📚 I: Laws of Probability | I：概率基本定律
Probability theory provides a rigorous framework for quantifying uncertainty. In IB Mathematics, mastering the fundamental laws of probability is essential for solving problems ranging from simple chance experiments to complex statistical inference. This article unpacks the core rules—addition, multiplication, conditional probability, independence, the law of total probability, and Bayes' theorem—with clear explanations, examples, and exam-focused insights.
概率论为量化不确定性提供了严谨的框架。在 IB 数学中，掌握概率的基本定律对于解决从简单的随机实验到复杂的统计推断等各种问题至关重要。本文将深入解读核心法则——加法法则、乘法法则、条件概率、独立性、全概率公式和贝叶斯定理，并附有清晰的解释、示例和应试要点。
1. Sample Space and Events | 样本空间与事件
An experiment is any process that yields a set of possible outcomes. The sample space S is the collection of all these outcomes. An event s any subset of S. For instance, tossing a fair coin gives S = {Head, Tail}, and the event “getting a head” a fair die, P(rolling a number greater than 4) = n({5,6}) / n({1,2,3,4,5,6}) = 2/6 = 1/3.
试验是指任何会产生一组可能结果的过程。样本空间 S 是所有可能结果的集合。事件 E 是样本空间的任意子集。例如，抛掷一枚均匀硬币，其样本空间为 S = {正面, 反面}，事件“出现正面”为 E = {正面}。当所有结果等可能时，事件的古典概率为 P(E) = n(E) / n(S)，其中 n(·) 表示结果的数量。掷一颗均匀骰子，P(点数大于4) = n({5,6}) / n({1,2,3,4,5,6}) = 2/6 = 1/3。
2. Axioms of Probability | 概率的公理
Modern probability rests on three axioms introduced by A.N. Kolmogorov. (1) Non‑negativity: For any event A, P(A) ≥ 0. (2) Normalisation: P(S) = 1, where S is the entire sample space. (3) Countable additivity: For any sequence of mutually exclusive events A₁, A₂, …, P(A₁ ∪ A₂ ∪ …) = P(A₁) + P(A₂) + … ility of the union of disjoint events is simply the sum of their individual probabilities.
现代概率论建立在 A.N. 科尔莫戈罗夫提出的三条公理之上。(1) 非负性：对于任意事件 A，P(A) ≥ 0。(2) 规范性：P(S) = 1，其中 S 是整个样本空间。(3) 可列可加性：对于任意两两互斥的事件序列 A₁, A₂, …，有 P(A₁ ∪ A₂ ∪ …) = P(A₁) + P(A₂) + … 。这些公理保证了概率如测度般运作，取值在 0 与 1 之间，且两两互斥事件的并的概率等于各自概率之和。
3. Complement Rule | 补事件法则
The complement of an event A, denoted A' or Aᶜ the probability of the complement is P(A′) = 1 − P(A). This rule is extremely handy: when “at least one” the probability of passing an exam is 0.72, the probability of failing is 1 − 0.72 = 0.28.
事件 A 的补事件记为 A′ 或 Aᶜ，由所有不在 A 中但属于样本空间 S 的结果组成。由于 A 与它的补事件互斥且恰好覆盖整个样本空间，补事件的概率为 P(A′) = 1 − P(A)。这条法则非常实用：当题目涉及“至少有一个”这类问题时，计算补事件往往能让问题简化。例如，若通过考试的概率为 0.72，则未通过的概率为 1 − 0.72 = 0.28。
4. Addition Law for Mutually Exclusive Events | 互斥事件的加法法则
Two events A and B are mutually exclusive (disjoint) if they cannot occur simultaneously, i.e. A ∩ B = ∅. Under the additivity axiom, their union satisfies P(A ∪ B) = P(A) + P(B). For example, when drawing one card from a standard deck, the events “drawing a heart” and “drawing a spade” are mutually exclusive, so P(heart ∪ 52 = 26/52 = 1/2. This straightforward rule forms the basis of more general addition laws.
若两个事件 A 和 B 不能同时发生，即 A ∩ B = ∅，则称它们为互斥事件（不相交）。根据可加性公理，它们的并的概率满足 P(A ∪ B) = P(A) + P(B)。例如，从一副标准扑克牌中抽一张牌，事件“抽到红心”与“抽到黑桃”是互斥的，故 P(红心 ∪ 黑桃) = 13/52 + 13/52 = 26/52 = 1/2。这一简洁的法则是更一般加法公式的基础。
5. General Addition Law | 一般加法法则
When events are not mutually exclusive, simply adding probabilities double‑counts their intersection. The general addition law corrects this: P(A ∪ B) = P(A) + P(B) − P(A ∩ B) nts, 40 study Mathematics, 30 study Physics, and 15 study both. The probability that a randomly chosen student studies Mathematics or Physics is P(M ∪ 0.55. This rule is often visualised with Venn diagrams, where the overlap is subtracted to avoid overcounting.
当事件并非互斥时，直接相加会重复计算它们的交集。一般加法法则对此进行了修正：P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。例如，某班有 100 名学生，40 人学习数学，30 人学习物理，15 人同时学习两科。随机选出一名学生，他学习数学或物理的概率为 P(M ∪ P) = 40/100 + 30/100 − 15/100 = 55/100 = 0.55。该法则常通过韦恩图展示，减去重叠部分以免重复计数。
6. Conditional Probability | 条件概率
Conditional probability quantifies the chance of event A occurring given that event B has already occurred, denoted P(A | B). Provided P(B) > 0, it is defined as P(A | B) = P(A ∩ B) / P(B). The condition “given B” ace to B. For example, a bag contains 3 red and 7 blue marbles. If a red marble is drawn and removed, the probability that a second draw is also red (without replacement) is P(red on 2nd | arbles remain among 9. Understanding conditional probability is crucial for grasping independence and Bayes' theorem.
条件概率量化了在事件 B 已经发生的条件下事件 A 发生的可能性，记为 P(A | B)。在 P(B) > 0 的前提下，其定义为 P(A | B) = P(A ∩ B) / P(B)。“给定 B”这个条件意味着将样本空间限制为 B。例如，袋中有 3 个红球和 7 个蓝球。若第一次抽到红球并移除（不放回），第二次再抽到红球的概率为 P(第二次红 | 第一次红) = 2/9，因为剩余 9 个球中只有 2 个红球。理解条件概率是掌握独立性和贝叶斯定理的关键。
7. Multiplication Law | 乘法法则
Rearranging the definition of conditional probability yields the multiplication law (general product rule): P(A ∩ B) = P(A) × P(B | A) = P(B) × P(A | B). This law is invaluable for calculating joint probabilities in multi‑ g two cards from a deck without replacement. The probability of getting two aces is P(Ace₁ ∩ Ace₂) = P(Ace₁) × P(Ace₂ | this sequential multiplication, where each branch multiplies probabilities along the path.
将条件概率的定义重新整理就得到了乘法法则（一般乘法定理）：P(A ∩ B) = P(A) × P(B | A) = P(B) × P(A | B)。该法则对于计算多阶段试验中的联合概率极为重要。考虑从一副扑克牌中不放回地抽取两张牌。抽到两张 A 的概率为 P(第一张A ∩ 第二张A) = P(第一张A) × P(第二张A | 第一张A) = (4/52) × (3/51) = 12/2652 ≈ 0.0045。树状图直观地表示这种顺序乘法，每条路径上的概率相乘。
8. Independence of Events | 事件的独立性
Two events A and B are independent if the occurrence of one does not affect the probability of the other. There are two equivalent definitions: (i) P(A | B) = P(A) when P(B) > 0, and (ii) P(A ∩ B) = P(A) × P(B) endent processes; thus P(rolling a 6 and getting heads) = (1/6) × (1/2) = 1/12. Crucially, mutually exclusive events are not independent (unless one event has probability zero) because if A and B cannot happen together, knowing that B occurred gives A probability 0, not P(A). In IB exams, students commonly confuse independence with mutual exclusivity—always check P(A ∩ B) against the product P(A)P(B).
若一个事件的发生不影响另一个事件发生的概率，则称两个事件 A 与 B 相互独立。有两个等价定义：(i) 当 P(B) > 0 时，P(A | B) = P(A)；(ii) P(A ∩ B) = P(A) × P(B)。后者更一般，是检验独立性的标准。例如，掷一颗均匀骰子与抛一枚均匀硬币是相互独立的过程；因此 P(掷出6且正面朝上) = (1/6) × (1/2) = 1/12。极为重要的是，互斥事件并不独立（除非其中一个概率为 0），因为若 A 与 B 不能同时发生，已知 B 发生会使 A 的概率变为 0，而非 P(A)。在 IB 考试中，学生常常混淆独立性与互斥性——一定要检查 P(A ∩ B) 是否等于 P(A)P(B)。
9. Law of Total Probability | 全概率公式
sample space is partitioned into a set of mutually exclusive and exhaustive events B₁, B₂, …, Bₙ (i.e. B₁ ∪ B₂ ∪ … ∪ Bₙ = S and Bᵢ ∩ Bⱼ = ∅ for i≠j), the probability of any event A can be expressed as a weighted sum: P(A) = Σ P(A | Bᵢ) × P(Bᵢ) er i from 1 to n. This law is particularly powerful when the conditional probabilities P(A | easier to find than P(A) directly. For instance, a factory has three machines producing 50%, 30% and 20% of items with defect rates 2%, 3% and 5% 29. Tree diagrams often help apply this law: sum the probabilities across all paths that lead to A.
当样本空间被划分为一组两两互斥且穷尽的完备事件组 B₁, B₂, …, Bₙ（即 B₁ ∪ B₂ ∪ … ∪ Bₙ = S 且 i≠j 时 Bᵢ ∩ Bⱼ = ∅），任意事件 A 的概率可表示为加权和：P(A) = Σ P(A | Bᵢ) × P(Bᵢ)，对 i 从 1 到 n 求和。当条件概率 P(A | Bᵢ) 比直接求 P(A) 更容易时，该公式尤为有力。例如，某工厂有三台机器，分别生产 50%、30% 和 20% 的产品，次品率分别为 2%、3% 和 5%。随机抽一件产品为次品的概率为 P(D) = (0.02)(0.50) + (0.03)(0.30) + (0.05)(0.20) = 0.010 + 0.009 + 0.010 = 0.029。树状图有助于应用此法则：将所有通向 A 的路径概率相加即可。
10. Bayes’ Theorem | 贝叶斯定理
Bayes’ Theorem reverses conditional conditioning. If {A₁, A₂, …, Aₙ} is a partition of S and B is any event, then for a particular Aₖ, P(Aₖ | B) = [P(B | Aₖ) × P(Aₖ)] / P(B), where P(B) can be expanded using the law of total probability: P(B) = Σ P(B | Aᵢ)P(Aᵢ). This theorem is the cornerstone of “inverse probability” y data above, given a product is defective, what is the probability it came from machine 1? P(Machine1 | d medical testing. In IB, always verify you have a valid partition and that the target conditional probability is well‑defined.
贝叶斯定理逆转了条件的方向。如果 {A₁, A₂, …, Aₙ} 是样本空间 S 的一个划分，B 是任意事件，那么对于某个特定的 Aₖ，有 P(Aₖ | B) = [P(B | Aₖ) × P(Aₖ)] / P(B)，其中 P(B) 可利用全概率公式展开：P(B) = Σ P(B | Aᵢ)P(Aᵢ)。该定理是“逆概率”推理的基石。例如，沿用上例工厂数据，已知抽到一件次品，它来自机器 1 的概率是多少？P(机器1 | D) = (0.02 × 0.50) / 0.029 ≈ 0.3448。贝叶斯定理利用新证据更新先验信念，广泛应用于统计学、机器学习和医学检验。在 IB 考试中，务必确认划分有效且所求条件概率有意义。
11. Visualising Laws: Venn Diagrams and Tree Diagrams | 可视化法则：韦恩图与树状图
IB problems often require translating worded scenarios into visual models. Venn diagrams excel at displaying set relationships, intersections, and complements—making the general addition law and conditional probability intuitive. Tree diagrams are ideal for sequential events with branches labelled by probabilities; they illustrate the multiplication law and law of total probability beautifully. When constructing a tree, ensure branch probabilities after each node sum to 1. To find P(A), multiply along branches and sum across paths. Both diagrams serve as powerful checking tools: if your computed probabilities lie outside [0,1], your model is flawed.
IB 的题目常要求将文字情境转化为可视化模型。韦恩图擅长展示集合关系、交集与补集——使得一般加法法则和条件概率变得直观。树状图则适合表示带有分支概率的顺序事件，完美地演示了乘法法则和全概率公式。绘制树状图时，要确保每个节点之后的分支概率之和为 1。求 P(A) 时，沿分支相乘再跨路径求和。这两种图都是强有力的检验工具：若计算出的概率落在 [0,1] 之外，你的模型就有问题。
12. Common Pitfalls and IB Exam Tips | 常见误区与 IB 应试技巧
Several misconceptions trip up IB candidates. Mutual exclusivity vs. independence: if P(A) > 0 and P(B) > 0, mutually exclusive events cannot be independent. Always check P(A ∩ B) to decide independence. Incorrect denominator in conditional probability: P(A | B) uses P(B) as the denominator, not P(S). Complex unions without subtraction: forgetting the −P(A ∩ B) term leads to probabilities >1. Assuming events are independent without justification: independence must be stated or derived, not guessed. In exams, show clear working with proper notation, use diagrams to clarify logic, and present answers as exact fractions or decimals to three significant figures unless stated otherwise. Practise past‑paper questions that combine these laws, and you will build robust probabilistic intuition.
几个常见的误区会让 IB 考生失分。互斥性与独立性混淆：若 P(A) > 0 且 P(B) > 0，互斥事件绝不可能是独立的。务必检查 P(A ∩ B) 以判别独立性。条件概率分母错误：P(A | B) 的分母是 P(B)，而非 P(S)。复杂并集忘记扣除交集：漏掉 −P(A ∩ B) 这一项会导致概率超过 1。无依据地假设事件独立：独立性必须由题意给出或推导得出，而非猜测。考试时，应写出清晰的计算步骤并使用恰当的符号，用图表辅助逻辑思维，答案除非特别说明，应以精确分数或三位有效数字的小数表示。多加练习融合这些法则的历年真题，你将建立起稳健的概率直觉。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图