📚 IB Math: Foundations of Probability Theory and Core Concepts | IB数学：概率论基础与核心概念梳理
Probability is the mathematical language of uncertainty. In the IB Mathematics curriculum — whether you are taking Analysis and Approaches (AA) or Applications and Interpretation (AI) — probability forms a central pillar of statistics, decision-making, and real-world modelling.
概率是描述不确定性的数学语言。在IB数学课程中，无论你修读的是分析与方法（AA）还是应用与解释（AI），概率都是统计学、决策制定和现实建模的核心支柱。
1. Random Experiments and Sample Space | 随机试验与样本空间
A random experiment is a process whose outcome cannot be predicted with certainty before it is performed. Examples include tossing a fair coin, rolling a six-sided die, or measuring the lifetime of a battery.
随机试验是指在进行之前无法确定其结果的过程。例如抛一枚均匀硬币、掷一颗六面骰子，或测量电池寿命。
The set of all possible outcomes of a random experiment is called the sample space, denoted by \(S\) or \(\Omega\). An individual outcome is called a sample point.
随机试验所有可能结果的集合称为样本空间，通常记为S或Ω。单个结果称为样本点。
For example, rolling a fair die yields the sample space S = {1, 2, 3, 4, 5, 6}. When the sample space is finite or countably infinite, we call it a discrete sample space.
例如，掷一颗均匀骰子的样本空间为S = {1, 2, 3, 4, 5, 6}。当样本空间是有限集或可数无限集时，我们称之为离散样本空间。
2. Events and Event Operations | 事件与事件运算
An event is a subset of the sample space. If an outcome belongs to that subset, we say the event has occurred. For instance, landing on an even number when rolling a die is the event A = {2, 4, 6}.
事件是样本空间的子集。若某个结果属于该子集，则称该事件发生。例如，掷骰子的结果为偶数这一事件可写为A = {2, 4, 6}。
Key operations on events mirror set theory:
事件的关键运算与集合论完全对应：
- Union A ∪ B: the event that A or B (or both) occurs.
- 并集A ∪ B：A或B（或两者）发生的事件。
- Intersection A ∩ B: the event that both A and B occur.
- 交集A ∩ B：A和B同时发生的事件。
- Complement A′ or Aᶜ: the event that A does not occur.
- 补事件A′或Aᶜ：A不发生的事件。
- Mutually exclusive: A ∩ B = ∅ means A and B cannot occur together.
- 互斥：A ∩ B = ∅，表示A和B不可能同时发生。
The empty set ∅ is the impossible event, and the full sample space S is the certain event.
空集∅表示不可能事件，而整个样本空间S表示必然事件。
3. Definitions of Probability | 概率的定义
The classical definition assumes all outcomes in a finite sample space are equally likely. If event A contains n(A) favourable outcomes and S contains n(S) equally likely outcomes, then:
古典定义假设有限样本空间中的所有结果出现的可能性相等。若事件A包含n(A)个有利结果，S包含n(S)个等可能结果，则：
P(A) = n(A) / n(S)
This definition works well for dice, coins, and cards, but it fails when outcomes are not equally likely.
这一定义在骰子、硬币和扑克牌问题中非常有效，但当结果并非等可能时便会失效。
The frequency definition interprets probability as the limit of relative frequency as the number of trials tends to infinity. The formal foundation, however, rests on the axiomatic definition due to Kolmogorov.
频率定义将概率解释为试验次数趋于无穷时相对频率的极限。然而，真正的数学基础建立在柯尔莫哥洛夫提出的公理化定义之上。
4. Axioms and Basic Rules | 概率公理与基本法则
Kolmogorov’s axioms provide the foundation for all probability calculations:
柯尔莫哥洛夫公理为所有概率计算提供了基础：
- Axiom 1: 0 ≤ P(A) ≤ 1 for every event A.
- 公理1：对任意事件A，有0 ≤ P(A) ≤ 1。
- Axiom 2: P(S) = 1.
- 公理2：P(S) = 1。
- Axiom 3: If events are pairwise mutually exclusive, then the probability of their union equals the sum of their probabilities.
- 公理3：若事件两两互斥，则它们并集的概率等于各自概率之和。
From these axioms we derive the complement rule and the addition rule:
由这些公理可推导出补事件法则与加法法则：
P(A′) = 1 − P(A)
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
When A and B are mutually exclusive, the term P(A ∩ B) = 0, so the formula simplifies to P(A ∪ B) = P(A) + P(B).
当A与B互斥时，P(A ∩ B) = 0，公式简化为P(A ∪ B) = P(A) + P(B)。
5. Counting Principles and Probability | 计数原理与概率
Many probability problems require counting outcomes systematically. The fundamental counting principle states that if task 1 has m ways and task 2 has n ways, then the sequence of tasks has m × n ways.
许多概率问题需要系统地计数结果。乘法计数原理指出：如果任务1有m种方式、任务2有n种方式，那么依次完成这两个任务共有m × n种方式。
Permutations count arrangements where order matters:
排列计算顺序有影响时的安排数：
P(n, r) = n! / (n − r)!
Combinations count selections where order does not matter:
组合计算顺序无影响时的选取数：
C(n, r) = n! / [r! (n − r)!]
For example, picking 3 students from 10 to form a committee involves C(10, 3) = 120 possible committees. If one particular student must be included, the count reduces to C(9, 2) = 36.
例如，从10名学生中选出3人组成委员会，共有C(10, 3) = 120种可能。若某一名特定学生必须入选，则组合数降为C(9, 2) = 36。
6. Conditional Probability | 条件概率
Conditional probability measures the likelihood of event A given that event B has already occurred. It is denoted P(A|B) and defined as:
条件概率衡量在事件B已经发生的前提下，事件A发生的可能性，记为P(A|B)，其定义为：
P(A|B) = P(A ∩ B) / P(B), provided P(B) > 0
Rearranging this definition gives the multiplication rule:
重排这一定义即可得到乘法法则：
P(A ∩ B) = P(A|B) × P(B) = P(B|A) × P(A)
In IB problems, tree diagrams are extremely helpful for tracking conditional probabilities across successive events.
在IB题目中，树形图对追踪连续事件中的条件概率极为有用。
7. Independent Events | 独立事件
Events A and B are independent if the occurrence of one does not affect the probability of the other. Mathematically, independence is defined by:
若事件A的发生不影响事件B发生的概率，则称A与B相互独立。数学上，独立性定义为：
P(A ∩ B) = P(A) × P(B)
Equivalently, P(A|B) = P(A) and P(B|A) = P(B), provided the relevant denominators are positive.
等价地，在相应分母为正的条件下，有P(A|B) = P(A)且P(B|A) = P(B)。
Independence and mutual exclusivity are fundamentally different concepts. In fact, if A and B are mutually exclusive and both have positive probability, they cannot be independent, because P(A ∩ B) = 0 while P(A)P(B) > 0.
独立与互斥是两个根本不同的概念。事实上，若A与B互斥且概率均为正，则它们不可能独立，因为此时P(A ∩ B) = 0，而P(A)P(B) > 0。
8. Bayes’ Theorem | 贝叶斯定理
Bayes’ theorem connects P(A|B) to P(B|A). For two events, it states:
贝叶斯定理将P(A|B)与P(B|A)联系起来。对于两个事件，它表述为：
P(A|B) = [P(B|A) × P(A)] / P(B)
Using the total probability rule, P(B) can be expanded. If the events A₁, A₂, …, Aₙ form a partition of the sample space and each has positive probability, then the general form is:
利用全概率公式，P(B)可以展开。若事件A₁, A₂, …, Aₙ构成样本空间的一个分割且各自概率为正，则贝叶斯定理的一般形式为：
P(Aᵢ|B) = [P(B|Aᵢ) P(Aᵢ)] / [Σⱼ P(B|Aⱼ) P(Aⱼ)]
Bayes’ theorem is particularly important in diagnostic testing, spam filtering, and decision theory — all contexts explored in IB AI applications.
贝叶斯定理在医学诊断检验、垃圾邮件过滤和决策理论中尤为重要——这些都是IB AI课程中所探讨的实际应用场景。
9. Discrete Random Variables | 离散型随机变量
In IB Mathematics, a key concept is the random variable itself.
在IB数学中，随机变量本身是一个关键概念。
A discrete random variable X takes values x₁, x₂, … with probabilities p₁, p₂, … that satisfy two conditions:
离散型随机变量X取值为x₁, x₂, …，对应概率为p₁, p₂, …，且满足两个条件：
- 0 ≤ pᵢ ≤ 1 for every i;
- 所有pᵢ满足0 ≤ pᵢ ≤ 1；
- Σ pᵢ = 1 over all possible values of X.
- 对所有可能的X值，有Σ pᵢ = 1。
The probability distribution of X can be given by a table, a piecewise formula, or a graph.
X的概率分布可以用表格、分段公式或图形来表示。
The cumulative distribution function is defined as F(x) = P(X ≤ x), and for a discrete variable, it is a step function that accumulates probability up to the value x.
累积分布函数定义为F(x) = P(X ≤ x)。对于离散变量，它是一个阶梯函数，逐段累加不超过x的概率。
10. Expectation and Variance | 期望与方差
The expected value of a discrete random variable is the probability-weighted average of its possible values:
离散型随机变量的期望值是它所有可能取值的概率加权平均：
E(X) = Σ xᵢ pᵢ
The variance measures the spread of the distribution around the mean:
方差衡量分布围绕均值的离散程度：
Var(X) = E(X²) − [E(X)]²
where E(X²) = Σ xᵢ² pᵢ. The standard deviation is the square root of the variance.
其中E(X²) = Σ xᵢ² pᵢ。标准差为方差的正平方根。
Key linear transformation rules:
关键的线性变换法则：
E(aX + b) = aE(X) + b
Var(aX + b) = a²Var(X)
11. Key Distributions: Binomial and Normal | 二项分布与正态分布
If X counts the number of successes in n independent trials, each with success probability p, then X follows a binomial distribution: X ~ B(n, p). Its probability mass function is:
若X表示n次独立试验中的成功次数，每次成功概率均为p，则X服从二项分布：X ~ B(n, p)。其概率质量函数为：
P(X = k) = C(n, k) pᵏ (1 − p)ⁿ⁻ᵏ, for k = 0, 1, …, n
Its expectation and variance are:
其期望与方差分别为：
E(X) = np, Var(X) = np(1 − p)
IB AA and AI both require proficiency with the binomial distribution — in AA, approximately 12–15% of the HL exam is devoted to probability and statistics.
IB AA和AI课程都要求学生熟练掌握二项分布——在AA中，概率与统计约占HL考试内容的12%–15%。
The normal distribution is the most important continuous distribution. Its density function is symmetric and bell-shaped, and a standard normal variable Z has mean 0 and variance 1. Standardisation converts any normal variable X ~ N(μ, σ²) into Z:
正态分布是最重要的连续型分布。其密度函数对称且呈钟形，标准正态变量Z的均值为0、方差为1。标准化可将任意正态变量X ~ N(μ, σ²)转换为Z：
Z = (X − μ) / σ
The empirical rule states that approximately 68% of values lie within one standard deviation of the mean, 95% within two, and 99.7% within three.
经验法则指出，约68%的数据落在均值的一个标准差之内，95%落在两个标准差之内，99.7%落在三个标准差之内。
12. Common Mistakes and Exam Tips | 常见错误与考试技巧
Many students lose marks by using the addition rule without subtracting the intersection when events are not mutually exclusive.
许多学生因为事件不互斥却直接用加法法则、忘记减去交集而丢分。
A frequent misconception is treating conditional probability P(A|B) as equal to P(A ∩ B). Remember that P(A|B) always involves division by P(B), so the sample space is restricted.
一个常见误区是将条件概率P(A|B)等同于P(A ∩ B)。请记住，P(A|B)始终需要除以P(B)，因此样本空间被限制缩小了。
When drawing a tree diagram, always label each branch with the correct conditional probability, and multiply along branches, then sum across disjoint final outcomes.
在画树形图时，务必为每条分支标注正确的条件概率，沿分支相乘，再对不同末梢结果求和。
Check that all probabilities in a distribution sum to 1 — this is the fastest way to detect arithmetic errors.
检查分布中所有概率之和是否为1——这是发现算术错误最快的方式。
Finally, interpret the context. A probability of 0.95 in a medical test does not automatically mean the patient has the disease; the prior probability matters through Bayes’ theorem.
最后，一定要结合情境进行解读。医学检验中0.95的概率并不自动意味着患者患病；先验概率需要通过贝叶斯定理发挥作用。
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