📚 IB Math: Probability Key Points | IB 数学：概率 考点精讲
Probability is a fundamental topic in IB Mathematics, bridging abstract theory with real-world applications. This article summarizes the key concepts and exam techniques you need to master, including basic rules, conditional probability, distributions, and more. Whether you're taking Analysis & Approaches or Applications & Interpretation, this guide will help you consolidate your understanding and avoid common pitfalls.
概率是 IB 数学的基础主题，连接抽象理论与实际应用。本文总结你需要掌握的核心概念和考试技巧，涵盖基本规则、条件概率、分布等内容。无论你学习分析与方法（AA）还是应用与解释（AI），本指南都将帮助你巩固理解并规避常见陷阱。
1. Basic Probability Rules | 概率基本规则
qually likely, then P(A) = n(A)/n(S), where n(A) is the number of outcomes favorable to A.
事件 A 的概率衡量该事件发生的可能性。如果样本空间 S 中所有结果等可能，则 P(A) = n(A)/n(S)，其中 n(A) 是对 A 有利的结果数。
P(A) = n(A) / n(S)
le states that the probability of an event not occurring is one minus its probability: 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). When A and B are disjoint, A ∩ B = ∅ so P(A ∪ B) = P(A) + P(B).
互补规则指出事件不发生的概率等于 1 减去其概率：P(A') = 1 − P(A)。对任意两事件 A 和 B 的加法规则是 P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。当 A 与 B 互斥时，A ∩ B = ∅，因此 P(A ∪ B) = P(A) + P(B)。
2. Mutually Exclusive & Independent Events | 互斥事件与独立事件
s are mutually exclusive (or disjoint) if they cannot happen simultaneously. Formally, P(A ∩ nce of one event does not affect the probability of the other. For independent events, P(A ∩ B) = P(A) × P(B).
两个事件如果不可能同时发生，则称它们为互斥（或不相交）事件。正式地，P(A ∩ B) = 0。独立性意味着一个事件的发生不影响另一个事件发生的概率。对于独立事件，P(A ∩ B) = P(A) × P(B)。
It is crucial not to confuse these two concepts. Mutually exclusive events with non-zero probabilities are never independent, because knowing one occurs makes the probability of the other zero. Independence is tested by verifying whether P(A ∩ B) equals the product P(A)P(B).
关键是不要混淆这两个概念。具有非零概率的互斥事件永远不会独立，因为知道一个事件发生会使另一个事件发生的概率变为零。独立性通过验证 P(A ∩ B) 是否等于乘积 P(A)P(B) 来检验。
3. Conditional Probability | 条件概率
Conditional probability P(A|B) is the probability of A occurring given that B has occurred. It is defined as P(A|B) = P(A ∩ B) / P(B), provided P(B) > 0. Rearranging gives the multiplication rule: P(A ∩ B) = P(A|B) P(B) = P(B|A) P(A).
条件概率 P(A|B) 是在 B 已发生的条件下 A 发生的概率。其定义为 P(A|B) = P(A ∩ B) / P(B)，其中 P(B) > 0。移项可得乘法规则：P(A ∩ B) = P(A|B) P(B) = P(B|A) P(A)。
P(A|B) = P(A ∩ B) / P(B)
Conditional probabilities are often calculated from a two-way table or a tree diagram. Always check the condition that restricts the sample space to B, and then find the likelihood of A within that reduced space.
条件概率通常通过双向表格或树状图来计算。务必检查将样本空间限制在 B 内的条件，然后在该缩小后的空间里求出 A 的概率。
4. Probability Tree Diagrams | 概率树状图
Tree diagrams are powerful tools for multi-stage experiments. Branches represent possible outcomes at each stage, with probabilities written along the branches. To find the probability of a sequence of events, multiply the probabilities along the corresponding path. If multiple paths lead to the same final outcome, add their probabilities.
树状图是多阶段实验的有力工具。分支代表每一阶段可能的结果，概率写在分支上。求一系列事件发生的概率时，将相应路径上的概率相乘。如果多条路径导致相同最终结果，则将它们的概率相加。
When sampling without replacement, the probabilities on the second set of branches depend on the first outcome, reflecting conditional probabilities. Always sum the probabilities of all leaves to 1 as a quick validity check.
在不放回抽样中，第二组分支上的概率取决于第一个结果，反映条件概率。始终将所有叶结点概率之和为 1 作为快速验算。
5. Venn Diagrams & Set Notation | 韦恩图与集合论
Venn diagrams give a visual representation of events as subsets of a sample space S. Key set operations include union (A ∪ B), intersection (A ∩ B), and complement (A′ or Ac). They are especially useful for applying the inclusion–exclusion principle.
韦恩图将事件表示为样本空间 S 的子集。关键集合运算包括并集（A ∪ B）、交集（A ∩ B）和补集（A′ 或 Ac）。它们在应用容斥原理时特别有用。
n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
ach region by the total number of outcomes. Three-event problems extend the principle: n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(B ∩ C) − n(A ∩ C) + n(A ∩ B ∩ C).
您可以通过将每个区域的元素数除以总结果数，从韦恩图中提取概率。三事件问题扩展了该原理：n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(B ∩ C) − n(A ∩ C) + n(A ∩ B ∩ C)。
6. Discrete Random Variables & Probability Distributions | 离散随机变量与概率分布
the corresponding probabilities P(X = x). The probabilities must sum to 1: Σ P(X = x) = 1.
离散随机变量 X 为随机实验的每个结果赋予一个数值。其概率分布列出所有可能取值 x 及相应的概率 P(X = x)。所有概率之和必须为 1：Σ P(X = x) = 1。
[(X − μ)²] = Σ (x − μ)² P(X = x) = E(X²) − [E(X)]². The standard deviation is σ = √Var(X).
X 的期望值（均值）为 E(X) = μ = Σ x P(X = x)。方差衡量离散程度：Var(X) = E[(X − μ)²] = Σ (x − μ)² P(X = x) = E(X²) − [E(X)]²。标准差为 σ = √Var(X)。
7. Binomial Distribution | 二项分布
The binomial distribution models the number of successes in n independent trials, each with the same probability of success p. We write X ~ B(n, p). The probability of exactly k successes is:
二项分布用于描述 n 次独立试验中成功的次数，每次试验成功的概率 p 相同。记作 X ~ B(n, p)。恰好 k 次成功的概率为：
P(X = k) = C(n, k) · pᵏ · (1 − p)ⁿ⁻ᵏ
where C(n, k) = n! / (k! (n − k)! al coefficient. The mean is E(X) = np and the variance is Var(X) = np(1 − p). Use your GDC's binompdf for exact probabilities and binomcdf for cumulative probabilities P(X ≤ k).
其中 C(n, k) = n! / (k! (n − k)!) 是二项式系数。均值为 E(X) = np，方差为 Var(X) = np(1 − p)。使用图形计算器的 binompdf 求单个概率，binomcdf 求累积概率 P(X ≤ k)。
8. Normal Distribution | 正态分布
The normal distribution is a continuous probability distribution defined by its mean μ and variance σ². We write X ~ N(μ, σ²). The standard normal distribution Z ~ N(0, 1) is obtained by standardizing:
正态分布是由均值 μ 和方差 σ² 定义的连续概率分布。记作 X ~ N(μ, σ²)。通过标准化可得到标准正态分布 Z ~ N(0, 1)：
Z = (X − μ) / σ
Use the normalcdf function on your calculator to find probabilities such as P(a < X < b). Use invNorm to find the value corresponding to a given cumulative probability. The empirical rule states that approximately 68% of data lie within 1σ of the mean, 95% within 2σ, and 99.7% within 3σ.
使用计算器上的 normalcdf 函数求概率，如 P(a < X < b)。使用 invNorm 求给定累积概率对应的值。经验规则表明约 68% 的数据落在均值 ±1σ 内，95% 落在 ±2σ 内，99.7% 落在 ±3σ 内。
9. Expectation, Variance & Standard Deviation | 期望、方差与标准差
Linear transformations of a random variable affect its mean and variance in predictable ways. For constants a and b:
随机变量的线性变换以可预测的方式影响其均值和方差。对于常数 a 和 b：
E(aX + b) = a E(X) + b
Var(aX + b) = a² Var(X)
(X) + Var(Y). The standard deviation σ is always the positive square root of the variance.
如果 X 和 Y 是独立的随机变量，则 E(X + Y) = E(X) + E(Y)，Var(X + Y) = Var(X) + Var(Y)。标准差 σ 始终是方差的正平方根。
These properties are essential when working with portfolio returns, measurement combinations, or any situation involving linear combinations of independent outcomes.
这些性质在处理投资组合收益、测量值组合或任何涉及独立结果线性组合的情形时至关重要。
10. Bayes' Theorem (HL) | 贝叶斯定理（高级）
Bayes' theorem provides a way to update the probability of an event based on new evidence. For two events A and B with P(B) > 0,
贝叶斯定理提供了一种根据新证据更新事件概率的方法。对于 P(B) > 0 的两个事件 A 和 B，
P(A|B) = [P(B|A) P(A)] / P(B)
where P(B) = P(B|A) P(A) + P(B|A') P(A' g, machine learning, and any scenario where prior knowledge is refined with observed data.
其中 P(B) = P(B|A) P(A) + P(B|A') P(A')，用于分母被划分的情形。该定理广泛应用于诊断测试、机器学习以及任何用观测数据修正先验知识的情景。
A typical IB HL question might give the prevalence of a disease (prior probability) and the accuracy of a test (conditional probabilities), then ask for the probability that a patient actually has the disease given a positive test result.
一个典型的 IB HL 问题可能会给出疾病流行率（先验概率）和检测的准确度（条件概率），然后要求计算测试呈阳性时患者实际患病的概率。
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