Probability Tree Diagrams | 概率树状图

📚 Probability Tree Diagrams | 概率树状图

Probability is a measure of how likely an event is to occur. When dealing with multiple events, it is often helpful to represent all possible outcomes visually. A probability tree diagram is one such visual representation, where each branch represents a possible outcome and its probability. In this lesson, we will learn how to construct tree diagrams, apply the multiplication rule, and solve problems involving both independent and dependent events.

概率是衡量事件发生可能性的尺度。当处理多个事件时，用可视化的方式表示所有可能结果往往很有帮助。概率树状图就是这样一种可视化表示方法，每条分支代表一个可能结果及其概率。在本课中，我们将学习如何绘制树状图、应用乘法法则，并解决涉及独立事件和非独立事件的问题。

1. What is a Probability Tree Diagram? | 什么是概率树状图？

A probability tree diagram is a branching diagram that shows all possible outcomes of repeated events. Each branch is labelled with the probability of that outcome happening. The end of each branch shows the combined outcome, and the probability of that outcome is the product of the probabilities along the path.

概率树状图是一种分支图，用于显示重复事件的所有可能结果。每个分支都标注该结果发生的概率。每个分支末端显示组合结果，该结果的概率是路径上各概率的乘积。

For example, imagine tossing a coin once. The tree starts with a single point. From this point, two branches emerge: one labelled ‘Heads’ (probability 1/2) and the other labelled ‘Tails’ (probability 1/2). This is the simplest tree diagram with just one event.

例如，想象抛一枚硬币一次。树从一个单点开始。从该点分出两条分支：一条标注”正面”（概率1/2），另一条标注”反面”（概率1/2）。这是只有一个事件的最简单树状图。

2. The Multiplication Rule | 乘法法则

When two or more events occur, the probability that they all happen is found by multiplying the probabilities along the corresponding branches. This is known as the multiplication rule for probability.

当两个或多个事件发生时，它们全部发生的概率可以通过将相应分支上的概率相乘而得到。这就是概率的乘法法则。

P(A and B) = P(A) × P(B given A)

For independent events, where the outcome of one event does not affect the other, the rule simplifies to P(A and B) = P(A) × P(B). In a tree diagram, this means that the probabilities on the second-level branches are the same for every first-level branch.

对于独立事件，即一个事件的结果不影响另一个事件，该法则简化为 P(A 且 B) = P(A) × P(B)。在树状图中，这意味着第二层分支的概率对每个第一层分支都是相同的。

3. Independent vs Dependent Events | 独立事件与非独立事件

Independent events are events whose outcomes do not influence each other. For example, tossing a coin and rolling a dice are independent. Dependent events, on the other hand, are affected by a previous outcome. Drawing a card from a deck without replacement is a classic example because the second probability depends on the first draw.

独立事件是指结果互不影响的事件。例如，掷硬币和掷骰子是独立的。而非独立事件则受先前结果的影响。从一副牌中不放回地抽取卡片是一个典型例子，因为第二次抽取的概率取决于第一次抽取的结果。

Tree diagrams naturally handle both types. For dependent events, we use conditional probabilities, which are placed on the second-level branches based on the outcome of the first level.

树状图天然地处理这两种类型。对于非独立事件，我们使用条件概率，这些条件概率根据第一层的结果放置在第二层的分支上。

4. Drawing the First Set of Branches | 绘制第一层分支

To draw a tree diagram, start by listing every possible outcome of the first event. For each outcome, draw a branch from the starting point. Label each branch with the outcome and its probability.

要绘制树状图，先列出第一个事件的所有可能结果。对于每个结果，从起点绘制一条分支。在每条分支上标注结果及其概率。

Remember that the sum of the probabilities on all branches from the same node must equal 1. This principle allows you to find a missing probability by subtracting the known probabilities from 1.

记住，从同一节点出发的所有分支的概率之和必须等于1。这一原则允许你通过从1减去已知概率来找到缺失的概率。

P(Not A) = 1 – P(A)

5. Adding Subsequent Branches | 添加后续分支

If the experiment involves more than one event, continue drawing branches from the end of each first-level branch. For independent events, the probabilities on these new branches are the same regardless of which first-level branch they come from. For dependent events, the probabilities are conditional, so they may differ.

如果试验涉及多个事件，则从每个第一层分支末端继续绘制分支。对于独立事件，这些新分支上的概率无论来自哪个第一层分支都是相同的。对于非独立事件，概率是条件概率，因此可能不同。

Each complete path from the start to a final endpoint represents one combined outcome. The probability of that combined outcome is the product of the probabilities along the path.

从起点到最终端点的每条完整路径代表一个组合结果。该组合结果的概率是路径上各概率的乘积。

6. Sum of Probabilities | 概率之和

One powerful check is that the sum of the probabilities of

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