📚 IB Mathematics: Core Concepts of Probability | IB数学：概率核心知识点梳理
Probability is a fundamental branch of mathematics that quantifies uncertainty. In the IB Mathematics curriculum (both Analysis and Approaches and Applications and Interpretation), probability forms the backbone of statistical reasoning and decision-making. This article consolidates the core concepts you must master, from basic definitions to advanced distributions.
概率是数学中量化不确定性的基本分支。在IB数学课程中（无论是分析与方法，还是应用与解释），概率都是统计推理和决策的基石。本文系统梳理你必须掌握的核心概念，从基础定义到进阶分布。
1. Sample Space and Events | 样本空间与事件
The sample space, denoted as S, is the set of all possible outcomes of a random experiment. An event e space. For example, when rolling a fair six-sided die, S = {1, 2, 3, 4, 5, 6}. The event "getting an even number" is E = {2, 4, 6}.
样本空间，记为 S，是随机试验所有可能结果的集合。一个事件是样本空间的任意子集。例如，掷一枚公平的六面骰子，S = {1, 2, 3, 4, 5, 6}。“掷出偶数”这一事件为 E = {2, 4, 6}。
Key set operations are essential: union (A ∪ B) means "A or B occurs", intersection (A ∩ B) means "A and B both occur", and complement A′ means "A does not occur". The empty set ∅ represents an impossible event, while the sample space S represents a certain event.
关键的集合运算非常重要：并集 (A ∪ B) 表示“A或B发生”，交集 (A ∩ B) 表示“A和B同时发生”，补集 A′ 表示“A不发生”。空集 ∅ 表示不可能事件，样本空间 S 表示必然事件。
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
2. The Probability Axioms | 概率公理
oms: (1) P(A) ≥ 0 for any event A; (2) P(S) = 1; (3) For mutually exclusive events A₁, A₂, …, P(A₁ ∪ A₂ ∪ …) = P(A₁) + P(A₂) + …. These axioms guarantee a consistent mathematical framework.
概率是从事件到实数的函数 P，满足三条公理：(1) 对任意事件 A，P(A) ≥ 0；(2) P(S) = 1；(3) 对互斥事件 A₁, A₂, …，P(A₁ ∪ A₂ ∪ …) = P(A₁) + P(A₂) + …。这些公理保证了数学框架的一致性。
A direct consequence is P(A′) = 1 − P(A). Also, if A ⊂ B, then P(A) ≤ P(B). All probabilities lie between 0 and 1 inclusive.
一个直接推论是 P(A′) = 1 − P(A)。此外，若 A ⊂ B，则 P(A) ≤ P(B)。所有概率都在 0 到 1 之间（含端点）。
3. 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. It is defined as:
条件概率是指在已知事件 B 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), provided P(B) > 0
Rearranging this gives the multiplication rule: P(A ∩ B) = P(A | is essential when analyzing sequential events, such as drawing cards without replacement.
重新整理该式可得乘法法则：P(A ∩ B) = P(A | B) × P(B)。这在分析连续事件（如不放回抽牌）时至关重要。
IB exam questions often require you to interpret a tree diagram or a two-way table to find a conditional probability. Remember that conditioning "shrinks" the sample space to the given event.
IB考试中常要求你根据树状图或二维表来计算条件概率。切记，条件化相当于把样本空间“缩小”到已知事件上。
4. 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 formal definition is:
如果事件 A 的发生不影响事件 B 发生的概率，则称 A 与 B 相互独立。其正式定义为：
P(A ∩ B) = P(A) × P(B)
Equivalently, if P(B) > 0, independence also means P(A | B) = P(A). Note that mutually exclusive events (A ∩ B = ∅) are generally not independent, except when one of them has probability 0.
等价地，若 P(B) > 0，独立性也可表示为 P(A | B) = P(A)。注意，互斥事件（A ∩ B = ∅）通常不独立，除非其中之一的概率为 0。
Common IB questions: "Two events are independent; one has P(A)=0.4 and P(A∪B)=0.7, find P(B)." Always apply the product rule in such cases.
常见IB题型：“设两事件独立，已知 P(A)=0.4，P(A∪B)=0.7，求 P(B)。”此类问题应直接使用乘法法则。
5. Bayes' Theorem | 贝叶斯定理
Bayes' theorem relates conditional probabilities in reverse direction. For events A₁, A₂, …, Aₙ that partition the sample space, and an event B with P(B) > 0:
贝叶斯定理用于反向计算条件概率。若 A₁, A₂, …, Aₙ 构成样本空间的一个划分，且 P(B) > 0，则：
P(Aᵢ | B) = P(B | Aᵢ) × P(Aᵢ) / Σⱼ P(B | Aⱼ) × P(Aⱼ)
In IB, Bayes' theorem is typically applied with two or three partitions. A tree diagram is the most reliable tool: the numerator is the path leading through Aᵢ to B; the denominator is the sum of all paths that end at B.
在IB课程中，贝叶斯定理通常用于两个或三个划分的情形。树状图是最可靠的工具：分子是经过 Aᵢ 到达 B 的路径概率；分母是所有到达 B 的路径概率之和。
For a two-case example, if a test for a disease has 95% sensitivity and 90% specificity, and the prevalence is 1%, Bayes' theorem allows you to compute the probability that a positive test actually means the person has the disease.
以一个二元情形为例：若某疾病的检测灵敏度为95%，特异度为90%，患病率为1%，则贝叶斯定理可用于计算检测阳性者真正患病的概率。
6. Discrete Random Variables | 离散随机变量
A discrete random variable X takes a countable set of values, each with a certain probability. Its probability distribution a function P(X = x) = p(x), satisfying p(x) ≥ 0 and Σ p(x) = 1 over all possible values x.
离散随机变量 X 取可数集合中的值，每个值对应一定概率。其概率分布由函数 P(X = x) = p(x) 给出，满足 p(x) ≥ 0，且对所有可能取值 x，Σ p(x) = 1。
You should be able to represent a distribution using a table, a graph, or a piecewise formula. IB questions often ask you to find an unknown constant by using the fact that the total probability is 1.
你需要能使用表格、图像或分段公式来表示分布。IB题目常要求你利用总概率为1的性质来求未知常数。
7. Expected Value and Variance | 期望与方差
The expected value (mean) of a discrete random variable X is denoted by E(X) and calculated as:
期望值（均值）记为 E(X)，计算公式为：
E(X) = Σ x · P(X = x)
The variance measures the spread of the distribution. It is defined as:
方差衡量分布的离散程度，其定义为：
Var(X) = E(X²) − [E(X)]²
the variance. Useful properties include E(aX + b) = aE(X) + b and Var(aX + b) = a²Var(X).
其中 E(X²) = Σ x² · P(X = x)。标准差是方差的正平方根。常用性质包括 E(aX + b) = aE(X) + b，Var(aX + b) = a²Var(X)。
In IB, you must also interpret E(X) as the long-run average and use it for decision-making in real-world contexts, such as expected profit in a game.
在IB课程中，你还需要将 E(X) 理解为长期平均值，并用于实际情境中的决策，例如游戏中的期望收益。
8. The Binomial Distribution | 二项分布
A random variable X follows a binomial distribution X ~ B(n, p) when it counts the number of successes in n independent Bernoulli trials, each with success probability p. Its probability mass function is:
当随机变量 X 表示 n 次独立伯努利试验中的成功次数，且每次成功概率为 p 时，X 服从二项分布 X ~ B(n, p)。其概率质量函数为：
P(X = x) = C(n, x) pˣ (1 − p)ⁿ⁻ˣ, x = 0, 1, …, n
Here C(n, x) is the binomial coefficient "n choose x". The mean and variance are:
其中 C(n, x) 是组合数“n 选 x”。二项分布的均值与方差为：
E(X) = np, Var(X) = np(1 − p)
In IB, make sure you can identify the parameters n and p correctly. For example, "the number of defective items in a sample of 10" is binomial only if defects occur independently with constant probability.
在IB中，务必正确识别参数 n 和 p。例如，“10件样本中的次品数”仅在次品独立出现且概率恒定时才服从二项分布。
9. The Poisson Distribution | 泊松分布
A random variable X follows a Poisson distribution X ~ Po(λ) when it counts the number of occurrences of a rare event in a fixed interval of time or space, with mean rate λ. The probability mass function is:
当随机变量 X 表示在固定时间或空间区间内某稀有事件发生的次数，且平均发生率为 λ 时，X 服从泊松分布 X ~ Po(λ)。其概率质量函数为：
P(X = x) = e⁻λ λˣ / x!, x = 0, 1, 2, …
For a Poisson distribution, the mean and variance are both equal to λ:
泊松分布的均值与方差都等于 λ：
E(X) = λ, Var(X) = λ
an also approximate the binomial distribution when n is large and p is small, with λ = np.
泊松分布常用于建模每小时接到的电话数、排队到达的人数或DNA片段中的突变数。当 n 很大且 p 很小时，它也可以用 λ = np 近似二项分布。
10. Normal Distribution and Continuity Correction | 正态分布与连续性校正
The normal distribution is a continuous probability distribution with probability density function:
正态分布是一种连续概率分布，其概率密度函数为：
f(x) = (1 / (σ√2π)) · e^(−(x − μ)² / (2σ²))
eviation. A normal variable has mean μ and variance σ². The standard normal distribution Z ~ N(0, 1) is obtained by the transformation Z = (X − μ) / σ.
其中 μ 是均值，σ 是标准差。正态变量均值为 μ，方差为 σ²。通过变换 Z = (X − μ) / σ 可得到标准正态分布 Z ~ N(0, 1)。
stribution B(n, p) by a normal distribution when np ≥ 5 and n(1 − p) ≥ 5. When doing so, a continuity correction is required. For example, P(X ≤ k) becomes P(Y ≤ k + 0.5) where Y ~ N(np, np(1 − p)).
在IB中，当 np ≥ 5 且 n(1 − p) ≥ 5 时，可以用正态分布近似离散的二项分布 B(n, p)。此时需要 连续性校正。例如，P(X ≤ k) 转化为 P(Y ≤ k + 0.5)，其中 Y ~ N(np, np(1 − p))。
11. Sampling and the Central Limit Theorem | 抽样与中心极限定理
For a random sample X₁, …, Xₙ from any distribution with mean μ and variance σ², the sample mean X̄ has expected value μ and variance σ²/n. The Central Limit Theorem (CLT) states that for sufficiently large n (usually n ≥ 30), the distribution of X̄ is approximately normal:
对于来自任意分布（均值为 μ，方差为 σ²）的随机样本 X₁, …, Xₙ，样本均值 X̄ 的期望值为 μ，方差为 σ²/n。中心极限定理指出，当 n 足够大（通常 n ≥ 30）时，X̄ 的分布近似正态：
X̄ ≈ N(μ, σ²/n)
This theorem justifies why many real-world averages are approximately normal, and it is a key concept in statistical inference. In IB, you should be able to compute probabilities involving sample means using the CLT.
该定理解释了为什么许多现实中的平均值近似服从正态分布，是统计推断的关键概念。在IB中，你需要能利用中心极限定理计算涉及样本均值的概率。
12. Common Pitfalls and Exam Tips | 常见误区与考试技巧
Do not confuse "mutually exclusive" with "independent". Mutually exclusive means P(A ∩ B) = 0; independent means P(A ∩ B) = P(A)P(B).
不要混淆“互斥”与“独立”。互斥意味着 P(A ∩ B) = 0；独立意味着 P(A ∩ B) = P(A)P(B)。
Always check that probabilities sum to 1 when dealing with distributions or tree diagrams.
在处理分布或树状图时，务必检查所有概率之和是否为 1。
Use the correct formula for conditional probability: P(A | B) = P(A ∩ B) / P(B), not P(A)/P(B) unless independent.
正确使用条件概率公式：P(A | B) = P(A ∩ B) / P(B)，不能写成 P(A)/P(B)，除非独立。
Read the problem carefully: "with replacement" implies independence; "without replacement" implies dependence.
仔细审题：“有放回”意味着独立；“无放回”意味着不独立。
For normal approximation to binomial, always apply continuity correction.
用正态近似二项分布时，务必进行连续性校正。
Mastering probability requires practice. Familiarise yourself with tree diagrams, Venn diagrams, and tables, and always state the distribution you are using. With these core concepts, you will be well-prepared for IB probability questions.
掌握概率需要勤加练习。熟悉树状图、韦恩图和表格，并始终说明你使用的分布。掌握这些核心概念后，你就能从容应对IB概率题。
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