Core Methods for Probability Calculations in A-Level Mathematics | A-Level 数学：概率计算的核心方法

📚 Core Methods for Probability Calculations in A-Level Mathematics | A-Level 数学：概率计算的核心方法

Probability is a central topic in A-Level Mathematics, requiring a clear understanding of definitions, rules, and counting techniques. This article presents the core methods you need to solve probability questions reliably and efficiently.

概率是 A-Level 数学的核心主题，要求清晰理解定义、规则和计数技巧。本文整理了在考试中稳定、高效求解概率问题所需的核心方法。

1. Basic Definitions and Axioms of Probability | 概率的基本定义与公理

Probability measures how likely an event is to occur. For any event \(A\), the probability \(P(A)\) satisfies: \(0 \le P(A) \le 1\). The sum of probabilities of all mutually exclusive and exhaustive outcomes in a sample space is 1.

概率衡量事件发生的可能性。对任意事件 \(A\)，概率 \(P(A)\) 满足：\(0 \le P(A) \le 1\)。样本空间中所有互斥且穷尽的结果的概率之和为 1。

The probability of an impossible event is 0: \(P(\emptyset)=0\).

不可能事件的概率为 0：\(P(\emptyset)=0\)。
The probability of a certain event is 1: \(P(S)=1\), where \(S\) is the sample space.

必然事件的概率为 1：\(P(S)=1\)，其中 \(S\) 是样本空间。
Probabilities can be interpreted as long-run relative frequencies or as equally likely outcomes when outcomes are symmetric.

概率既可以解释为长期相对频率，也可以在所有结果等可能时按等可能结果计算。

For equally likely outcomes: \(P(A) = \frac{|A|}{|S|}\)

对于等可能结果：\(P(A) = \frac{|A|}{|S|}\)

2. Sample Space and Events | 样本空间与事件

The sample space \(S\) is the set of all possible outcomes of a random experiment. An event is a subset of the sample space. Representing the sample space clearly is often the first step in solving a probability problem.

样本空间 \(S\) 是随机试验所有可能结果的集合。事件是样本空间的子集。清晰地表示样本空间通常是解决概率问题的第一步。

Use Venn diagrams to visualise unions, intersections, and complements of events.

使用维恩图可视化事件的并、交和补。
Use two-way tables for problems involving two categorical variables.

使用二维表处理涉及两个分类变量的概率问题。
Use list or systematic listing to avoid missing outcomes in small sample spaces.

在小样本空间中使用列举法或系统列举，避免遗漏结果。

Example: Tossing two fair coins has sample space \(\{HH, HT, TH, TT\}\). The event “at least one head” is \(\{HH, HT, TH\}\).

例如：抛两枚均匀硬币的样本空间为 \(\{HH, HT, TH, TT\}\)。事件“至少一个正面”为 \(\{HH, HT, TH\}\)。

3. Complement and Addition Rules | 互补法则与加法法则

The complement of event \(A\), written \(A’\), is the set of outcomes not in \(A\). The complement rule states \(P(A’) = 1 – P(A)\). The addition rule connects the probability of a union with the probabilities of the individual events and their intersection.

事件 \(A\) 的补事件记为 \(A’\)，是不在 \(A\) 中的结果集合。互补法则指出 \(P(A’) = 1 – P(A)\)。加法法则将并事件的概率与各事件及其交的概率联系起来。

Addition rule: \(P(A \cup B) = P(A) + P(B) – P(A \cap B)\)

加法法则：\(P(A \cup B) = P(A) + P(B) – P(A \cap B)\)

If \(A\) and \(B\) are mutually exclusive, then \(P(A \cap B)=0\), so \(P(A \cup B)=P(A)+P(B)\).

如果 \(A\) 与 \(B\) 互斥，则 \(P(A \cap B)=0\)，因此 \(P(A \cup B)=P(A)+P(B)\)。
The complement rule is especially useful when “at least one” appears in the question.

互补法则在问题中出现“至少一个”时尤其有用。

At least one: \(P(\text{at least one}) = 1 – P(\text{none})\)

至少一个：\(P(\text{至少一个}) = 1 – P(\text{一个也没有})\)

4. Conditional Probability | 条件概率

Conditional probability measures the probability of one event given that another event has occurred. It is written \(P(A|B)\), read as “probability of \(A\) given \(B\)”.

条件概率衡量在已知另一个事件发生的情况下某一事件发生的概率。记作 \(P(A|B)\)，读作“在 \(B\) 条件下 \(A\) 的概率”。

\(P(A|B) = \frac{P(A \cap B)}{P(B)}\), provided \(P(B)>0\)

\(P(A|B) = \frac{P(A \cap B)}{P(B)}\)，前提是 \(P(B)>0\)

Conditional probabilities can also be read directly from tree diagrams or two-way tables by restricting the sample space.

条件概率也可以通过缩小样本空间直接从树状图或二维表中读取。
When computing \(P(A|B)\), always check whether the denominator \(P(B)\) is given or must be calculated.

计算 \(P(A|B)\) 时，务必检查分母 \(P(B)\) 是已知的还是需要计算的。

Example: In a class of 30 students, 18 study physics and 12 study both physics and mathematics. Given that a randomly chosen student studies physics, the probability they also study mathematics is \(12/18 = 2/3\).

例如：一个班有 30 名学生，其中 18 人学习物理，12 人同时学习物理和数学。已知随机选出的学生学物理，则他也学数学的概率是 \(12/18 = 2/3\)。

5. Multiplication Rule and Independent Events | 乘法法则与独立事件

The multiplication rule is derived from the definition of conditional probability. It allows us to find the probability of an intersection by multiplying conditional probabilities.

乘法法则由条件概率的定义推导而来。它通过相乘条件概率来求交事件的概率。

\(P(A \cap B) = P(A)P(B|A) = P(B)P(A|B)\)

Two events \(A\) and \(B\) are independent if the occurrence of one does not affect the probability of the other. In that case, \(P(A|B)=P(A)\) and the multiplication rule simplifies.

两个事件 \(A\) 和 \(B\) 独立，是指一个事件的发生不影响另一个事件发生的概率。此时 \(P(A|B)=P(A)\)，乘法法则简化为：

Independent events: \(P(A \cap B) = P(A)P(B)\)

独立事件：\(P(A \cap B) = P(A)P(B)\)

Do not confuse “mutually exclusive” with “independent”. Mutually exclusive events cannot both occur; independent events can.

不要混淆“互斥”与“独立”。互斥事件不可能同时发生；独立事件可以同时发生。
For three independent events, \(P(A \cap B \cap C)=P(A)P(B)P(C)\).

对于三个独立事件，\(P(A \cap B \cap C)=P(A)P(B)P(C)\)。

6. Tree Diagrams and Systematic Listing | 树状图与系统列举

Tree diagrams are powerful tools for multi-stage probability problems. Each branch represents a possible outcome, and the probability written on each branch is usually a conditional probability.

树状图是处理多阶段概率问题的强大工具。每条分支代表一个可能结果，分支上标注的概率通常是条件概率。

Multiply along branches to find the probability of a sequence of events.

沿分支相乘可得到事件序列发生的概率。
Add results from different branches when the question asks for “or” or “at least”.

当问题涉及“或”或“至少”时，将不同分支的结果相加。
Label every branch clearly and check that all probabilities at each node sum to 1.

清晰标记每条分支，并检查每个节点处的概率之和为 1。

Example: A bag contains 3 red and 5 blue balls. Two balls are drawn without replacement. The probability that both are red is \(\frac{3}{8} \times \frac{2}{7} = \frac{3}{28}\).

例如：一个袋子中有 3 个红球和 5 个蓝球。不放回地抽取两个球。两个球都是红球的概率是 \(\frac{3}{8} \times \frac{2}{7} = \frac{3}{28}\)。

7. Permutations and Combinations in Probability | 排列与组合在概率计算中的应用

For equally likely outcomes, counting the number of favorable outcomes and the total number of outcomes is essential. Permutations count ordered arrangements, while combinations count unordered selections.

对于等可能结果，计算有利结果数和总结果数至关重要。排列计数有序排列，组合计数无序选取。

\(^{n}P_{r} = \frac{n!}{(n-r)!}\), \(^{n}C_{r} = \binom{n}{r} = \frac{n!}{r!(n-r)!}\)

\(^{n}P_{r} = \frac{n!}{(n-r)!}\)，\(^{n}C_{r} = \binom{n}{r} = \frac{n!}{r!(n-r)!}\)

Use combinations when the order of selection does not matter, such as selecting a committee from a group.

当选取顺序不重要时使用组合，例如从一组人中选取委员会。
Use permutations when the order matters, such as arranging students in seats.

当顺序重要时使用排列，例如安排学生入座。
In probability, the probability of selecting a specific hand of cards is \(1 / {^{52}C_{5}}\).

在概率中，抽到特定一手牌的概率是 \(1 / {^{52}C_{5}}\)。

Example: From 6 boys and 4 girls, choose 3 students at random. The probability that all chosen are boys is \(\frac{^{6}C_{3}}{^{10}C_{3}} = \frac{20}{120} = \frac{1}{6}\).

例如：从 6 名男生和 4 名女生中随机选 3 人。选出的全是男生的概率是 \(\frac{^{6}C_{3}}{^{10}C_{3}} = \frac{20}{120} = \frac{1}{6}\)。

8. Law of Total Probability | 全概率公式

The law of total probability allows us to calculate the probability of an event \(B\) by splitting the sample space into mutually exclusive parts. If \(A_1, A_2, \dots, A_n\) form a partition of the sample space, then for any event \(B\):

全概率公式通过将样本空间划分为互斥部分来计算事件 \(B\) 的概率。如果 \(A_1, A_2, \dots, A_n\) 构成样本空间的一个划分，则对任意事件 \(B\)：

\(P(B) = P(A_1)P(B|A_1) + P(A_2)P(B|A_2) + \cdots + P(A_n)P(B|A_n)\)

This formula is especially useful when the sample space is naturally divided into groups, such as factories producing items or gender groups in a survey.

该公式在样本空间自然分组时特别有用，例如工厂生产的产品或调查中的性别分组。
Tree diagrams often implement the law of total probability by summing the relevant terminal branches.

树状图通常通过将所有相关末端分支相加来实现全概率公式。

Example: Two machines produce items. Machine A produces 60% of items and has a 2% defect rate; Machine B produces the rest with a 5% defect rate. The overall defect probability is \(0.6 \times 0.02 + 0.4 \times 0.05 = 0.032\).

例如：两台机器生产零件。机器 A 生产 60% 的零件，次品率为 2%；机器 B 生产其余零件，次品率为 5%。总次品概率为 \(0.6 \times 0.02 + 0.4 \times 0.05 = 0.032\)。

9. Bayes’ Theorem | 贝叶斯定理

Bayes’ theorem is used to reverse the direction of conditioning. It tells us the posterior probability \(P(A_i|B)\) after observing \(B\), based on prior probabilities and likelihoods.

贝叶斯定理用于逆转条件的方向。它在观察到 \(B\) 之后，基于先验概率和似然给出后验概率 \(P(A_i|B)\)。

\(P(A_i|B) = \frac{P(A_i)P(B|A_i)}{P(B)}\), where \(P(B)\) is found by the law of total probability

\(P(A_i|B) = \frac{P(A_i)P(B|A_i)}{P(B)}\)，其中 \(P(B)\) 由全概率公式求得

Bayes’ theorem is frequently tested with tree diagrams or two-way tables. Identifying \(A_i\) as the “cause” and \(B\) as the “effect” helps.

贝叶斯定理常与树状图或二维表结合考查。将 \(A_i\) 识别为“原因”、\(B\) 识别为“结果”有助于解题。
Always compute \(P(B)\) first if it is not directly given.

如果 \(P(B)\) 未直接给出，务必先计算它。

Example: In the previous machine example, if an item is defective, the probability it came from machine A is \(\frac{0.6 \times 0.02}{0.032} = 0.375\).

例如：在前面的机器例子中，如果零件是次品，它来自机器 A 的概率是 \(\frac{0.6 \times 0.02}{0.032} = 0.375\)。

10. Random Variables and Expectation | 随机变量与期望

In many probability problems, we assign numerical values to outcomes. A random variable \(X\) maps outcomes to numbers, and its probability distribution lists each value \(x\) with its probability \(P(X=x)\).

在许多概率问题中，我们为结果赋予数值。随机变量 \(X\) 将结果映射为数值，其概率分布列出每个取值 \(x\) 及其概率 \(P(X=x)\)。

The sum of all probabilities in a discrete distribution is 1: \(\sum P(X=x) = 1\).

离散分布中所有概率之和为 1：\(\sum P(X=x) = 1\)。
The expected value is the weighted average of outcomes: \(E(X) = \sum xP(X=x)\).

期望值是结果的加权平均：\(E(X) = \sum xP(X=x)\)。
The variance is \(Var(X) = E[(X-\mu)^2] = \sum (x-\mu)^2 P(X=x)\).

方差是 \(Var(X) = E[(X-\mu)^2] = \sum (x-\mu)^2 P(X=x)\)。

Example: For a fair die, \(E(X) = \frac{1}{6}(1+2+3+4+5+6) = 3.5\).

例如：对于公平骰子，\(E(X) = \frac{1}{6}(1+2+3+4+5+6) = 3.5\)。

Key formulas: \(E(aX+b)=aE(X)+b\), \(Var(aX+b)=a^2Var(X)\)

关键公式：\(E(aX+b)=aE(X)+b\)，\(Var(aX+b)=a^2Var(X)\)

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Core Methods for Probability Calculations in A-Level Mathematics | A-Level 数学：概率计算的核心方法