Combined Events and Probability | 组合事件与概率

📚 Combined Events and Probability | 组合事件与概率

Combined events are at the heart of Edexcel A-Level Statistics. Understanding how to combine probabilities using addition and multiplication rules, and distinguishing between independent and mutually exclusive events, is essential for tackling exam questions on probability.

组合事件是爱德思 A-Level 统计学的核心内容。掌握如何运用加法法则和乘法法则来组合概率，并区分独立事件与互斥事件，对于解答概率相关考题至关重要。

1. Sample Space and Events | 样本空间与事件

A sample space, usually denoted by S, is the set of all possible outcomes of a random experiment. An event is any subset of the sample space. For example, when rolling a fair six-sided die, S = {1, 2, 3, 4, 5, 6} and the event “rolling an even number” is E = {2, 4, 6}.

样本空间通常用 S 表示，是随机实验所有可能结果的集合。事件是样本空间的任意子集。例如，抛掷一枚均匀六面骰子时，S = {1, 2, 3, 4, 5, 6}，事件”掷出偶数”为 E = {2, 4, 6}。

The probability of an event A is written P(A) and satisfies 0 ≤ P(A) ≤ 1. If all outcomes are equally likely, then P(A) = number of outcomes in A ÷ total number of outcomes in S.

事件 A 的概率记作 P(A)，满足 0 ≤ P(A) ≤ 1。如果所有结果等可能，则 P(A) = A 中的结果数 ÷ 样本空间 S 中的总结果数。

2. Probability Rules Recap | 概率法则回顾

For any event A, the complement of A, written A′ or Aᶜ, is the event that A does not occur. Its probability is given by P(A′) = 1 − P(A). This is known as the complement rule.

对于任何事件 A，A 的补集记作 A′ 或 Aᶜ，表示 A 不发生的事件。其概率由 P(A′) = 1 − P(A) 给出，这称为补集法则。

If A and B are two events, the intersection A ∩ B means both A and B occur. The union A ∪ B means A occurs or B occurs or both. These set operations are fundamental when combining events.

如果 A 和 B 是两个事件，交集 A ∩ B 表示 A 和 B 同时发生。并集 A ∪ B 表示 A 发生或 B 发生或两者都发生。这些集合运算是组合事件的基础。

P(A ∪ B) = probability that at least one of A or B occurs
P(A ∩ B) = probability that both A and B occur
P(A′) = probability that A does not occur

P(A ∪ B) = A 或 B 至少一个发生的概率
P(A ∩ B) = A 和 B 同时发生的概率
P(A′) = A 不发生的概率

3. Mutually Exclusive Events | 互斥事件

Two events A and B are mutually exclusive (or disjoint) if they cannot occur at the same time. In set notation, A ∩ B = ∅, where ∅ is the empty set.

两个事件 A 和 B 如果不可能同时发生，则称它们为互斥事件（或不相交事件）。用集合符号表示为 A ∩ B = ∅，其中 ∅ 是空集。

For mutually exclusive events, the addition rule simplifies because P(A ∩ B) = 0. Therefore, P(A ∪ B) = P(A) + P(B). For example, when drawing one card from a standard deck, the events “drawing a heart” and “drawing a spade” are mutually exclusive.

对于互斥事件，由于 P(A ∩ B) = 0，加法法则可以简化。因此 P(A ∪ B) = P(A) + P(B)。例如，从一副标准扑克牌中抽一张牌，事件”抽到红桃”和”抽到黑桃”是互斥的。

4. Independent Events | 独立事件

Two events A and B are independent if the occurrence of one does not affect the probability of the other. Mathematically, A and B are independent if P(A ∩ B) = P(A) × P(B).

两个事件 A 和 B 如果一个事件的发生不影响另一个事件发生的概率，则称它们为独立事件。数学上，A 和 B 独立当且仅当 P(A ∩ B) = P(A) × P(B)。

Independence also means P(A | B) = P(A) when P(B) > 0, and P(B | A) = P(B) when P(A) > 0. For example, tossing a fair coin twice: the result of the first toss does not affect the result of the second toss.

独立性还意味着当 P(B) > 0 时，P(A | B) = P(A)；当 P(A) > 0 时，P(B | A) = P(B)。例如，连续抛一枚均匀硬币两次：第一次的结果不影响第二次的结果。

P(A ∩ B) = P(A) × P(B) for independent events

独立事件满足 P(A ∩ B) = P(A) × P(B)

5. The Addition Rule | 加法法则

The general addition rule for any two events A and B is P(A ∪ B) = P(A) + P(B) − P(A ∩ B). The subtraction of P(A ∩ B) prevents double-counting outcomes that belong to both events.

任意两个事件 A 和 B 的一般加法法则是 P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。减去 P(A ∩ B) 是为了避免重复计算同时属于两个事件的结果。

If events are mutually exclusive, then P(A ∩ B) = 0, so the rule reduces to P(A ∪ B) = P(A) + P(B). This is often tested in Edexcel papers using Venn diagrams.

如果事件是互斥的，则 P(A ∩ B) = 0，因此法则简化为 P(A ∪ B) = P(A) + P(B)。爱德思考试中经常用维恩图来考查这一内容。

General: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Mutually exclusive: P(A ∪ B) = P(A) + P(B)

一般情况：P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
互斥事件：P(A ∪ B) = P(A) + P(B)

6. Conditional Probability | 条件概率

Conditional probability is the probability of event A occurring given that event B has already occurred. It is written P(A | B) and defined as P(A | B) = P(A ∩ B) / P(B), provided P(B) > 0.

条件概率是指在事件 B 已经发生的条件下，事件 A 发生的概率。记作 P(A | B)，定义为 P(A | B) = P(A ∩ B) / P(B)，前提是 P(B) > 0。

Similarly, P(B | A) = P(A ∩ B) / P(A) when P(A) > 0. Rearranging these formulas gives the multiplication rule for dependent events: P(A ∩ B) = P(A) × P(B | A).

类似地，当 P(A) > 0 时，P(B | A) = P(A ∩ B) / P(A)。重新整理这些公式即可得到非独立事件的乘法法则：P(A ∩ B) = P(A) × P(B | A)。

P(A | B) = P(A ∩ B) ÷ P(B)

7. The Multiplication Rule | 乘法法则

The multiplication rule connects joint probability to conditional probability. For any two events A and B, P(A ∩ B) = P(A) × P(B | A). This is the general form and always holds.

乘法法则将联合概率与条件概率联系起来。对于任意两个事件 A 和 B，P(A ∩ B) = P(A) × P(B | A)。这是一般形式，始终成立。

If A and B are independent, then P(B | A) = P(B), so the rule simplifies to P(A ∩ B) = P(A) × P(B). This is a key distinction that examiners expect you to identify from context or data.

如果 A 和 B 是独立的，则 P(B | A) = P(B)，因此法则简化为 P(A ∩ B) = P(A) × P(B)。这是考官期望你根据上下文或数据识别的一个关键区别。

Relationship	Formula
General intersection	P(A ∩ B) = P(A) × P(B \| A)
Independent events	P(A ∩ B) = P(A) × P(B)
Mutually exclusive events	P(A ∩ B) = 0

关系	公式
一般交集	P(A ∩ B) = P(A) × P(B \| A)
独立事件	P(A ∩ B) = P(A) × P(B)
互斥事件	P(A ∩ B) = 0

8. Tree Diagrams | 树形图

Tree diagrams are useful for multi-stage experiments where probabilities change from stage to stage. Each branch is labelled with a probability, and the probabilities on branches from the same node must sum to 1.

树形图适用于多阶段实验，其中每一阶段的概率会发生变化。每条分支都标有概率，从同一节点出发的所有分支的概率之和必须为 1。

To find the probability of a complete path, multiply the probabilities along the branches of that path. To find the probability of an event that includes more than one path, add the probabilities of the relevant paths.

求一条完整路径的概率时，将该路径上各分支的概率相乘。求包含多条路径的事件的概率时，将相关路径的概率相加。

For example, in a bag with 3 red and 2 blue balls, if two balls are drawn without replacement, the probability of drawing a red then a blue is (3/5) × (2/4) = 6/20 = 3/10. The denominator changes because the first ball is not replaced.

例如，一个袋中有 3 个红球和 2 个蓝球，如果不放回地抽取两个球，先红后蓝的概率为 (3/5) × (2/4) = 6/20 = 3/10。分母发生变化是因为第一个球没有被放回。

9. Venn Diagrams and Two-Way Tables | 维恩图与双向表

Venn diagrams visually represent sample spaces and relationships between events. The rectangle represents the sample space S, and circles inside represent events. Overlapping circles show intersections.

维恩图可视化地表示样本空间和事件之间的关系。矩形表示样本空间 S，内部的圆表示事件。重叠的圆表示交集。

Two-way tables (also called contingency tables) are another way to organise probabilities for two categorical variables. They are especially useful for calculating conditional probabilities and testing independence.

双向表（也称列联表）是组织两个分类变量概率的另一种方法。它们对于计算条件概率和检验独立性特别有用。

When using a two-way table, always check that the row and column totals sum correctly. Conditional probabilities can be read directly by restricting attention to the appropriate row or column.

使用双向表时，务必检查行合计和列合计是否正确。条件概率可以通过将注意力限制在相应的行或列上来直接读取。

10. Common Exam Pitfalls | 常见考试易错点

Many marks are lost by confusing mutually exclusive events with independent events. Mutually exclusive events cannot happen together, whereas independent events do not influence each other’s probabilities. They are different concepts.

许多分数因混淆互斥事件与独立事件而丢失。互斥事件不可能同时发生，而独立事件不会影响彼此的概率。它们是两个不同的概念。

Another common mistake is using P(A ∪ B) = P(A) + P(B) without checking whether A and B are mutually exclusive. If they are not, you must subtract P(A ∩ B).

另一个常见错误是在没有检验 A 和 B 是否互斥的情况下直接使用 P(A ∪ B) = P(A) + P(B)。如果它们不互斥，则必须减去 P(A ∩ B)。

Also, when using tree diagrams, remember to adjust probabilities after each branch when sampling without replacement. Forgetting to change the denominator is a frequent source of error.

此外，在使用树形图时，如果是不放回抽样，请记住在每条分支后调整概率。忘记改变分母是常见的错误来源。

Finally, always check that your final probability is between 0 and 1. If you get a probability greater than 1 or negative, revisit your formula and calculations.

最后，始终检查最终概率是否在 0 和 1 之间。如果得到大于 1 或负数的概率，请重新检查你的公式和计算。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Combined Events and Probability | 组合事件与概率