Probability Formulae | 概率公式汇总与考点精讲

📚 Probability Formulae | 概率公式汇总与考点精讲

Probability formulae underpin almost every statistics question in the Edexcel A-Level Mathematics specification. Mastering these formulae allows you to convert a written scenario into a calculation and to justify which rule you are using. This revision guide covers the essential probability formulae, how they link together, and how to choose the right one under exam conditions.

概率公式是 Edexcel A-Level 数学大纲中几乎所有统计题的基础。掌握这些公式能让你把文字情境转化为计算，并说明你使用的是哪一条法则。本复习指南涵盖核心概率公式、它们之间的联系，以及如何在考试中选用正确公式。

1. Basic Probability Laws | 基本概率法则

For any event A in a sample space S, the probability satisfies 0 ≤ P(A) ≤ 1. The probability of the whole sample space is P(S) = 1, and the probability of the impossible empty set is P(∅) = 0. The complement rule states that P(A′) = 1 − P(A).

对于样本空间 S 中的任意事件 A，概率满足 0 ≤ P(A) ≤ 1。整个样本空间的概率为 P(S) = 1，不可能事件的概率为 P(∅) = 0。补事件法则为 P(A′) = 1 − P(A)。

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

The general addition rule is written above. We subtract P(A ∩ B) because the intersection is counted in both P(A) and P(B), so it must be removed once to avoid double counting. If A and B are mutually exclusive, the intersection is empty, so the rule simplifies to P(A ∪ B) = P(A) + P(B).

一般加法法则如上所示。我们要减去 P(A ∩ B)，因为交集在 P(A) 和 P(B) 中都被计入，必须减去一次以避免重复计算。如果 A 与 B 互斥，交集为空，因此法则简化为 P(A ∪ B) = P(A) + P(B)。

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2. Mutually Exclusive and Independent Events | 互斥事件与独立事件

Mutually exclusive events cannot occur at the same time. For example, a card cannot be both red and black, so P(red ∩ black) = 0. When two events are mutually exclusive, the addition rule becomes P(A ∪ B) = P(A) + P(B).

互斥事件指不能同时发生的事件。例如，一张牌不可能既是红色又是黑色，因此 P(红色 ∩ 黑色) = 0。当两个事件互斥时，加法法则变为 P(A ∪ B) = P(A) + P(B)。

Independent events mean that the occurrence of one event does not affect the probability of the other. In symbols, A and B are independent if P(A | B) = P(A), or equivalently:

独立事件指一个事件的发生不影响另一个事件的概率。用符号表示，A 与 B 独立当且仅当 P(A | B) = P(A)，等价地：

P(A ∩ B) = P(A) × P(B)

Students often confuse independent events with mutually exclusive events. If A and B are mutually exclusive and both have positive probability, then they cannot be independent, because P(A | B) = 0 while P(A) > 0.

学生常把独立事件与互斥事件混淆。如果 A 与 B 互斥且概率都为正，那么它们不可能独立，因为 P(A | B) = 0 而 P(A) > 0。

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3. Conditional Probability Formula | 条件概率公式

The conditional probability of A given B is defined by:

在已知事件 B 发生的条件下，事件 A 发生的概率定义为：

P(A | B) = P(A ∩ B) / P(B)

This formula is valid only when P(B) > 0. It measures how likely A is within the reduced sample space of B. In a two-way table, find the marginal total for B first, then divide the cell frequency for A ∩ B by that total. In a Venn diagram, the denominator is the entire region representing B.

该公式仅在 P(B) > 0 时成立。它衡量在 B 的缩减样本空间中 A 发生的可能性。在双向表中，先找出 B 的边缘总数，再用 A ∩ B 的单元格频数除以该总数。在维恩图中，分母就是代表 B 的整个区域。

For example, if P(A ∩ B) = 0.2 and P(B) = 0.4, then P(A | B) = 0.2 / 0.4 = 0.5.

例如，如果 P(A ∩ B) = 0.2 且 P(B) = 0.4，则 P(A | B) = 0.2 / 0.4 = 0.5。

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4. Multiplication Rule | 乘法法则

Rearranging the conditional probability formula gives the multiplication rule:

把条件概率公式变形即可得到乘法法则：

P(A ∩ B) = P(A) × P(B | A) = P(B) × P(A | B)

This rule is used extensively in probability tree diagrams. On a tree diagram, multiply probabilities along one path to find the joint probability of that path. If the event can occur by more than one route, add the probabilities of all relevant paths.

该法则广泛用于概率树图。在树图中，沿一条路径相乘各分支概率，即可得到该路径的联合概率。如果事件可以通过多条路径发生，就把所有相关路径的概率相加。

If A and B are independent, then P(B | A) = P(B), so the multiplication rule becomes P(A ∩ B) = P(A) × P(B), which is the special case used for independent events.

如果 A 与 B 独立，则 P(B | A) = P(B)，因此乘法法则变为 P(A ∩ B) = P(A) × P(B)，这就是独立事件的特殊形式。

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5. The Law of Total Probability | 全概率公式

When an event A can happen through several mutually exclusive and exhaustive routes B₁, B₂, …, Bₙ, the total probability of A is:

当事件 A 可以通过若干互斥且穷尽的事件 B₁、B₂、…、Bₙ 发生时，A 的全概率为：

P(A) = Σ P(Bᵢ) × P(A | Bᵢ)

The events Bᵢ must form a partition of the sample space: they are mutually exclusive and together cover all possible outcomes. In tree diagrams, this means adding all the path probabilities that end at A.

事件 Bᵢ 必须构成样本空间的一个划分：它们互斥且共同覆盖所有可能结果。在树图中，这意味着把所有到达 A 的路径概率相加。

For example, when a diagnostic test can be positive for people with or without a disease, the overall probability of a positive result is found

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