📚 Understanding Probability: Combined Events and Tree Diagrams | 理解概率：复合事件与树状图
Probability is the branch of mathematics that deals with how likely events are to happen. In KS3 Cambridge Mathematics, you will meet simple probability before progressing to combined events, which involve two or more things happening. This article explains key concepts such as sample spaces, independence, mutual exclusivity, and how to use tree diagrams to calculate probabilities systematically. Master these topics and you will be well-prepared for your checkpoint exam and beyond.
概率是数学中研究事件发生可能性的分支。在KS3剑桥数学课程中，你会先学习简单概率，再进入复合事件——即涉及两个或以上事情发生的情况。本文解释样本空间、独立性、互斥性等关键概念，以及如何使用树状图系统地计算概率。掌握这些主题，你将能从容应对 checkpoint 考试及以后的挑战。
1. What Is Probability? | 什么是概率？
Probability measures the chance that a particular event will occur. It is expressed as a number between 0 and 1, where 0 means the event is impossible, and 1 means the event is certain. You can also express probability as a fraction, decimal, or percentage. For example, when flipping a fair coin, the probability of getting heads is 1/2, or 0.5, or 50%.
概率衡量特定事件发生的可能性。它用0到1之间的数字表示，0代表事件不可能发生，1代表事件必然发生。你也可以用分数、小数或百分比表示概率。例如，抛一枚均匀硬币时，得到正面的概率是1/2，即0.5或50%。
2. Probability Scale | 概率标度
The probability scale is a number line from 0 to 1. Events can be placed on this line according to their likelihood. An event with a probability close to 0 is very unlikely; an event with a probability close to 1 is very likely. Words like ‘impossible’, ‘unlikely’, ‘evens chance’, ‘likely’, and ‘certain’ correspond to positions on the scale. For instance, rolling a 7 on a standard dice has probability 0, while rolling a number less than 7 has probability 1.
概率标度是一条从0到1的数轴。根据可能性的大小，事件可以标在这条数轴上。概率接近0的事件非常不太可能发生；接近1的事件非常可能发生。“不可能”、“不太可能”、“等可能”、“很可能”、“必然”等词语对应标度上的位置。比如，在一颗标准骰子上掷出7的概率为0，掷出小于7的数的概率为1。
3. Simple Events vs Combined Events | 简单事件与复合事件
A simple (or single) event is an outcome from one action, such as rolling a 4 on a dice. A combined event involves two or more separate actions or outcomes, like rolling a dice and flipping a coin, or drawing two cards from a deck. When we calculate probabilities for combined events, we need to consider all possible outcomes together. This leads us to concepts like sample space and tree diagrams. The key is to identify whether the events are independent or dependent.
简单事件（或单一事件）是一次行为产生的结果，例如掷骰子得到4。复合事件涉及两个或更多独立行为或结果，比如掷骰子同时抛硬币，或从一副牌中抽两张牌。计算复合事件概率时，我们需要同时考虑所有可能的结果。这就引出了样本空间和树状图等概念。关键在于判断事件是独立还是相依。
4. Listing Outcomes: Sample Space Diagrams | 列出结果：样本空间图
A sample space is a list or diagram of all possible outcomes of an experiment. For two events, you can use a sample space table (two-way table). For example, rolling two dice: the sample space has 36 ordered pairs (1,1), (1,2), …, (6,6). You can then count the favourable outcomes for an event, such as ‘sum of 7’: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). Probability = number of favourable outcomes / total number of outcomes = 6/36 = 1/6.
样本空间是实验所有可能结果的列表或图表。对于两个事件，可用样本空间表（二维表）。例如掷两枚骰子：样本空间有36个有序对 (1,1)、(1,2)、…、(6,6)。然后你可以计算符合事件的结果数，比如“和为7”：(1,6)、(2,5)、(3,4)、(4,3)、(5,2)、(6,1)。概率 = 有利结果数 / 总结果数 = 6/36 = 1/6。
A table helps visualise this clearly. Here is a partial sample space for two dice:
下面用部分表格直观展示两枚骰子的情况：
| Dice 1 / Dice 2 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| 1 | (1,1) | (1,2) | (1,3) | (1,4) | (1,5) | (1,6) |
| 2 | (2,1) | (2,2) | (2,3) | (2,4) | (2,5) | (2,6) |
| 3 | (3,1) | (3,2) | (3,3) | (3,4) | (3,5) | (3,6) |
Each cell is equally likely, making it easy to count cells where the sum is 7. Always check if outcomes are equally likely before using P(event) = favourable / total.
每个格子等可能发生，这样很容易数出和为7的格子。务必在使用 P(事件)=有利/总数 前确认结果是否等可能。
5. The Multiplication Rule for Independent Events | 独立事件的乘法法则
Two events are independent if the outcome of one does not affect the outcome of the other. For independent events A and B, the probability that both occur is the product of their individual probabilities. This is known as the multiplication rule or the AND rule.
如果一件事的结果不影响另一件事的结果，这两个事件就是独立事件。对于独立事件A和B，两者同时发生的概率等于各自概率的乘积。这被称为乘法法则或“与”法则。
P(A and B) = P(A) x P(B)
For example, the probability of getting heads on a coin (1/2) and rolling a 6 on a dice (1/6) is (1/2) x (1/6) = 1/12. This rule only applies to independent events. It also extends to more than two events: P(A and B and C) = P(A) x P(B) x P(C).
例如，得到硬币正面（1/2）且掷骰子得6（1/6）的概率是 (1/2) x (1/6) = 1/12。这个法则仅适用于独立事件。它还可以扩展到两个以上的事件：P(A 和 B 和 C) = P(A) x P(B) x P(C)。
6. Mutually Exclusive Events | 互斥事件
Events are mutually exclusive if they cannot happen at the same time. For example, when rolling a dice, getting an odd number and getting an even number are mutually exclusive. For mutually exclusive events, the probability of either event occurring is the sum of their probabilities. This is the OR rule.
如果两个事件不可能同时发生，则它们是互斥事件。例如掷骰子时，得到奇数和得到偶数是互斥的。对于互斥事件，任一事件发生的概率等于各自概率之和。这是“或”法则。
P(A or B) = P(A) + P(B)
If events are not mutually exclusive, you must subtract the overlap: P(A or B) = P(A) + P(B) – P(A and B). This prevents double-counting outcomes that are in both events. Understanding the difference between AND and OR rules is fundamental for combined probability.
如果事件不互斥，则必须减去重叠部分：P(A 或 B) = P(A) + P(B) – P(A 和 B)。这样可以避免重复计算同时属于两个事件的结果。理解“与”规则和“或”规则的区别，是解决复合概率的基础。
7. Tree Diagrams Unfolded | 树状图解析
A tree diagram is a visual way to show all possible outcomes of combined events, especially when there are two or more stages. Each branch represents a possible outcome at a stage, and the probability is written along the branch. The outcomes are written at the ends of the branches. Tree diagrams help us apply the multiplication rule along branches and the addition rule across branches. When constructing a tree diagram, the probabilities on branches from any point must sum to 1.
树状图是一种展示复合事件所有可能结果的直观方法，尤其适用于两个或更多阶段的情况。每条分支代表某一阶段的一个可能结果，概率写在分支上。结果写在分支末端。树状图帮助我们在分支上应用乘法法则，在分支间应用加法法则。构建树状图时，从同一点出发的所有分支概率之和必须等于1。
For a two-stage experiment, you start at a point, draw branches for the first event, then from each first-event outcome draw more branches for the second event. This gives a complete picture of all paths.
对于两阶段试验，从一个点开始，为第一个事件画出分支，然后从每个第一事件结果再为第二个事件画出分支。这样就完整呈现出所有路径。
8. Calculating Probabilities from Tree Diagrams | 由树状图计算概率
To find the probability of a specific combined outcome, multiply the probabilities along the branches that lead to that outcome. For example, a bag contains 3 red and 2 blue counters. We pick one counter, replace it, then pick another. The tree diagram has two stages, each with branches for red (3/5) and blue (2/5). The probability of picking red then blue is (3/5) x (2/5) = 6/25. Since there are two paths for one red and one blue (RB and BR), the total probability for ‘one red and one blue in any order’ is 6/25 + 6/25 = 12/25.
要计算某个特定复合结果的概率，沿着导向该结果的所有分支路径将概率相乘。例如，一个袋子有3个红球和2个蓝球。我们抽取一个球并放回，再抽一个。树状图有两个阶段，每个阶段都有红（3/5）和蓝（2/5）两条分支。先红后蓝的概率是 (3/5) x (2/5) = 6/25。因为达成一红一蓝有两条路径（红蓝和蓝红），所以“任意顺序一红一蓝”的总概率为 6/25 + 6/25 = 12/25。
Here is a summary table of the possible paths and probabilities for this replacement example:
以下是该放回抽取例子的路径与概率汇总表：
| Path (1st then 2nd) | Calculation | Probability |
|---|---|---|
| Red, Red | (3/5) x (3/5) | 9/25 |
| Red, Blue | (3/5) x (2/5) | 6/25 |
| Blue, Red | (2/5) x (3/5) | 6/25 |
| Blue, Blue | (2/5) x (2/5) | 4/25 |
9. Tree Diagrams with Non-Replacement | 无放回情况下的树状图
When items are not replaced, the probabilities for the second event change depending on what happened first. For example, from the same bag (3 red, 2 blue), picking without replacement: the first pick probability is red 3/5, blue 2/5. If a red is taken first, the bag now contains 2 red and 2 blue, so second pick red becomes 2/4, blue 2/4. If a blue was taken first, the second pick has red 3/4, blue 1/4. The tree diagram will reflect these changing probabilities. This illustrates dependent events, where the outcome of one event affects the probability of the next.
当物品不被放回时，第二次事件的概率会根据第一次的结果改变。例如，同样的袋子（3红2蓝），无放回抽取：第一次抽到红的概率为3/5，蓝为2/5。如果第一次抽到红，袋子变成2红2蓝，第二次抽到红的概率为2/4，蓝为2/4。如果第一次抽到蓝，第二次抽到红为3/4，蓝为1/4。树状图会体现这些变化的概率。这体现了相依事件，即一个事件的结果会影响下一个事件的概率。
Always redraw probabilities on the second-stage branches based on the changed composition. Then multiply along paths as before. The overall tree still organises all final outcomes.
务必根据改变后的构成重新计算并标注第二阶段分支的概率。然后像之前一样沿路径相乘。
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