📚 Probability Formulae | 概率公式
Probability formulae form the backbone of A-Level Mathematics in the Edexcel specification. They allow us to model uncertainty, calculate likelihoods in complex scenarios, and connect conditional relationships through logical structure. Mastering these formulas is essential for tackling problems involving events, tree diagrams, discrete random variables, and statistical inference.
概率公式是 Edexcel A-Level 数学的核心基础。它们帮助我们模拟不确定性、计算复杂情境下的可能性，并通过逻辑结构将条件关系联系起来。掌握这些公式对于解决涉及事件、树图、离散随机变量和统计推断的问题至关重要。
1. Basic Probability Rules | 基本概率法则
. The probability of the entire sample space S is P(S) = 1. The complement of A, denoted A', has probability P(A') = 1 − P(A).
对任意事件 A，概率 P(A) 满足 0 ≤ P(A) ≤ 1。样本空间 S 的概率为 P(S) = 1。A 的补集记为 A'，其概率为 P(A') = 1 − P(A)。
The addition rule for any two events A and B is: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). This avoids double-counting outcomes that belong to both events.
任意两个事件 A 和 B 的加法法则为：P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。这样可以避免重复计算同时属于两个事件的结果。
2. Conditional Probability | 条件概率
ability of event A occurring given that event B has already occurred. It is defined as P(A|B) = P(A ∩ B) / P(B), provided P(B) > 0. This formula re-scales the probability space to B.
条件概率衡量在事件 B 已发生的条件下事件 A 发生的概率。其定义为 P(A|B) = P(A ∩ B) / P(B)，前提是 P(B) > 0。该公式将概率空间重新缩放至 B。
It is important to interpret the vertical bar correctly: P(A|B) reads "the probability of A given B". Conditional probability forms the basis for multiplication rules and Bayes' theorem.
正确理解竖线很重要：P(A|B) 读作“给定 B 时 A 的概率”。条件概率是乘法法则和贝叶斯定理的基础。
3. Multiplication Rule and Independent Events | 乘法法则与独立事件
From the definition of conditional probability, we obtain the multiplication rule: P(A ∩ B) = P(B) × P(A|B) = P(A) × P(B|A). This allows us to calculate joint probabilities in sequential experiments.
根据条件概率的定义，我们可以得到乘法法则：P(A ∩ B) = P(B) × P(A|B) = P(A) × P(B|A)。这让我们能够计算序贯试验中的联合概率。
of one does not affect the probability of the other. Formally, independence satisfies P(A|B) = P(A) and P(B|A) = P(B), or equivalently P(A ∩ B) = P(A) × P(B).
如果两个事件 A 和 B 的发生互不影响，则它们独立。形式上，独立性满足 P(A|B) = P(A) 且 P(B|A) = P(B)，或者等价地 P(A ∩ B) = P(A) × P(B)。
For a set of mutually independent events A₁, A₂, … , Aₙ, the probability of all occurring is the product of their individual probabilities.
对于一组相互独立的事件 A₁, A₂, … , Aₙ，全部发生的概率等于各自概率的乘积。
4. Mutually Exclusive Events and the Addition Rule | 互斥事件与加法法则
mutually exclusive (or disjoint) if they cannot happen at the same time. In this case, P(A ∩ B) = 0, and the addition rule simplifies to P(A ∪ B) = P(A) + P(B).
如果事件 A 和 B 不可能同时发生，则称它们互斥（或不交）。此时 P(A ∩ B) = 0，加法法则简化为 P(A ∪ B) = P(A) + P(B)。
For several mutually exclusive events A₁, A₂, … , Aₙ that partition the sample space, the sum of their probabilities is 1. This property is used in the law of total probability.
对于划分样本空间的多个互斥事件 A₁, A₂, … , Aₙ，它们的概率之和为 1。该性质在全概率公式中非常有用。
5. Probability Tree Diagrams | 概率树图
Tree diagrams visually represent sequences of events and their conditional probabilities. Each branch is labelled with a probability, and the probabilities on branches from a single node sum to 1.
树图能够直观地表示事件序列及其条件概率。每条分支标有概率，从同一个节点出发的分支概率之和为 1。
To find the probability of a particular path, we multiply the probabilities along the branches. For combined events across different paths, we add the path probabilities if the paths correspond to mutually exclusive outcomes.
要计算某一条路径的概率，我们将该路径上各分支的概率相乘。如果不同路径对应互斥的结果，则将这些路径的概率相加，得到组合事件的概率。
Tree diagrams are especially useful for applying the multiplication rule P(A ∩ B) = P(A) × P(B| peatedly, and they provide a clear structure for solving conditional probability problems.
树图特别适合反复应用乘法法则 P(A ∩ B) = P(A) × P(B|A)，并为解决条件概率问题提供了清晰的结构。
6. Law of Total Probability | 全概率公式
Suppose events B₁, B₂, … , Bₙ form a partition of the sample space, meaning they are mutually exclusive and exhaustive. For any event A, the probability of A can be expressed as:
假设事件 B₁, B₂, … , Bₙ 构成样本空间的一个划分，即它们互斥且完备。对于任意事件 A，其概率可以表示为：
P(A) = Σ P(Bᵢ) × P(A|Bᵢ)
其中 Σ 表示对所有 i 求和。这就是全概率公式，它将 A 的概率分解为各个条件下概率的加权平均。
This formula is extremely powerful when the conditional probabilities P(A|Bᵢ) are easy to obtain, but P(A) directly is not. It underpins many applications in diagnostic testing and risk analysis.
当条件概率 P(A|Bᵢ) 容易获得而直接求 P(A) 较难时，该公式非常强大。它在诊断测试和风险分析中有广泛的应用基础。
7. Bayes’ Theorem | 贝叶斯定理
Bayes’ theorem reverses the direction of conditional probability. It uses prior knowledge about related events to update the probability of an event after new evidence is observed. The formula for a partition B₁, B₂, … , Bₙ is:
贝叶斯定理反转了条件概率的方向。它利用相关事件的先验知识，在观测到新证据后更新某事件的概率。对于划分 B₁, B₂, … , Bₙ，其公式为：
P(Bₖ|A) = [P(A|Bₖ) × P(Bₖ)] / [Σ P(Bᵢ) × P(A|Bᵢ)]
分母就是全概率公式中的 P(A)。这个定理可以解释为：后验概率正比于似然函数乘以先验概率。
In simpler cases with only two complementary events B and B', Bayes’ theorem becomes P(B|A) = [P(A|B)P(B)] / [P(A|B)P(B) + P(A|B')P(B')]. This form frequently appears in exam questions on medical testing or false positives.
在只有两个互补事件 B 和 B' 的简单情形中，贝叶斯定理变为 P(B|A) = [P(A|B)P(B)] / [P(A|B)P(B) + P(A|B')P(B')]。这种形式经常出现在医学检测或假阳性的考题中。
8. Discrete Probability Distributions | 离散概率分布
A discrete random variable X takes a countable number of values x₁, x₂, … ᵢ). The probability distribution must satisfy Σ P(X = xᵢ) = 1 and P(X = xᵢ) ≥ 0 for all i.
离散随机变量 X 取可数个值 x₁, x₂, …，每个值对应概率 P(X = xᵢ)。概率分布必须满足 Σ P(X = xᵢ) = 1，且对所有 i 有 P(X = xᵢ) ≥ 0。
ility that X is at most a certain value. It is often used to find probabilities of ranges.
累积分布函数 F(x) = P(X ≤ x) 给出了 X 不超过某个值的概率，常用于求区间概率。
Common discrete distributions in the Edexcel syllabus include the discrete uniform distribution and the binomial distribution, though the latter is treated separately with its own probability mass formula.
Edexcel 考试大纲中常见的离散分布包括离散均匀分布和二项分布，不过二项分布另有单独的概率质量公式处理。
9. Expected Value and Variance | 期望与方差
mean) of a discrete random variable X is E(X) = μ = Σ xᵢ P(X = xᵢ). It represents the long‑term average outcome if the experiment is repeated many times.
离散随机变量 X 的期望值（或均值）为 E(X) = μ = Σ xᵢ P(X = xᵢ)。它表示当试验重复多次时的长期平均结果。
A useful computational formula is Var(X) = E(X²) − [E(X)]², where E(X²) = Σ xᵢ² P(X = xᵢ).
X 的方差衡量分布的离散程度，定义为 Var(X) = E[(X − μ)²] = Σ (xᵢ − μ)² P(X = xᵢ)。一个实用的计算式为 Var(X) = E(X²) − [E(X)]²，其中 E(X²) = Σ xᵢ² P(X = xᵢ)。
ures are essential for understanding the reliability of predictions and hypothesis testing.
标准差是方差的平方根：σ = √Var(X)。这些度量对于理解预测的可靠性和假设检验至关重要。
10. Using Probability Formulae in Problem Solving | 概率公式在解题中的应用
Examination questions often require combining several formulae. Start by defining events clearly, using notation like A, B, A', B'. Identify whether events are independent or mutually exclusive, and choose the appropriate rule.
考试题目经常需要综合运用多个公式。首先明确地定义事件，使用 A, B, A', B' 等符号。判断事件是独立还是互斥，然后选择相应的法则。
When a problem involves sequential stages or conditional information, draw a tree diagram or use the law of total probability to break down complex probabilities into manageable parts. Always check that probabilities sum to 1 at each branch or partition.
当题目涉及多个阶段或条件信息时，可以绘制树图或使用全概率公式，将复杂概率分解为易于处理的部分。务必检查每个分支或划分处的概率之和是否为 1。
For “reverse conditional” problems, apply Bayes’ theorem after finding P(A) via total probability. Keep your working systematic to avoid errors in substitution.
对于“反向条件”问题，先用全概率公式求出 P(A)，再应用贝叶斯定理。保持解题过程有条不紊，以避免代入错误。
11. Common Mistakes and How to Avoid Them | 常见错误及避免方法
Many students confuse P(A|B) with P(B| read the condition carefully. Another error is assuming independence without justification—use the multiplicative test P(A ∩ B) = P(A)P(B) to verify.
许多学生容易混淆 P(A|B) 和 P(B|A)；务必仔细阅读条件。另一个常见错误是在没有依据的情况下假设独立——应使用乘法检验 P(A ∩ B) = P(A)P(B) 进行验证。
When using the addition rule, remember to subtract the intersection P(A ∩ B) unless events are mutually exclusive. In tree diagrams, forgetting to add the probabilities of different relevant paths is a frequent slip.
使用加法法则时，除非事件互斥，否则要记得减去交集 P(A ∩ B)。在树图中，忘记将不同相关路径的概率相加是常见失误。
With probability distributions, ensure that your probabilities sum exactly to 1 and that no probability is negative. For variance, use the computational formula carefully to avoid rounding errors in long calculations.
处理概率分布时，要确保概率之和恰好为 1，且没有负值。计算方差时，仔细使用计算式以避免在冗长计算中出现舍入误差。
12. Summary and Exam Tips | 总结与考试技巧
Probability formulae in Edexcel A-Level Mathematics revolve around conditional probability, multiplication and addition rules, total probability, Bayes’ theorem, and the expectation and variance of discrete random variables. Mastering the logical connections between these concepts is the key to success.
Edexcel A-Level 数学中的概率公式主要围绕条件概率、乘法和加法法则、全概率公式、贝叶斯定理以及离散随机变量的期望和方差展开。掌握这些概念之间的逻辑联系是成功的关键。
In the exam, show clear notation, state the formula you are using, and interpret results in context. Practice past paper questions that mix conditional probability with discrete distributions, and always check your answers for consistency.
在考试中，要清楚标示符号，写明所使用的公式，并结合上下文解释结果。多做将条件概率和离散分布混合在一起的历年真题，并始终检查答案的合理性。
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