📚 Mutually Exclusive and Independent Events | 互斥事件与独立事件
In Edexcel A-Level Mathematics, probability questions often require you to decide whether two events are mutually exclusive, independent, both, or neither. These two concepts describe different relationships between events, and mixing them up is one of the most common errors in Statistics.
在 Edexcel A-Level 数学中，概率题经常要求你判断两个事件是互斥、独立、两者都是，还是都不是。这两个概念描述事件之间不同的关系，混淆它们是统计学中最常见的错误之一。
1. Core Definitions | 核心定义
Two events A and B are mutually exclusive if they cannot happen at the same time. On a Venn diagram, their regions do not overlap, so
两个事件 A 和 B 互斥，指的是它们不能同时发生。在维恩图上，它们的区域不重叠，因此
P(A ∩ B) = 0
Two events A and B are independent if the occurrence of one does not affect the probability of the other. The formal condition is
两个事件 A 和 B 独立，指的是一个事件的发生不影响另一个事件的概率。正式的判断条件是
P(A ∩ B) = P(A) × P(B)
or equivalently P(A|B) = P(A) when P(B) > 0.
或者等价地，当 P(B) > 0 时，P(A|B) = P(A)。
2. Mutually Exclusive Events and the Addition Law | 互斥事件与加法法则
For mutually exclusive events A and B, the probability that A or B occurs is simply the sum of their individual probabilities.
对于互斥事件 A 和 B，A 或 B 发生的概率就是它们各自概率的和。
P(A ∪ B) = P(A) + P(B)
This is a special case of the addition law. For example, when rolling a fair six-sided die, the events "roll a 2" and "roll a 5" are mutually exclusive, so
这是加法法则的特殊情形。例如，掷一枚公平的六面骰子时，事件 "掷出 2" 和 "掷出 5" 互斥，因此
P(2 or 5) = 1/6 + 1/6 = 2/6 = 1/3
3. The General Addition Law for Any Two Events | 任意两事件的加法法则
If two events are not mutually exclusive, adding P(A) and P(B) double-counts the overlap, so we subtract P(A ∩ B).
如果两个事件不是互斥的，把 P(A) 和 P(B) 相加会重复计算重叠部分，因此要减去 P(A ∩ B)。
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
This formula always holds, including for mutually exclusive events where the last term is zero.
这个公式始终成立，包括互斥事件的情形，此时最后一项为零。
Example: In a class of 30 students, 12 study Biology, 8 study Chemistry, and 4 study both. The probability that a randomly chosen student studies at least one of these subjects is
例题：在一个 30 名学生的班级中，12 名学习生物，8 名学习化学，4 名两门都学。随机选一名学生，至少学习其中一门的概率为
P(B ∪ C) = 12/30 + 8/30 − 4/30 = 16/30 = 8/15
4. Independent Events and the Multiplication Law | 独立事件与乘法法则
When events are independent, the probability that both A and B occur is the product of their individual probabilities.
当事件独立时，事件 A 和 B 同时发生的概率等于它们各自概率的乘积。
P(A ∩ B) = P(A) × P(B)
For example, if the probability that Alice passes a test is 0.8 and the probability that Ben passes the same test is 0.7, and their results are independent, then
例如，如果 Alice 通过测试的概率是 0.8，Ben 通过测试的概率是 0.7，并且他们的成绩相互独立，那么
P(both pass) = 0.8 × 0.7 = 0.56
Independence often comes from real-world assumptions: flipping two coins, rolling two dice, or drawing with replacement.
独立性通常来自现实假设：抛两枚硬币、掷两颗骰子，或有放回地抽取。
5. Using the Multiplication Law in Tree Diagrams | 树图中的乘法法则
In Edexcel questions, you often multiply along branches of a probability tree only when the trials are independent or when conditional probabilities are known.
在 Edexcel 考题中，只有当各次试验独立，或者已知条件概率时，你才可以在概率树的各分支上相乘。
he tree shows all outcomes. For instance, tossing a biased coin twice with P(Heads) = 0.6:
对于两次独立试验，树状图显示所有结果。例如，抛一枚偏硬币两次，P(正面) = 0.6：
- P(HH) = 0.6 × 0.6 = 0.36 | P(两正) = 0.6 × 0.6 = 0.36
- P(HT) = 0.6 × 0.4 = 0.24 | P(正反) = 0.6 × 0.4 = 0.24
- P(TH) = 0.4 × 0.6 = 0.24 | P(反正) = 0.4 × 0.6 = 0.24
- P(TT) = 0.4 × 0.4 = 0.16 | P(两反) = 0.4 × 0.4 = 0.16
Multiplying along branches is justified by the multiplication law for independent events.
沿分支相乘的依据是独立事件的乘法法则。
6. Conditional Probability Link | 条件概率联系
Independence can also be checked using conditional probability. If A and B are independent and P(B) > 0, then
独立性也可以用条件概率来检验。如果 A 和 B 独立且 P(B) > 0，那么
P(A|B) = P(A)
This means knowing that B has happened gives no extra information about A.
这意味着知道 B 已经发生并不会提供关于 A 的额外信息。
From the general multiplication law,
由一般乘法法则，
P(A ∩ B) = P(B) × P(A|B)
If P(A|B) = P(A), this reduces to the independence formula.
如果 P(A|B) = P(A)，它就简化为独立性的公式。
7. Can Events Be Both Mutually Exclusive and Independent? | 事件可以既互斥又独立吗？
This is a favourite exam discussion point. If two events are mutually exclusive, then P(A ∩ B) = 0. If they are also independent, then P(A ∩ B) = P(A) × P(B). Equating gives
这是考试中喜欢讨论的一个点。如果两个事件互斥，则 P(A ∩ B) = 0。如果它们同时独立，则 P(A ∩ B) = P(A) × P
Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图