📚 IB & OCR Mathematics: Probability Revision Guide | IB OCR 数学：概率考点精讲
Probability is a cornerstone of the IB and OCR mathematics syllabi, providing the language to quantify uncertainty and make informed predictions. From the basic axioms to the complexity of Bayesian inference and normal approximations, mastering probability unlocks success in both analysis and applications papers. This guide distils the essential concepts, formulas, and exam techniques you need, organised into clear, paired English–Chinese explanations.
概率是 IB 与 OCR 数学课程体系的基石，为量化不确定性与做出理性预测提供了语言。从基本公理到贝叶斯推断与正态近似的复杂性，掌握概率能帮助你攻克分析与应用试卷。本指南提炼了核心概念、公式与应试技巧，采用中英对照方式组织，便于透彻理解。
1. Basic Probability Concepts | 概率基本概念
umber of outcomes. All probabilities satisfy 0 ≤ P(A) ≤ 1, and the sum of probabilities of all possible outcomes in a sample space S is 1.
概率衡量事件 A 发生的可能性，记作 P(A)。对于等可能结果，P(A) = 有利结果数 / 总结果数。所有概率均满足 0 ≤ P(A) ≤ 1，且样本空间 S 中所有可能结果的概率之和为 1。
- Complement rule: P(not A) = 1 − P(A)
- 互补律：P(非 A) = 1 − P(A)
- Addition rule for any two events: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
- 一般加法公式：P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
2. Mutually Exclusive & Independent Events | 互斥事件与独立事件
Two events are mutually exclusive if they cannot occur simultaneously, so P(A ∩ B) = 0 and P(A ∪ ccurrence of one does not affect the probability of the other. For independent events, P(A ∩ B) = P(A) × P(B). Grasping this distinction is critical for avoiding common exam errors.
两个事件互斥是指它们不能同时发生，此时 P(A ∩ B) = 0 且 P(A ∪ B) = P(A) + P(B)。而独立意味着一个事件的发生不影响另一事件的发生概率。对于独立事件，P(A ∩ B) = P(A) × P(B)。厘清这一区别对避免考试中的常见错误至关重要。
| Concept | Mutually Exclusive | Independent |
|---|---|---|
| Condition | A ∩ B = ∅ | P(A ∩ B) = P(A)P(B) |
| Addition | P(A ∪ B) = P(A) + P(B) | P(A ∪ B) = P(A) + P(B) − P(A)P(B) |
注意：互斥事件一定不独立（除非其中一个概率为零）；独立事件通常不互斥。
3. Conditional Probability | 条件概率
Conditional probability P(A|B) is the probability of A given that B has occurred, defined as P(A|B) = P(A ∩ B) / P(B), provided P(B) > 0. Rearranging yields the multiplication rule: P(A ∩ B) = P(A|B) P(B) = P(B|A) P(A). This formula is the foundation of tree diagrams and Bayesian analysis.
条件概率 P(A|B) 表示在 B 已发生的前提下 A 发生的概率，定义为 P(A|B) = P(A ∩ B) / P(B)，前提是 P(B) > 0。变形后得到乘法法则：P(A ∩ B) = P(A|B) P(B) = P(B|A) P(A)。该公式是概率树和贝叶斯分析的基础。
In problems involving multiple stages, always identify which event is the condition and ensure you use the reduced sample space of the given event.
在多阶段问题中，务必明确条件事件，并确保使用条件事件所对应的缩减样本空间。
4. Probability Trees & Venn Diagrams | 概率树与维恩图
Tree diagrams help visualise sequential events. Each branch is labelled with a conditional probability; the probability of a path is the product of its branch probabilities. Probabilities of different outcomes are found by summing relevant path probabilities. Venn diagrams, on the other hand, display sets and their intersections, making union and intersection relationships immediate.
概率树有助于将顺序事件可视化。每条分支标注条件概率；一条路径的概率等于沿途分支概率的乘积。不同结果的概率则通过对相关路径求和得到。维恩图则展示集合及其交并关系，使并集与交集一目了然。
For tree diagrams, remember that probabilities on branches from a single node must sum to 1. Use Venn diagrams effectively to represent P(A ∪ B), P(A ∩ B), and complements, especially when dealing with three events.
对于概率树，切记从同一节点出发的分支概率之和必须为 1。在处理三个事件时，尤其要善于利用维恩图表示 P(A ∪ B)、P(A ∩ B) 及补集。
5. Bayes' Theorem | 贝叶斯定理
Bayes’ theorem reverses conditional probabilities and is particularly useful when you know P(B|A) but need P(A|B). The formula is:
P(A|B) = [P(B|A) × P(A)] / P(B)
贝叶斯定理用于逆转条件概率，当已知 P(B|A) 而需要 P(A|B) 时非常有用。公式为：
P(A|B) = [P(B|A) × P(A)] / P(B)
Here P(B) is often expanded using the law of total probability: P(B) = P(B|A)P(A) + P(B| cal testing, spam filtering, and any scenario where a hypothesis is updated with evidence.
此处 P(B) 通常借助全概率公式展开：P(B) = P(B|A)P(A) + P(B|非 A)P(非 A)。贝叶斯定理常见于医学检测、垃圾邮件过滤以及任何借助证据更新假设的场景。
6. Discrete Random Variables | 离散随机变量
babilities P(X = x), where Σ P(X = x) = 1. The expected value (mean) is E(X) = Σ x·P(X = x), and the variance is Var(X) = E(X²) − [E(X)]² or Σ (x − μ)² P(X = x).
离散随机变量 X 取可数个值。其概率分布由取值与对应概率 P(X = x) 的表格定义，满足 Σ P(X = x) = 1。期望值（均值）为 E(X) = Σ x·P(X = x)，方差为 Var(X) = E(X²) − [E(X)]² 或 Σ (x − μ)² P(X = x)。
ities using ΣP=1, compute E(X) and Var(X), and then apply these to linear transformations.
在 IB 和 OCR 考试中，常要求利用 ΣP=1 求出未知概率，计算 E(X) 与 Var(X)，并将其应用于线性变换。
7. Binomial Distribution | 二项分布
umber of successes in n independent trials, each with constant success probability p. If X ~ B(n, p), then P(X = k) = ⁿCₖ pᵏ (1−p)ⁿ⁻ᵏ, where ⁿCₖ = n! / [k!(n−k)!]. The mean is E(X) = np and variance Var(X) = np(1−p).
二项分布用来描述 n 次独立试验中成功的次数，每次成功概率 p 恒定。若 X ~ B(n, p)，则 P(X = k) = ⁿCₖ pᵏ (1−p)ⁿ⁻ᵏ，其中 ⁿCₖ = n! / [k!(n−k)!]。均值 E(X) = np，方差 Var(X) = np(1−p)。
Conditions for a binomial model: fixed number of trials, two possible outcomes per trial, independent trials, and constant probability. In questions, validate these assumptions before proceeding.
二项模型的条件包括：试验次数固定、每次试验只有两种结果、试验相互独立、概率恒定。做题时，应先核验这些假设再开始计算。
8. Normal Distribution | 正态分布
ll-shaped curve. Probabilities correspond to areas under the curve. To find P(X ≤ a) for X ~ N(μ, σ²), standardise using Z = (X − μ) / σ, giving Z ~ N(0,1). Use statistical tables or calculator functions to find Φ(z).
正态分布 N(μ, σ²) 是连续、对称的钟形曲线，概率对应曲线下的面积。当 X ~ N(μ, σ²) 时，先将 X 标准化：Z = (X − μ) / σ，则 Z ~ N(0,1)。然后使用统计表或计算器求 Φ(z)。
Common tasks include finding the probability that X lies between two values, finding a quantile given a probability (inverse normal), or solving for μ or σ.
常见题型包括求 X 落在两值之间的概率、由概率求分位数（逆正态），以及求解 μ 或 σ。
9. Normal Approximation to Binomial | 正态近似二项
propriate when np > 5 and n(1−p) > 5. Apply a continuity correction: for an integer k, use P(X ≤ k) ≈ P(Y < k + 0.5) where Y ~ N(np, np(1−p)).
当 n 很大时，二项分布 B(n, p) 可用正态分布 N(μ, σ²) 近似，其中 μ = np，σ² = np(1−p)。适用范围为 np > 5 且 n(1−p) > 5。需进行连续性校正：对整数 k，采用 P(X ≤ k) ≈ P(Y < k + 0.5)，其中 Y ~ N(np, np(1−p))。
This technique is particularly useful in hypothesis testing and exam questions where exact binomial calculations are tedious.
在假设检验以及精确二项计算过于繁琐的考题中，这一技巧尤为实用。
10. Expectation Algebra | 期望代数
pendent random variables X and Y, E(X + Y) = E(X) + E(Y) and Var(X ± Y) = Var(X) + Var(Y).
随机变量的线性变换遵循简洁的代数法则。对于常数 a 和 b：E(aX + b) = aE(X) + b，Var(aX + b) = a² Var(X)。对于独立的随机变量 X 和 Y，有 E(X + Y) = E(X) + E(Y) 且 Var(X ± Y) = Var(X) + Var(Y)。
These properties enable efficient computation of means and variances for sample sums and means, crucial for the Central Limit Theorem as well.
这些性质使样本总和与样本均值的期望和方差计算十分高效，对中心极限定理的应用同样至关重要。
11. Combinations & Permutations in Probability | 排列组合在概率中的应用
Many probability problems require counting outcomes using permutations (order matters) and combinations (order doesn't matter). The number of ways to arrange n distinct items is n!. The number of ways to choose r items from n is ⁿCᵣ. For arrangements with repeated items, divide by the factorial of each repetition count.
许多概率问题需要借助排列（顺序重要）和组合（顺序无关）来计数。n 个不同物体的排列数为 n!。从 n 个中选出 r 个的组合数为 ⁿCᵣ。对于含有重复物体的排列，需除以各重复数量的阶乘。
Typical IB/OCR problems include selecting committees, arranging letters, or calculating lottery probabilities. Always decide whether order matters before applying a counting method.
IB/OCR 的典型问题包括选举委员会、字母排列或彩票概率计算。在应用计数方法前，务必先判断顺序是否重要。
12. Common Pitfalls & Exam Tips | 常见错误与应考技巧
- Pitfall: Confusing P(A|B) with P(B|A). Always identify which event is the condition.
- 常见错误：混淆 P(A|B) 与 P(B|A)。务必明确条件事件。
- Pitfall: Adding probabilities for non‑mutually exclusive events without subtracting the intersection.
- 常见错误：对非互斥事件直接相加概率而不减去交集。
- Pitfall: Using binomial formulas when trials are not independent or p is not constant.
- 常见错误：试验不独立或 p 不恒定时仍使用二项公式。
- Tip: Draw a tree or Venn diagram for conditional probability problems to clarify the relationships.
- 应考技巧：面对条件概率问题，先画出概率树或维恩图以理清关系。
- Tip: For normal distribution questions, always sketch the bell curve and shade the required region.
- 应考技巧：做正态分布题时，务必画出钟形曲线并涂黑目标区域。
- Tip: In binomial-to-normal approximation, never forget the continuity correction (±0.5).
- 应考技巧：二项转正态近似时，切记连续性校正（±0.5）。
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