Sample Space, Events and Probability Calculations | 样本空间、事件与概率计算

📚 Sample Space, Events and Probability Calculations | 样本空间、事件与概率计算

Probability is the mathematical language of uncertainty. In the IB Mathematics curriculum, a solid grasp of sample spaces, events, and the rules of probability is essential for solving problems across statistics, finance, and the natural sciences. This revision article covers the core definitions, axioms, and calculation techniques that appear regularly in both Analysis & Approaches and Applications & Interpretation examinations.

概率是描述不确定性的数学语言。在IB数学课程中，牢固掌握样本空间、事件以及概率计算规则，是解决统计、金融和自然科学中各类问题的关键。本复习文章涵盖分析与方法（AA）以及应用与解释（AI）考试中反复出现的基本定义、公理和计算技巧。

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1. Sample Space and Events | 样本空间与事件

The sample space, usually denoted by S or Ω (omega), is the set of all possible outcomes of a random experiment. An event is any subset of the sample space, and events are usually denoted by capital letters such as A, B, or E. For example, when rolling a fair six-sided die once, the sample space is S = {1, 2, 3, 4, 5, 6}.

样本空间通常用S或Ω（omega）表示，它是随机试验中所有可能结果的集合。事件是样本空间的任意子集，通常用大写字母如A、B或E表示。例如，掷一枚公平的六面骰子一次时，样本空间为S = {1, 2, 3, 4, 5, 6}。

The event “rolling an even number” corresponds to the subset E = {2, 4, 6}. If the outcome of the experiment belongs to E, we say that event E has occurred. A single outcome, such as {5}, is called an elementary event. In IB questions, always identify the sample space first; many errors arise from an incomplete or incorrectly defined sample space.

事件”掷出的点数为偶数”对应子集E = {2, 4, 6}。若试验结果属于E，我们就说事件E发生了。单个结果，例如{5}，称为基本事件。在IB题目中，务必首先明确样本空间；许多错误源于样本空间定义不完整或不正确。

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2. Axioms of Probability | 概率公理

Probability is a function that assigns a real number between 0 and 1 to every event. The three fundamental axioms are as follows: (1) For any event A, P(A) ≥ 0; (2) The probability of the entire sample space is P(S) = 1; (3) For a sequence of mutually exclusive events A₁, A₂, A₃, …, the probability of their union equals the sum of their individual probabilities.

概率是一个函数，它为每一个事件赋予0到1之间的实数。三个基本公理如下：(1) 对任意事件A，有P(A) ≥ 0；(2) 整个样本空间的概率为P(S) = 1；(3) 对一列互斥事件A₁、A₂、A₃……，它们并集的概率等于各自概率之和。

These axioms imply that 0 ≤ P(A) ≤ 1 and that P(∅) = 0. In examination problems, a probability value outside the interval [0, 1] immediately signals a computational mistake. Moreover, in any complete probability distribution, the sum of all probabilities must equal exactly 1, an excellent self-check after completing a calculation.

这些公理蕴含着0 ≤ P(A) ≤ 1以及P(∅) = 0。在考试题目中，如果概率值超出区间[0, 1]，立即说明计算有误。此外，在任何一个完整的概率分布中，所有概率之和必须恰好等于1，这是完成计算后极佳的自检方法。

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3. Complementary Events | 补事件

The complement of an event A, written Aᶜ or A’, is the set of all outcomes in the sample space that are not in A. Because A and Aᶜ together form a complete partition of the sample space, we obtain the fundamental identity P(Aᶜ) = 1 − P(A).

事件A的补事件，记作Aᶜ或A’，是样本空间中所有不属于A的结果的集合。由于A与Aᶜ共同构成样本空间的一个完整划分，因此我们得到基本恒等式P(Aᶜ) = 1 − P(A)。

This identity is especially powerful when the probability of the complement is easier to compute. For example, the probability of rolling at least one 6 in five throws of a fair die is found more efficiently by computing the probability of no 6s and subtracting from 1: P(at least one 6) = 1 − (5/6)⁵. IB examiners frequently set questions where direct enumeration is tedious but the complement is trivial.

当补事件的概率更容易计算时，这一恒等式尤为有力。例如，掷一枚公平骰子五次，至少出现一次6的概率，可通过先计算没有出现6的概率再用1减去而高效求得：P(至少一次6) = 1 − (5/6)⁵。IB命题者经常设置这样的题目：直接枚举非常繁琐，但补事件的计算却十分简单。

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4. Mutually Exclusive Events and the Addition Rule | 互斥事件与加法法则

Two events A and B are mutually exclusive (or disjoint) if they have no outcomes in common, meaning A ∩ B = ∅. If A and B are mutually exclusive, they cannot occur simultaneously. The general addition rule for any two events is P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

如果事件A和B没有任何共同结果，即A ∩ B = ∅，则称A与B互斥（或不相容）。若A与B互斥，则二者不可能同时发生。对任意两个事件，一般加法法则为P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。

When A and B are mutually exclusive, the intersection term vanishes, and the rule simplifies to P(A ∪ B) = P(A) + P(B). A common pitfall in IB examinations is to use the simplified formula without checking whether the two events truly overlap. Always verify whether A ∩ B is empty before applying the simplified version.

当A与B互斥时，交集项消失，法则简化为P(A ∪ B) = P(A) + P(B)。IB考试中一个常见陷阱是：不先检查两个事件是否真的有重叠，就直接使用简化公式。在使用简化版本之前，务必核实A ∩ B是否为空集。

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

加法法则：P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

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5. Independent Events and the Multiplication Rule | 独立事件与乘法法则

Two events A and B are independent if the occurrence of one does not change the probability of the other occurring. Mathematically, independence is defined by the multiplication rule P(A ∩ B) = P(A) × P(B). If this equality holds, the events are independent; if it fails, they are dependent.

如果事件A的发生不改变事件B发生的概率，则称A与B相互独立。数学上，独立性由乘法法则P(A ∩ B) = P(A) × P(B)来定义。若该等式成立，则两事件独立；若等式不成立，则它们是相依的。

Independence and mutual exclusivity are frequently confused, but they are fundamentally different concepts. Mutually exclusive events cannot occur together, while independent events can. In fact, if two events have positive probabilities and are mutually exclusive, they cannot be independent, because P(A ∩ B) = 0 while P(A) × P(B) > 0.

独立性与互斥性常常被混淆，但二者是本质不同的概念。互斥事件不能同时发生，而独立事件可以同时发生。事实上，若两个概率为正的事件互斥，则它们不可能独立，因为此时P(A ∩ B) = 0，而P(A) × P(B) > 0。

For three independent events, the rule extends naturally: P(A ∩ B ∩ C) = P(A) × P(B) × P(C). This is widely used in multi-stage experiments such as repeated independent trials where outcomes do not affect one another.

对三个相互独立的事件，法则自然地推广为：P(A ∩ B ∩ C) = P(A) × P(B) × P(C)。这一公式广泛应用于多阶段试验，例如各次结果互不影响地重复独立试验。

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6. Conditional Probability | 条件概率

The conditional probability of event A given that event B has occurred is defined as P(A|B) = P(A ∩ B) / P(B), provided P(B) > 0. Intuitively, we are restricting our attention to the outcomes contained in B and asking what proportion of them also lie in A.

在事件B已发生的条件下，事件A的条件概率定义为P(A|B) = P(A ∩ B) / P(B)，前提是P(B) > 0。直观上，我们将关注范围限制在B所包含的结果中，并考察其中有多大比例也属于A。

If A and B are independent, then P(A|B) = P(A), meaning that knowledge of B provides no additional information about A. The conditional probability formula can also be rearranged into the multiplication rule P(A ∩ B) = P(A|B) × P(B) = P(B|A) × P(A), which is invaluable when navigating a tree diagram.

若A与B独立，则P(A|B) = P(A)，即知道B的发生并不能为A提供额外信息。条件概率公式还可变形为乘法法则P(A ∩ B) = P(A|B) × P(B) = P(B|A) × P(A)，这在借助树形图解题时极为重要。

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7. Tree Diagrams | 树形图

Tree diagrams are a powerful visual tool for multistage experiments. Each branch is labelled with a conditional probability, and the probability of following a complete path is the product of the probabilities along that path. For example, suppose a bag contains 3 red and 2 blue balls. Drawing two balls without replacement, the probability of drawing red first and blue second is (3/5) × (2/4) = 3/10.

树形图是多阶段试验中强有力的可视化工具。每条分支都标有条件概率，而沿完整路径行进的总概率等于该路径上所有概率的乘积。例如，设袋中装有3个红球和2个蓝球，不放回地连续抽取两个球，先取红球再取蓝球的概率为(3/5) × (2/4) = 3/10。

Notice that the second-stage probabilities change after the first draw, reflecting the conditional nature of the process. When drawing with replacement, by contrast, the probabilities remain identical at each stage. IB candidates should write all branch probabilities directly on the diagram and then enumerate the relevant paths before combining their probabilities.

注意：第一次抽取后，第二阶段的概率会随之改变，这体现了过程中条件性的本质。相比之下，若采用放回抽取，则每一阶段的概率保持不变。IB考生应在树形图上直接标注所有分支概率，先列出相关路径，然后再对这些概率进行合并。

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8. Venn Diagrams and Two-Way Tables | 文氏图与二维表

Venn diagrams are indispensable for visualising intersections, unions, and complements. Each region of the diagram corresponds to a distinct combination of membership in the events involved. A two-way table, on the other hand, organises the entire sample space into rows and columns, allowing joint and marginal probabilities to be read off directly.

文氏图对于直观理解交集、并集和补集不可或缺。图中每个区域对应事件归属的一种特定组合。另一方面，二维表将整个样本空间按行与列组织起来，使联合概率和边缘概率可以直接读出。

In IB Paper 2, a GDC can compute probabilities quickly, but a clearly drawn Venn diagram often reveals the structure of a problem more effectively than a purely algebraic approach. Candidates should be comfortable translating between set notation, Venn regions, and table entries, since examiners frequently present the same information in multiple representations.

在IB Paper 2中，图形计算器（GDC）可以快速计算概率，但一张绘制清晰的文氏图往往比纯代数方法更能揭示问题的结构。考生应熟练地在集合记号、文氏图区域和表格条目之间进行转换，因为命题者经常以多种表示方式呈现同一信息。

Property | 性质	Mutually Exclusive | 互斥	Independent | 独立
Can both occur? | 能同时发生吗？	No | 不能	Yes | 能
Key condition | 关键条件	P(A ∩ B) = 0	P(A ∩ B) = P(A)P(B)

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9. Counting and Probability | 计数与概率

When all outcomes in a sample space are equally likely, the probability of an event is the number of favourable outcomes divided by the total number of outcomes. Counting techniques—permutations and combinations—therefore lie at the heart of many probability problems.

当样本空间中所有结果等可能时，事件概率等于有利结果数除以结果总数。因此，排列与组合等计数方法处于许多概率问题的心脏位置。

For example, the probability of being dealt exactly two aces in a five-card poker hand from a standard 52-card deck is C(4,2) × C(48,3) / C(52,5). Here C(n,r) denotes the number of ways to choose r objects from n without regard to order. IB candidates should be fluent in using the nCr and nPr functions on their GDC and must be careful about whether the question involves ordered or unordered selections.

例如，从一副标准52张扑克牌中发五张牌，恰好有两张A的概率为C(4,2) × C(48,3) / C(52,5)。这里C(n,r)表示从n个对象中不考虑顺序地选取r个对象的方法数。IB考生应熟练使用GDC上的nCr和nPr功能，并务必分清题目涉及的是有序还是无序选取。

P(A) = (number of favourable outcomes) / (total number of equally likely outcomes)

P(A) = 有利结果数 / 等可能结果总数

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10. Bayes’ Theorem | 贝叶斯定理

Bayes’ theorem allows us to reverse a conditional probability. If A₁, A₂, …, Aₙ form a partition of the sample space and B is any event with P(B) > 0, then the posterior probability P(Aᵢ|B) can be expressed in terms of the prior probabilities P(Aᵢ) and the likelihoods P(B|Aᵢ). In its simplest two-event form, the theorem states:

贝叶斯定理使我们能够将条件概率反向计算。若A₁、A₂、…、Aₙ构成样本空间的一个划分，且B是满足P(B) > 0的任意事件，则后验概率P(Aᵢ|B)可以用先验概率P(Aᵢ)和似然度P(B|Aᵢ)来表达。在最简单的双事件形式下，定理表述为：

P(A|B) = [P(B|A) × P(A)] / P(B)

贝叶斯定理：P(A|B) = [P(B|A) × P(A)] / P(B)

The denominator can be expanded using the law of total probability: P(B) = P(B|A) × P(A) + P(B|Aᶜ) × P(Aᶜ). This expansion is particularly valuable in diagnostic testing problems, where the test’s sensitivity and specificity are given and we must find the probability that a patient actually has the disease given a positive test result.

分母可以使用全概率公式展开：P(B) = P(B|A) × P(A) + P(B|Aᶜ) × P(Aᶜ)。这一展开式在诊断检验问题中尤为有价值：题目给出检验的敏感度和特异度，要求我们在检验结果为阳性的条件下求患者真正患病的概率。

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11. Common Mistakes and Exam Strategies | 常见错误与考试策略

The most frequently encountered mistakes on IB probability examinations include the following: confusing mutually exclusive events with independent events; forgetting to subtract the intersection term in the addition rule; reversing the order in a conditional probability such as writing P(A|B) instead of P(B|A); and simplifying a fractional answer prematurely before using it in a further calculation.

IB概率考试中最常见的错误包括：混淆互斥事件与独立事件；在加法法则中忘记减去交集项；将条件概率中的先后顺序写反，例如把P(B|A)误写成P(A|B)；以及在后续计算使用之前过早化简分数答案。

• Sketch a Venn diagram or tree diagram before applying any formula. | 在套用任何公式前先画出文氏图或树形图。
• Check whether events are mutually exclusive or independent before choosing a rule. | 在选择法则前先判断事件是互斥还是独立。
• Verify that all probabilities in a complete distribution sum to 1. | 检验完整分布中所有概率之和是否为1。
• Write down every step clearly so that method marks are awarded even if the final answer is incorrect. | 清晰写出每个步骤，即使最终答案有误也能获得步骤分。

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