📚 Combined Events | 联合事件
In probability, a combined event is formed by linking two or more simple events using the connectives 'and' or 'or'. These composite events occur when we calculate the chance of either one outcome or another happening, or the chance of two outcomes happening together. The rules for combining probabilities depend on whether events are mutually exclusive, independent, or conditional.
在概率论中，联合事件是由两个或多个简单事件通过“且”或“或”连接而成的复合事件。当我们计算某一种结果或者另一种结果发生的概率，或者两个结果同时发生的概率时，就需要用到这些复合事件。组合概率的规则取决于事件是否互斥、独立或者条件相关。
1. Sample Space and Simple Events | 样本空间与简单事件
en denoted by S) is the set of all possible outcomes. For example, when rolling a fair six‑ , 4, 5, 6}. The probability of any simple event e is P(e) = 1/6, provided the die is fair.
简单事件是概率实验中的单个结果。样本空间（常用 S 表示）是所有可能结果的集合。例如，抛掷一个公平的六面骰子，样本空间为 S = {1, 2, 3, 4, 5, 6}。在骰子公平的前提下，任何简单事件 e 的概率为 P(e) = 1/6。
bined events are subsets of the sample space that contain more than one outcome. The event 'rolling an even number' of three simple events. To find P(A) we count the favourable outcomes and divide by the total number of outcomes in S.
联合事件是样本空间的子集，包含多个结果。事件“掷出偶数”为 A = {2, 4, 6}，这是三个简单事件的组合。计算 P(A) 时，我们统计有利结果的数量，再除以 S 中结果的总数。
2. Union and Intersection | 并集与交集
The union of two events A and B, written as A ∪ B, is the event that A occurs, B occurs, or both occur. The intersection A ∩ B is the event that both A and B occur simultaneously. Understanding these set operations is essential when combining probabilities because the addition and multiplication rules rely directly on union and intersection.
两个事件 A 和 B 的并集，记作 A ∪ B，表示 A 发生、B 发生或者两者同时发生的事件。交集 A ∩ B 表示 A 和 B 同时发生的事件。理解这些集合运算是组合概率的基础，因为加法法则和乘法法则都直接依赖于并集和交集。
Venn diagrams and two‑way tables are powerful tools for visualising unions and intersections. When we shade the region representing A ∪ B, we cover all outcomes in A, all outcomes in B, and the overlap. The notation P(A ∪ B) means the probability of the union of A and B.
维恩图和双向表是可视化并集和交集的强大工具。当我们给 A ∪ B 区域涂色时，就覆盖了 A 的所有结果、B 的所有结果以及重叠部分。记号 P(A ∪ B) 表示 A 和 B 并集的概率。
3. Addition Rule for Probabilities | 概率的加法法则
The general addition rule states: P(A ∪ B) = P(A) + P(B) – P(A ∩ B). This formula is valid for any two events A and B. The subtraction of P(A ∩ B) avoids double‑counting outcomes that belong to both A and B. For mutually exclusive events, P(A ∩ B) = 0, so the rule simplifies to P(A ∪ B) = P(A) + P(B).
一般加法法则为：P(A ∪ B) = P(A) + P(B) – P(A ∩ B)。该公式对任意两个事件 A 和 B 均成立。减去 P(A ∩ B) 可以避免重复计数同时属于 A 和 B 的结果。对于互斥事件，P(A ∩ B) = 0，于是公式简化为 P(A ∪ B) = P(A) + P(B)。
cannot happen at the same time. For example, when drawing one card from a deck, the events 'drawing a King' and 'drawing a Queen' ly are mutually exclusive because a single card cannot be both a King and a Queen. In this case, P(King ∪ Queen) = P(King) + P(Queen) = 4/52 + 4/52 = 8/52.
互斥事件不能同时发生。例如，从一副扑克牌中抽一张牌，事件“抽到 K”和“抽到 Q”互斥，因为一张牌不可能同时是 K 和 Q。在这种情况下，P(K ∪ Q) = P(K) + P(Q) = 4/52 + 4/52 = 8/52。
4. Multiplication Rule and Independence | 乘法法则与独立性
the intersection probability is given by the product of the individual probabilities: P(A ∩ B) = P(A) × P(B). This is the multiplication rule for independent events.
如果事件 A 的发生不影响事件 B 的概率，那么 A 和 B 就是相互独立的。对于独立事件，交集的概率等于各自概率的乘积：P(A ∩ B) = P(A) × P(B)。这就是独立事件的乘法法则。
To check for independence, we can test whether P(A ∩ B) = P(A) × P(B) holds. Alternatively, we can verify that P(A| e as the unconditional probability of A. If this equality fails, the events are dependent.
要检验独立性，可以验证 P(A ∩ B) = P(A) × P(B) 是否成立。或者验证 P(A|B) = P(A)，即已知 B 发生的条件下 A 的条件概率等于 A 的无条件概率。如果等式不成立，则两事件相关。
5. Conditional Probability | 条件概率
d of an event occurring given that another event has already occurred. The notation is P(A|B), read as 'probability of A given B', and is defined by the formula: P(A|B) = P(A ∩ ), provided P(B) > 0. This formula rearranges to give the general multiplication rule: P(A ∩ B) = P(A|B) × P(B).
条件概率衡量在另一事件已经发生的条件下，某事件发生的可能性。记作 P(A|B)，读作“在 B 发生的条件下 A 的概率”，其定义为：P(A|B) = P(A ∩ B) / P(B)，其中 P(B) > 0。该公式可变形为一般乘法法则：P(A ∩ B) = P(A|B) × P(B)。
Conditional probability helps update our beliefs after receiving new information. For instance, if we know a person is a smoker, the probability of developing a certain health condition may be higher than in the general population. On tree diagrams, second‑branch probabilities are often conditional.
条件概率可以在获得新信息后更新我们的认知。例如，如果我们知道一个人吸烟，那么患上某种疾病的概率可能会比普通人群更高。在树状图中，第二级分支上的概率通常就是条件概率。
6. Tree Diagrams for Combined Events | 联合事件的树状图
Tree diagrams are extremely useful for displaying sequences of events, especially when there are two or more stages. Each branch represents an outcome and is labelled with its probability. For independent events, the probabilities along branches remain the same at each stage. For conditional cases, they vary according to previous outcomes.
树状图非常适合展示事件的先后顺序，尤其是在有两个或更多阶段时。每个分支代表一种结果，并标记其概率。对于独立事件，各阶段分支上的概率保持不变。如果是条件概率，则根据先前的结果而改变。
To find the probability of a combined path, we multiply the probabilities along the branches. To find the probability of an event that occurs via several different paths, we add the path probabilities. This approach directly implements the multiplication and addition rules.
要找到某条组合路径的概率，我们将沿途各分支的概率相乘。要找到可通过多条不同路径实现的事件的概率，我们将各条路径的概率相加。这种方法直接体现了乘法和加法法则。
7. Probability Laws and Set Notation Practice | 概率定律与集合符号练习
Let us apply the rules with a two‑way table. Consider a class of 30 students: 18 study Chemistry (C), 12 study Physics (P), and 5 study both. The table can be drawn as:
让我们用一个双向表来应用这些规则。假设一个班级有 30 名学生：18 人学习化学 (C)，12 人学习物理 (P)，5 人同时学习两科。可绘制表格如下：
| P | not P | Total | |
|---|---|---|---|
| C | 5 | 13 | 18 |
| not C | 7 | 5 | 12 |
| Total | 12 | 18 | 30 |
From the table we compute: P(C ∩ P) = 5/30 = 1/6. P(C ∪ P) = P(C) + P(P) – P(C ∩ P) = 18/30 + 12/30 – 5/30 = 25/30 = 5/6. The conditional probability P(P|C) = P(P ∩ C)/P(C) = (5/30)/(18/30) = 5/18.
根据表格可计算：P(C ∩ P) = 5/30 = 1/6。P(C ∪ P) = P(C) + P(P) – P(C ∩ P) = 18/30 + 12/30 – 5/30 = 25/30 = 5/6。条件概率 P(P|C) = P(P ∩ C)/P(C) = (5/30)/(18/30) = 5/18。
Are the events independent? P(C) × P(P) = (18/30) × (12/30) = 216/900 = 0.24, while P(C ∩ se are not equal, studying Chemistry and Physics are not independent events in this class.
事件独立吗？P(C) × P(P) = (18/30) × (12/30) = 216/900 = 0.24，而 P(C ∩ P) = 5/30 ≈ 0.1667。两者不相等，因此在这个班级中，学习化学和物理不是独立事件。
8. More Complex Combined Events | 更复杂的联合事件
ts are combined, the same principles extend naturally. For independent events A, B, C: P(A ∩ B ∩ C) = P(A) × P(B) × P(C). The union formula for three events expands to: P(A ∪ B ∪ C) = P(A) + P(B) + P(C) – P(A ∩ B) – P(A ∩ C) – P(B ∩ C) + P(A ∩ B ∩ C).
当三个或更多事件组合时，同样的原则自然延伸。对于独立事件 A, B, C：P(A ∩ B ∩ C) = P(A) × P(B) × P(C)。三个事件的并集公式扩展为：P(A ∪ B ∪ C) = P(A) + P(B) + P(C) – P(A ∩ B) – P(A ∩ C) – P(B ∩ C) + P(A ∩ B ∩ C)。
often tackle such problems using tree diagrams with three levels or by constructing three‑way Venn diagrams. The key is to track intersections carefully and avoid double‑ h complements can sometimes simplify calculations, for example finding P(at least one) = 1 – P(none).
学生通常借助三层树状图或三环维恩图来解决此类问题。关键在于谨慎跟踪交集，避免重复计数。有时利用补集可以简化计算，例如 P(至少有一个发生) = 1 – P(一个都不发生)。
9. Common Mistakes with Combined Events | 联合事件中的常见错误
A frequent error is to add probabilities without checking for mutual exclusivity. If events are not mutually exclusive, simply adding P(A) + P(B) will overestimate the true probability because the intersection is counted twice. Always apply the general addition rule.
一个常见错误是未检查互斥性就直接相加概率。如果事件不互斥，仅仅计算 P(A) + P(B) 会高估真实概率，因为交集被计数了两次。务必使用一般加法法则。
Another mistake is to assume independence without justification. Many real‑world events are dependent, and multiplying probabilities blindly will yield incorrect results. Always verify independence using the definition or given data before applying the simple multiplication rule.
另一个错误是在没有依据的情况下假设独立性。许多现实事件是相关的，盲目相乘概率会产生错误结果。在应用简单乘法法则前，务必使用定义或给定数据验证独立性。
Additionally, when a question asks for the probability of 'A and B,' do not confuse it with 'A or B.' The wording 'and' means intersection, while 'or' (in the inclusive sense) means union. Careful reading is essential.
此外，当题目询问“A 和 B”的概率时，不要将其与“A 或 B”混淆。“且”表示交集，而“或”（包含性）表示并集。仔细审题至关重要。
10. Expected Value and Combined Probabilities | 期望值与联合概率
dom variable X with values xᵢ and probabilities pᵢ = P(X = xᵢ), the expected value is E(X) = Σ xᵢpᵢ. If Y = aX + b, then E(Y) = aE(X) + b.
期望值 E(X) 的概念与联合事件密切相关，尤其是当随机变量由多个结果组合定义时。对于离散型随机变量 X，取值为 xᵢ，概率为 pᵢ = P(X = xᵢ)，期望值为 E(X) = Σ xᵢpᵢ。若 Y = aX + b，则 E(Y) = aE(X) + b。
vents allows us to compute the total expected gain or loss. For instance, if a player wins £5 with probability 0.2, wins £1 with probability 0.5, and loses £2 with probability 0.3, the expected value is E = (5 × 0.2) + (1 × 0.5) + (–2 × 0.3) = 1 + 0.5 – 0.6 = 0.9. This means an average gain of £0.90 per game.
在机会游戏中，组合事件可以让我们计算总期望盈利或亏损。例如，一名玩家以 0.2 的概率赢 5 英镑，以 0.5 的概率赢 1 英镑，以 0.3 的概率输 2 英镑，则期望值为 E = (5 × 0.2) + (1 × 0.5) + (–2 × 0.3) = 1 + 0.5 – 0.6 = 0.9。这意味着平均每局赢 0.90 英镑。
11. Combined Events in Hypothesis Testing | 假设检验中的联合事件
In A‑Level statistics, combined events appear implicitly in hypothesis testing. When we calculate a p‑value for a binomial test, we sum the probabilities of several mutually exclusive outcomes — essentially a union of simple events. For a two‑tailed test, we combine probabilities from both tails of the distribution.
在 A‑Level 统计学中，联合事件隐含地出现在假设检验里。当我们计算二项检验的 p 值时，我们要求出几个互斥结局的概率之和——本质上是简单事件的并集。对于双尾检验，我们将分布左尾和右尾的概率组合起来。
Understanding how to add probabilities for non‑ s is the same skill required for combined events. If the test statistic is discrete, the p‑ observed) or P(X ≥ observed), and these are unions of mutually exclusive simple outcomes.
理解如何将不重叠区域的概率相加，正是处理联合事件所需的技能。如果检验统计量是离散的，p 值通常是 P(X ≤ 观测值) 或 P(X ≥ 观测值)，而这些就是互斥简单结果的并集。
12. Revision Checklist for Combined Events | 联合事件复习清单
- Define sample space and list outcomes systematically.
- Distinguish between union (or) and intersection (and).
- Memorise the general addition rule: P(A ∪ B) = P(A) + P(B) – P(A ∩ B).
- Test for mutual exclusivity by checking if P(A ∩ B) = 0.
- Test for independence by verifying P(A ∩ B) = P(A) × P(B) or P(A|B) = P(A).
- Apply the multiplication rule for independent events correctly.
- Use conditional probability formula P(A|B) = P(A ∩ B)/P(B).
- Draw and interpret tree diagrams and Venn diagrams accurately.
- Recognise when to add path probabilities and when to multiply along a branch.
- Watch out for wording: 'given that,' 'at least,' 'neither,' 'both.'
- 定义样本空间并系统性地列出结果。
- 区分并集（或）和交集（且）。
- 熟记一般加法法则：P(A ∪ B) = P(A) + P(B) – P(A ∩ B)。
- 通过检查 P(A ∩ B) = 0 来判断互斥性。
- 通过验证 P(A ∩ B) = P(A) × P(B) 或 P(A|B) = P(A) 来检验独立性。
- 正确应用独立事件的乘法法则。
- 使用条件概率公式 P(A|B) = P(A ∩ B)/P(B)。
- 准确绘制并解读树状图和维恩图。
- 识别何时应加总路径概率，何时应沿分支相乘。
- 注意关键词：“已知……”“至少”“都不”“同时发生”。
Mastering combined events provides the foundation for more advanced topics such as the binomial distribution, normal approximations, and conditional expectation.
掌握联合事件为进一步学习二项分布、正态近似和条件期望等高级主题奠定了基础。
Published by TutorHao | Probability & Statistics Revision Series | aleveler.com
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