marbles are drawn without replacement. The first-stage branches are P(Red) = 5/8 and P(Green) = 3/8. After a red marble is drawn, only 4 red and 3 green remain.

假设一个袋子中有 5 个红色弹珠和 3 个绿色弹珠，不放回地抽取两个。第一阶段的分支为 P(红) = 5/8，P(绿) = 3/8。抽出一个红色弹珠后，袋中只剩下 4 红 3 绿。

The four final outcomes and their probabilities are summarised in the table below.

四种最终结果及其概率汇总如下表。

Outcome	Calculation	Probability
Red, Red	5/8 × 4/7	20/56 = 5/14
Red, Green	5/8 × 3/7	15/56
Green, Red	3/8 × 5/7	15/56
Green, Green	3/8 × 2/7	6/56 = 3/28

Notice that the sum of all leaf probabilities is 1, which confirms that the tree has been constructed correctly.

注意，所有叶概率之和为 1，这验证了树形图构建的正确性。

7. The Product Rule along Branches | 分支乘法法则

The product rule states that the probability of two events both occurring is the product of the probability of the first event and the conditional probability of the second event given the first.

乘法法则指出：两个事件同时发生的概率，等于第一个事件的概率乘以在第一个事件发生条件下第二个事件的条件概率。

P(A and B) = P(A) × P(B|A)

P(A 且 B) = P(A) × P(B|A)

If A and B are independent, this simplifies to P(A and B) = P(A) × P(B), because P(B|A) = P(B).

若 A 与 B 独立，则该式简化为 P(A 且 B) = P(A) × P(B)，因为此时 P(B|A) = P(B)。

On a tree diagram, this appears naturally: every time you move along an additional branch, you multiply by that branch’s probability.

在树形图中，这一点体现得十分自然：每沿一条新的分支前进，你就乘以该分支的概率。

8. The Addition Rule for Combined Outcomes | 合并结果的加法法则

Often a question asks for the probability of an outcome that can be reached by more than one path, such as “getting exactly one head” after two tosses.

许多问题会询问可通过多条路径到达的结果的概率，例如抛两次硬币后“恰好出现一次正面”。

For two fair coins, exactly one head occurs along two paths: Head then Tail, and Tail then Head. Each path has probability ¼.

对于两枚均匀硬币，“恰好一次正面”沿两条路径发生：正然后反，以及反然后正。每条路径的概率均为 ¼。

P(exactly one head) = ¼ + ¼ = ½

P(恰好一次正面) = ¼ + ¼ = ½

This is the addition rule: because the two paths are mutually exclusive, we add their probabilities. Tree diagrams make such multi-path events easy to spot and compute.

这就是加法法则：由于两条路径互斥，我们将它们的概率相加。树形图使这类多路径事件更容易识别和计算。

9. Real-World Applications | 实际应用

Probability trees are not merely exam exercises; they are widely used in real-world decision making and risk analysis.

树形图不仅仅是考试题，它们被广泛用于现实世界的决策与风险分析中。

Medicine: doctors use tree diagrams to estimate the probability that a patient has a disease given a positive or negative test result.

医学：医生使用树形图来估计在检测结果为阳性或阴性的条件下，患者患病的概率。
Quality control: manufacturers model the probability that a product passes or fails successive inspection stages.

质量控制：制造商用树形图模拟产品在连续多个检验阶段中通过或未通过的概率。
Weather forecasting: meteorologists combine the probabilities of independent and dependent weather events over consecutive days.

天气预报：气象学家将连续几天内独立与相依天气事件的概率结合起来。
Finance: analysts use decision trees to evaluate the likely outcomes of an investment under different market conditions.

金融：分析师使用决策树来评估不同市场条件下某项投资的可能结果。

In every case, the tree turns a complex sequence of uncertain events into an organised calculation.

在任何情况下，树形图都将一串复杂的不确定事件转化为有条理的计算。

10. Common Mistakes and How to Avoid Them | 常见错误与规避方法

Students often lose marks on tree diagram questions due to a few repeated errors. Being aware of these can greatly improve accuracy.

学生在树形图题目中常因一些反复出现的错误而失分。了解这些错误可以大大提高准确率。

Mistake 1: Forgetting that probabilities change without replacement. Always update the numerator and the denominator.

错误一：忘记不放回时概率会变化。务必同时更新分子和分母。
Mistake 2: Failing to multiply along the branches. Some students add probabilities along a path; remember it is multiplication.

错误二：忘记沿分支相乘。有些学生沿路径相加；请记住，路径内是相乘。
Mistake 3: Adding probabilities that are not mutually exclusive. Only add the probabilities of distinct paths that cannot happen together.

错误三：把并不互斥的概率相加。只能相加互不相同且不会同时发生的路径概率。
Mistake 4: Skipping the final check. The sum of all leaf probabilities must equal 1; if it does not, the tree is wrong.

错误四：跳过最终检查。所有叶概率之和必须等于 1；若不为 1，则树形图有误。

A quick check after each problem saves marks and builds confidence.

每道题做完后快速检查一遍，既能保住分数，也能增强信心。

11. Worked Example | 完整例题

A bag contains 4 red and 6 blue pens. Two pens are drawn at random without replacement. Find the probability that both pens have the same colour.

一个袋子中有 4 支红色笔和 6 支蓝色笔。不放回地随机抽取两支笔。求两支笔颜色相同的概率。

First construct the tree. At the first stage, P(Red) = 4/10 and P(Blue) = 6/10. If the first pen is red, the second stage has P(Red) = 3/9 and P(Blue) = 6/9. If the first pen is blue, the second stage has P(Red) = 4/9 and P(Blue) = 5/9.

首先构建树形图。在第一阶段，P(红) = 4/10，P(蓝) = 6/10。若第一支是红色，则第二阶段 P(红) = 3/9，P(蓝) = 6/9；若第一支是蓝色，则第二阶段 P(红) = 4/9，P(蓝) = 5/9。

Both pens have the same colour in exactly two cases: Red then Red, or Blue then Blue. These paths are mutually exclusive, so we add them.

两支笔颜色相同恰有两种情况：红然后红，或蓝然后蓝。这两条路径互斥，因此我们将它们相加。

P(same colour) = (4/10 × 3/9) + (6/10 × 5/9) = 12/90 + 30/90 = 42/90 = 7/15

P(颜色相同) = (4/10 × 3/9) + (6/10 × 5/9) = 12/90 + 30/90 = 42/90 = 7/15

Therefore, the probability that both pens are the same colour is 7/15, or approximately 0.467.

因此，两支笔颜色相同的概率为 7/15，约为 0.467。

12. Summary and Revision Tips | 总结与复习建议

Probability tree diagrams turn multi-stage probability problems into structured drawings. Master the product rule for paths and the addition rule for combined events, and check that your tree sums to 1.

概率树形图将多阶段概率问题转化为结构化的图形。掌握路径的乘法法则与复合事件的加法法则，并检查你的树形图总概率是否为 1。

To revise effectively: practice drawing trees for independent events first, then move to “without replacement” questions, and finally attempt mixed conditional-probability problems.

高效复习方法：先练习独立事件的树形图，再过渡到“不放回”问题，最后尝试混合条件概率题目。

Examiners frequently test tree diagrams at both GCSE and A-Level. A neatly drawn tree with labelled probabilities is the clearest evidence of solid understanding.

考官在 GCSE 和 A-Level 中经常考查树形图。一棵画得整洁、概率标注清晰明了的树，是理解扎实最清晰的证明。

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Construction and Application of Probability Tree Diagrams | 概率树形图的构建与应用