📚 Probability and Tree Diagrams | 概率与树形图
Probability is the branch of mathematics that deals with how likely events are to happen. In everyday life we use words like 'impossible', 'unlikely', 'even chance', 'likely' and 'certain'. To make accurate predictions, however, we need numbers. Tree diagrams are a powerful visual tool that helps us list all possible outcomes and calculate probabilities for two or more stages – whether you are tossing coins, picking sweets from a bag, or forecasting the weather. This article builds your understanding step by step, from basic probability rules to drawing and reading probability trees, focusing on the Cambridge KS3 curriculum.
概率是数学中研究事件发生可能性大小的一个分支。在日常生活中我们常使用‘不可能’、‘不太可能’、‘对半几率’、‘很可能’和‘一定’这样的词。然而，要做出准确的预测，我们就需要数字。树形图是一种强大的视觉工具，能帮助我们列出所有可能的结果，并计算涉及两个或更多阶段的概率——无论你是在掷硬币、从袋子里取糖果，还是预报天气。本文将逐步建立你的理解，从基本的概率规则到绘制和解读概率树，紧扣剑桥 KS3 课程大纲。
1. What Is Probability? | 什么是概率？
Probability measures the chance that a specific event will occur. It is always a number between 0 and 1. An event with probability 0 is impossible (e.g. rolling a 7 on a standard six‑sided die). An event with probability 1 is certain (e.g. the sun rising tomorrow). Probabilities can also be expressed as fractions, decimals or percentages.
概率衡量某个特定事件发生的可能性大小。它永远是 0 到 1 之间的一个数。概率为 0 的事件是不可能发生的（例如在一个标准的六面骰子上掷出 7 点）。概率为 1 的事件是必然发生的（例如太阳明天升起）。概率还可以用分数、小数或百分数来表示。
For any event A, we write P(A) to mean ‘the probability of A’. The formula is:
对于任何事件 A，我们用 P(A) 表示‘A 的概率’。公式为：
P(A) = Number of favourable outcomes ÷ Total number of outcomes
P(A) = 有利结果数 ÷ 所有等可能结果总数
2. The Probability Scale | 概率尺度
We can place probabilities on a scale from 0 to 1. A probability close to 0 indicates an event is very unlikely, a probability of 0.5 represents an even chance, and a probability close to 1 suggests the event is almost certain. Below is a quick reference table showing common words and their numeric ranges.
我们可以将概率放在一条从 0 到 1 的尺度上。接近 0 的概率表示事件非常不太可能发生，0.5 的概率代表对半几率，接近 1 的概率则表示事件几乎必然发生。下面是一个参考表，列出了常用词语及其数值范围。
| Word | Probability as a fraction | Probability as a decimal |
|---|---|---|
| Impossible | 0 | 0 |
| Unlikely | 1/4 | 0.25 |
| Even chance | ½ | 0.5 |
| Likely | ¾ | 0.75 |
| Certain | 1 | 1 |
Remember that probabilities can never be negative or greater than 1. If you calculate a probability outside this range, you have made an error.
请记住，概率永远不会是负数，也不会大于 1。如果你计算出的概率超出了这个范围，那就出错了。
3. Experimental vs Theoretical Probability | 实验概率与理论概率
mental probability comes from actually carrying out an experiment or survey and recording what happens. It is found using:
理论概率是我们基于等可能结果所期望发生的情况。例如，抛一枚公平硬币时我们说 P(正面朝上) = ½。实验概率则来自实际进行实验或调查，并记录发生的情况。计算公式为：
Experimental probability = Number of times the event occurs ÷ Total number of trials
实验概率 = 事件发生的次数 ÷ 总试验次数
r of trials increases, experimental probability tends to get closer to the theoretical value.
如果你抛一枚硬币 100 次，得到 47 次正面，那么正面的实验概率就是 47/100 = 0.47。随着试验次数的增加，实验概率会越来越接近理论值。
4. Calculating Probabilities of Single Events | 计算单一事件的概率
l outcomes are equally likely, we can find P(A) by counting. For instance, with a fair six‑ ing an even number is 3/6 = ½. The probability of rolling a number less than 5 is 4/6 = ⅔.
当所有结果等可能时，我们可以通过计数来求 P(A)。例如，对于一个公平的六面骰子，可能的结果是 1, 2, 3, 4, 5, 6。掷出偶数的概率是 3/6 = ½。掷出小于 5 的数的概率是 4/6 = ⅔。
ment rule is very useful: P(not A) = 1 − P(A). So if P(rain) = 0.3, then P(no rain) = 0.7.
一定要检查实验中所有互斥的可能结果的概率之和是否等于 1。补集规则非常有用：P(非 A) = 1 − P(A)。因此，如果 P(下雨) = 0.3，则 P(不下雨) = 0.7。
5. Understanding Tree Diagrams | 理解树形图
A tree diagram shows all the possible outcomes of an experiment that consists of two or more steps. Each branch represents a possible outcome at that stage, and we write the probability along the branch. The branches from a single point always add up to 1. Tree diagrams help us apply the multiplication rule for compound events and keep our work organised.
树形图展示了一个包含两个或更多步骤的实验的所有可能结果。每根树枝代表该阶段的一个可能结果，我们把概率写在树枝旁边。从一个点分出的树枝的概率之和总是等于 1。树形图帮助我们对复合事件应用乘法规则，并使解题过程井井有条。
again labelled ½. The final outcomes are HH, HT, TH, TT, each with probability ½ × ½ = ¼.
例如，抛掷两枚硬币时，第一组树枝显示第一枚硬币的正反面（每种概率为 ½）。从每一个端点再画出第二组树枝表示第二枚硬币，同样标注 ½。最终结果是 HH、HT、TH、TT，每个的概率是 ½ × ½ = ¼。
6. Drawing a Tree Diagram | 绘制树形图
Follow these steps to draw a probability tree:
按照以下步骤绘制概率树形图：
Identify the stages of the experiment and the possible outcomes at each stage.
确定实验的阶段以及每个阶段可能的结果。
Draw a point on the left and draw one branch for each outcome of the first stage, labelling it with its probability.
在左侧画一个点，为第一阶段每一个结果画一根树枝，并标注其概率。
From the end of each first‑stage branch, draw branches for the second stage (and further stages if needed).
从每个第一阶段树枝的末端，画出第二阶段（如果需要还有后续阶段）的树枝。
Write the probability on each branch. Check that the probabilities from any single point sum to 1.
在每根树枝上写上概率。检查从同一点分出的概率之和是否为 1。
Label the final outcomes at the end of the last branches.
在最后树枝的末端标注最终结果。
e one ball, note its colour, and do not replace it. Then you take a second ball. The first‑stage probabilities are: P(Red)=3/5, P(Blue)=2/5. The second‑stage probabilities change depending on what was taken first. This is an example of ‘without replacement’, which we will explore later.
我们来画一个袋子中有 3 个红球和 2 个蓝球的树形图。你取出一个球，记下颜色，不放回，然后取出第二个球。第一阶段的概率是：P(红) = 3/5，P(蓝) = 2/5。第二阶段的概率取决于第一次取出了什么球。这是一个‘无放回’的例子，稍后我们会深入探讨。
7. Multiplying Probabilities – the ‘AND’ Rule | 乘法‘且’规则
When we want the probability that event A and event B both happen along a particular path of the tree, we multiply the probabilities on the branches. This is called the AND rule.
当我们要求沿着树形图某一条路径，事件 A 且事件 B 都发生的概率时，我们将这条路径上各树枝的概率相乘。这称为‘且’规则。
P(A and B) = P(A) × P(B given that A has happened)
P(A 且 B) = P(A) × P(已知 A 发生的前提下 B 发生的概率)
ndependent, the second probability is simply P(B). If they are dependent, we must use the conditional probability P(B|A).
在硬币的例子中，P(第一次正面且第二次正面) = ½ × ½ = ¼。当事件独立时，第二个概率就是简单的 P(B)。如果事件不独立，我们就必须使用条件概率 P(B|A)。
8. Adding Probabilities – the ‘OR’ Rule | 加法‘或’规则
If more than one path through the tree gives the desired outcome, we add the probabilities of those paths. This is the OR rule for mutually exclusive outcomes.
如果树形图中不止一条路径能得到所要求的结果，我们就把这些路径的概率加起来。这就是互斥结果的‘或’规则。
P(A or B) = P(A) + P(B) (if A and B cannot happen at the same time)
P(A 或 B) = P(A) + P(B) （当 A 和 B 不可能同时发生时）
robability is ¼ + ¼ = ½. Always check that you are adding probabilities of distinct paths.
对于两枚硬币，P(恰好一个正面)可以通过两种方式发生：HT 或 TH。每条路径的概率为 ¼，因此总概率为 ¼ + ¼ = ½。一定要确保你加的是不同路径的概率。
9. Independent Events | 独立事件
Two events are independent if the outcome of the first does not affect the probability of the second. Examples include rolling a die and then tossing a coin, or taking a sweet from a bag and replacing it before taking a second.
如果第一个事件的结果不影响第二个事件发生的概率，那么这两个事件就是独立的。例子包括先掷骰子再抛硬币，或者从袋子里取出一个糖果后放回再取第二次。
When events are independent, the tree diagram has identical second‑stage probabilities regardless of the first outcome. This makes calculations easier because we can simply multiply the same branch probabilities.
当事件独立时，不管第一阶段结果如何，树形图中第二阶段的概率都相同。这使得计算更加容易，因为我们只需乘以相同的树枝概率。
it twice. The tree has 4 branches each time, each labelled ¼. P(A then B) = ¼ × ¼ = 1/16.
例如：一个转盘被等分成四个扇形，分别标有 A、B、C、D。你转动转盘两次。每次都有 4 根树枝，各标有 ¼。P(先 A 后 B) = ¼ × ¼ = 1/16。
10. Probability with Replacement | 有放回概率
In problems where an item is selected and then replaced, the probabilities remain the same for each stage. This makes the events independent. Tree diagrams for 'with replacement' problems have identical branches for the second (and further) stages.
在每次选取物品后放回的问题中，每个阶段的概率保持不变。这使得事件相互独立。对于‘有放回’问题，树形图中第二阶段（及后续阶段）的树枝与第一阶段完全相同。
Worked example: A box contains 5 red and 3 green counters. A counter is taken, its colour noted, and replaced. Then a second counter is taken. Draw the tree and find P(both red) and P(one of each colour).
练习题示例：一个盒子里有 5 个红色和 3 个绿色筹码。取出一个，记下颜色，放回。然后取出第二个。画出树形图，并求 P(两个都是红色) 和 P(一红一绿)。
Stage 1: P(R)=5/8, P(G)=3/8.
第一阶段：P(R)=5/8，P(G)=3/8。
Stage 2: same probabilities from each node.
第二阶段：从每个节点出发的概率相同。
3/8 = 15/64; P(GR) = 3/8 × 5/8 = 15/64; so P(one of each) = 15/64 + 15/64 = 30/64 = 15/32.
P(RR) = 5/8 × 5/8 = 25/64。P(RG) = 5/8 × 3/8 = 15/64；P(GR) = 3/8 × 5/8 = 15/64；因此 P(一红一绿) = 15/64 + 15/64 = 30/64 = 15/32。
11. Probability without Replacement | 无放回概率
When items are not replaced, the events are dependent: the first outcome changes the probabilities for the second stage. The tree diagram reflects this by showing different probabilities on the second branches depending on what happened first.
当物品不放回时，事件是相关的：第一个结果会改变第二阶段的概率。树形图通过在第二阶段树枝上显示不同的概率来体现这一点，这些概率取决于第一阶段发生了什么。
, but without replacement: after taking one red, only 4 red remain out of 7. So the second‑stage probabilities from the red branch are P(R|R)=4/7, P(G|R)=3/7. From the green branch, P(R|G)=5/7, P(G|G)=2/7.
还是那 5 个红球和 3 个绿球，但这次不放回：取出一个红球后，剩下 4 个红球，总共 7 个。因此从红球树枝出发的第二阶段概率为 P(R|R)=4/7，P(G|R)=3/7。从绿球树枝出发，P(R|G)=5/7，P(G|G)=2/7。
× 3/7) + (3/8 × 5/7) = 15/56 + 15/56 = 30/56 = 15/28. This is slightly different from the 'with replacement' answer, showing the effect of removing an item.
现在 P(RR) = 5/8 × 4/7 = 20/56 = 5/14。P(一红一绿) = P(RG) + P(GR) = (5/8 × 3/7) + (3/8 × 5/7) = 15/56 + 15/56 = 30/56 = 15/28。这与‘有放回’的答案略有不同，说明了不放回的效果。
12. Solving Multi‑Stage Problems | 解决多阶段问题
tay systematic, draw all branches, and use the AND rule along each path and the OR rule between paths.
树形图可以扩展到三个或更多阶段。例如，抛掷三枚硬币：树形图会有 2 × 2 × 2 = 8 个末端。得到三个正面的概率是 ½ × ½ × ½ = ⅛。关键是要有条不紊，画出所有树枝，对每条路径使用‘且’规则，对不同路径使用‘或’规则。
tomorrow. If it rains today, the probability of rain tomorrow rises to 0.6. You can model this with a probability tree and answer questions like P(rain on exactly one day).
更复杂的问题可能涉及不同类型的混合实验。例如，天气预报给出今天和明天下雨的概率均为 0.4。如果今天下雨，明天下雨的概率上升到 0.6。你可以用概率树对此建模，并回答像 P(恰好一天下雨) 这样的问题。
Always label final outcomes clearly and double‑check that the total probabilities of all final outcomes sum to 1. This is an excellent way to catch mistakes.
务必清晰地标注最终结果，并仔细检查所有最终结果的概率之和是否为 1。这是发现错误的一个极好方法。
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