A-Level Mathematics: Probability Formulas Summary and Applications | A-Level数学：概率公式整理与运用

📚 A-Level Mathematics: Probability Formulas Summary and Applications | A-Level数学：概率公式整理与运用

Probability is a core topic in A-Level Mathematics, and mastering the key formulas is essential for solving exam questions effectively. This article summarises the most important probability rules, distributions, and their applications in a clear, revision-friendly format.

概率是A-Level数学的核心内容，熟练掌握关键公式是高效解题的基础。本文以清晰的复习形式，总结最重要的概率法则、分布及其应用。

1. Basic Probability Notation | 基本概率记号

Let A and B be events. The probability of event A is written P(A), where 0 ≤ P(A) ≤ 1. The complement of A is written A′ or Aᶜ, and P(Aᶜ) = 1 − P(A).

设A和B为事件。事件A的概率写作P(A)，满足0 ≤ P(A) ≤ 1。A的对立事件记作A′或Aᶜ，且P(Aᶜ) = 1 − P(A)。

For any two events, the addition rule connects the union and intersection:

对于任意两个事件，加法法则将并集与交集联系起来：

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

If A and B are mutually exclusive, then P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B).

若A与B互斥，则P(A ∩ B) = 0，因此P(A ∪ B) = P(A) + P(B)。

2. Conditional Probability | 条件概率

The conditional probability of A given B is the probability that A occurs, under the assumption that B has already occurred. The fundamental formula is:

在B已发生的条件下，A发生的概率称为条件概率。基本公式为：

P(A | B) = P(A ∩ B) / P(B), where P(B) > 0

This formula can be rearranged to give the multiplication rule:

该公式可以变形得到乘法法则：

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

Be careful: P(A | B) is not the same as P(B | A). They are equal only in special cases.

注意：P(A | B)与P(B | A)不同，它们仅在特殊情况下相等。

3. 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. The key conditions are:

若一个事件的发生不影响另一个事件的概率，则称事件A与B相互独立。关键条件为：

P(A ∩ B) = P(A) × P(B)

Equivalently, P(A | B) = P(A) and P(B | A) = P(B) when the relevant probabilities are non-zero.

等价地，当相关概率非零时，P(A | B) = P(A) 且 P(B | A) = P(B)。

Do not confuse independent events with mutually exclusive events. Mutually exclusive events cannot occur together, so they are strongly dependent unless one has probability 0.

不要将独立事件与互斥事件混淆。互斥事件不能同时发生，因此它们高度相关，除非其中一个概率为0。

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

If events B₁, B₂, …, Bₙ form a partition of the sample space, meaning they are mutually exclusive and their union is the whole sample space, then for any event A:

若事件B₁, B₂, …, Bₙ构成样本空间的一个划分，即它们互斥且并集为整个样本空间，则对任意事件A有：

P(A) = P(A | B₁)P(B₁) + P(A | B₂)P(B₂) + … + P(A | Bₙ)P(Bₙ)

This formula is especially useful when the probability of A is difficult to compute directly but is easy to compute within each partition part.

当A的概率难以直接计算，但在每个划分部分中容易计算时，该公式尤其有用。

For the simplest case with two partitions B and Bᶜ:

对于仅含两个划分B和Bᶜ的最简单情形：

P(A) = P(A | B)P(B) + P(A | Bᶜ)P(Bᶜ)

5. Bayes’ Theorem | 贝叶斯定理

Bayes’ theorem allows us to reverse a conditional probability. For a partition B₁, B₂, …, Bₙ and an event A with P(A) > 0:

贝叶斯定理允许我们反转条件概率。对于划分B₁, B₂, …, Bₙ和满足P(A) > 0的事件A：

P(Bᵢ | A) = P(A | Bᵢ)P(Bᵢ) / P(A)

The denominator P(A) is usually computed using the law of total probability.

分母P(A)通常用全概率公式计算。

In A-Level questions, Bayes’ theorem often appears in medical testing or quality-control contexts. Always identify clearly which event is the “cause” and which is the “observed result”.

在A-Level考题中，贝叶斯定理常出现在医学检测或质量控制的背景中。务必清楚识别哪个事件是”原因”，哪个是”观察结果”。

6. Discrete Random Variables | 离散随机变量

A discrete random variable X takes distinct values x₁, x₂, … with probabilities p₁, p₂, … such that each pᵢ ≥ 0 and Σpᵢ = 1.

离散随机变量X取不同的值x₁, x₂, …，对应概率p₁, p₂, …，满足每个pᵢ ≥ 0且Σpᵢ = 1。

The expected value (mean) of X is:

随机变量X的期望（均值）为：

E(X) = μ = Σ xᵢ P(X = xᵢ)

The variance is defined as:

方差定义为：

Var(X) = E[(X − μ)²] = E(X²) − [E(X)]²

Where E(X²) = Σ xᵢ² P(X = xᵢ). Remember to use the formula E(X²) − μ² for quicker calculation.

其中E(X²) = Σ xᵢ² P(X = xᵢ)。记住使用E(X²) − μ²可加快计算。

7. Linear Combinations of Variables | 变量的线性组合

For constants a and b, the expectation and variance of a linear transformation are:

对于常数a和b，线性变换的期望与方差为：

E(aX + b) = aE(X) + b

Var(aX + b) = a²Var(X)

For two independent variables X and Y:

对于两个独立变量X和Y：

E(X + Y) = E(X) + E(Y)

Var(X + Y) = Var(X) + Var(Y)

Also Var(X − Y) = Var(X) + Var(Y) for independent variables. Note that variances add, not subtract, when variables are independent.

独立变量同样有Var(X − Y) = Var(X) + Var(Y)。注意：变量独立时，方差是相加而不是相减。

8. Binomial Distribution | 二项分布

A binomial distribution models the number of successes in n independent trials, each with the same success probability p. Write X ~ B(n, p). The probability of exactly r successes is:

二项分布用于建模n次独立试验中的成功次数，每次试验成功概率p相同。记X ~ B(n, p)。恰好r次成功的概率为：

P(X = r) = C(n, r) × pʳ × (1 − p)ⁿ⁻ʳ

where C(n, r) = n! / [r!(n − r)!] is the binomial coefficient.

其中C(n, r) = n! / [r!(n − r)!]为二项式系数。

For X ~ B(n, p):

对于X ~ B(n, p)：

E(X) = np, Var(X) = np(1 − p)

Conditions: fixed number of trials, two outcomes, constant probability, and independence between trials.

条件：固定试验次数、两种结果、概率恒定、各次试验相互独立。

9. Poisson Distribution | 泊松分布

The Poisson distribution models the number of rare events occurring in a fixed interval of time or space. Write X ~ Po(λ), where λ is the average rate. The probability function is:

泊松分布用于建模在固定时间或空间间隔内稀有事件发生的次数。记X ~ Po(λ)，其中λ为平均发生率。概率函数为：

P(X = r) = e⁻ˡᵃᵐᵇᵈᵃ × λʳ / r!

For a Poisson distribution:

对于泊松分布：

E(X) = λ, Var(X) = λ

The equal mean and variance is a special property. If X ~ B(n, p) with n large and p small, then X can be approximated by Po(λ) where λ = np.

期望等于方差是其特殊性质。若X ~ B(n, p)，且n很大、p很小，则X可用Po(λ)近似，其中λ = np。

10. Normal Distribution | 正态分布

A continuous random variable X with mean μ and variance σ² is written X ~ N(μ, σ²). Its probability density function is symmetrical about the mean.

具有均值μ和方差σ²的连续随机变量X记作X ~ N(μ, σ²)。其概率密度函数关于均值对称。

To find probabilities, standardise to the standard normal variable Z:

为求概率，需标准化为标准正态变量Z：

Z = (X − μ) / σ ~ N(0, 1)

Then use the standard normal table to look up Φ(z) = P(Z ≤ z). For example:

然后使用标准正态分布表查Φ(z) = P(Z ≤ z)。例如：

P(X < a) = Φ((a − μ) / σ)

Remember to apply a continuity correction when approximating a discrete distribution, such as B(n, p) or Po(λ), by a normal distribution.

当使用正态分布近似离散分布（如B(n, p)或Po(λ)）时，记得应用连续性修正。

11. Normal Approximation to the Binomial | 正态近似二项分布

If X ~ B(n, p) and n is large such that np > 5 and n(1 − p) > 5, then X can be approximated by:

若X ~ B(n, p)，且n足够大，使得np > 5且n(1 − p) > 5，则X可近似为：

X ≈ N(np, np(1 − p))

When using this approximation, apply a continuity correction. For example:

使用该近似时，需进行连续性修正。例如：

P(X ≤ r) ≈ Φ((r + 0.5 − np) / √[np(1 − p)])

P(X ≥ r) ≈ 1 − Φ((r − 0.5 − np) / √[np(1 − p)])

This method is commonly tested in A-Level statistics papers.

此方法在A-Level统计试卷中经常考到。

12. Revision Tips and Common Pitfalls | 复习建议与常见陷阱

Here are some key reminders when applying probability formulas:

以下是应用概率公式时的一些关键提醒：

Always check that the probabilities in a distribution sum to 1.
Use P(A ∩ B) = P(A) + P(B) only for mutually exclusive events.
Use P(A ∩ B) = P(A)P(B) only for independent events.
When computing variance, use E(X²) − [E(X)]² and do not confuse E(X²) with [E(X)]².
For binomial and Poisson approximations to normal, always include the continuity correction.
Clearly state the distribution and parameters before writing any formula in exam answers.

始终检查分布中所有概率之和为1。

只有互斥事件才能使用P(A ∩ B) = P(A) + P(B)。

只有独立事件才能使用P(A ∩ B) = P(A)P(B)。

计算方差时，使用E(X²) − [E(X)]²，不要混淆E(X²)与[E(X)]²。

二项分布或泊松分布用正态近似时，务必加上连续性修正。

在考试答题中，先写明分布及其参数，再使用公式。

Mastering these formulas and understanding their conditions will greatly improve your accuracy and speed in probability questions. Practice past-paper questions to build confidence.

掌握这些公式并理解其适用条件，将大幅提高你在概率题中的准确性与速度。多做真题练习以增强信心。

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A-Level Mathematics: Probability Formulas Summary and Applications | A-Level数学：概率公式整理与运用