📚 Theoretical Probability | 理论概率
Published by TutorHao | IB Mathematics Revision Series | aleveler.com
Theoretical probability forms the backbone of predictive mathematics, allowing us to calculate the likelihood of events based purely on logical reasoning and the structure of the sample space. In the IB Mathematics curriculum, a solid grasp of theoretical probability is essential for tackling problems in both Analysis & Approaches (AA) and Applications & Interpretation (AI), from simple dice rolls to complex decision trees. This article walks you through the key concepts, formulas, and applications, always pairing English explanations with Chinese translations to support bilingual learning.
理论概率是预测性数学的基石，使我们能够纯粹基于逻辑推理和样本空间的结构来计算事件发生的可能性。在IB数学课程中，无论是分析与方法（AA）还是应用与解释（AI），牢固掌握理论概率对于解决从掷骰子到复杂决策树的各种问题都至关重要。本文将带你梳理关键概念、公式及其应用，并通过中英双语对照助力学习。
1. What is Theoretical Probability? | 什么是理论概率？
Theoretical probability, often called a priori probability, is the likelihood of an event occurring based on known possible outcomes, without conducting any actual experiment. It assumes all outcomes in the sample space are equally likely. In IB Mathematics, this concept is foundational: you predict that the probability of rolling a 3 on a fair six‑sided die is 1/6 because there is one favourable outcome among six equally likely possibilities.
理论概率，常被称为先验概率，是基于已知的可能结果、在不进行任何实际实验的情况下得出的某事件发生的可能性。它假设样本空间中所有结果等可能。在IB数学中，这是一个基础概念：你预测掷一枚均匀六面骰子得到3的概率是1/6，因为六个等可能的结果中有一个有利结果。
t P(Heads) = 0.5 without ever tossing it, simply because there are two symmetric outcomes.
理论概率不依赖于历史数据或试验，它源于一个完全公平的模型。当你抛一枚均匀硬币时，无需实际抛掷就能声明 P(正面) = 0.5，因为只存在两种对称的结果。
2. Basic Probability Formula | 基本概率公式
The fundamental formula for theoretical probability is:
理论概率的基本公式是：
P(A) = n(A) / n(S)
where n(A) is the number of favourable outcomes for event A, and n(S) is the total number of equally likely outcomes in the sample space S. In IB examinations, you will often be required to identify the correct sample space before applying this formula.
其中 n(A) 是事件 A 的有利结果数，n(S) 是样本空间 S 中所有等可能结果的总数。在IB考试中，经常要求你先识别正确的样本空间，再应用该公式。
For example, drawing a heart from a standard 52‑ press probabilities as fractions in simplest form, decimals, or percentages as instructed.
例如，从一副标准的52张扑克牌中抽出一张红心，n(A)=13，n(S)=52，因此 P(红心) = 13/52 = 1/4。始终根据题目要求将概率表示为最简分数、小数或百分数。
This formula only holds when the outcomes are equally likely. If the die is biased, theoretical probability shifts to a weighted model, which IB students study later using probability distributions.
该公式仅在结果等可能时成立。如果骰子不均匀，理论概率将转向加权模型，IB学生在后续学习中会通过概率分布来探究。
3. Equally Likely Outcomes | 等可能结果
Equally likely outcomes mean every simple event in the sample space has the same chance of occurring. In classical probability problems – coins, dice, spinners, and well‑shuffled cards – this assumption lets us count outcomes directly. IB questions often test whether you can justify why outcomes are equally likely, for instance by symmetry or randomisation.
等可能结果意味着样本空间中的每一个简单事件发生的概率都相同。在经典概率问题中——如硬币、骰子、转盘和洗匀的扑克牌——这一假设让我们能够直接计数结果。IB题目常考查你能否论证结果为何是等可能的，例如通过对称性或随机化。
ed pairs, rather than treating the sums 2 to 12 as equally likely, is a typical pitfall that IB examiners target.
考虑掷两枚均匀骰子。共有36个等可能的有序对 (1,1), (1,2), ..., (6,6)。和为7可以通过6种方式实现，因此 P(和为7) = 6/36 = 1/6。识别这些有序对，而不是误将和值2至12视为等可能，是IB考官常针对的典型错误。
4. Sample Spaces and Counting | 样本空间与计数
A sample space is the set of all possible outcomes of an experiment. For theoretical probability, defining the sample space precisely is the first step. IB students are expected to list samples spaces using systematic methods: grids for two dice, tree diagrams for multistage events, or tables for spinner games.
样本空间是一个实验所有可能结果的集合。对于理论概率而言，精确地定义样本空间是第一步。IB学生应掌握使用系统方法列出样本空间：用网格表示两枚骰子、用树状图表示多阶段事件，或用表格表示转盘游戏。
ance, choosing a shirt from 5 options and trousers from 4 options gives 5 × 4 = 20 outfits – a simple sample space of size 20.
计数结果数常常需要基本的组合数学。乘法原理指出，如果一个步骤有m种方式，另一个步骤有n种方式，那么两者一起有m×n种方式。例如，从5件衬衫和4条裤子中各选一件，共有5×4=20套搭配——这是一个大小为20的简单样本空间。
In IB Mathematics AA, students may also use permutations and combinations when order matters or does not matter, but the underlying principle remains: correctly defining n(S) is critical.
在IB数学AA中，当顺序重要或不重要时，学生可能还会用到排列和组合，但其根本原则不变：准确定义 n(S) 至关重要。
5. Complementary Events | 互补事件
The complement of an event A, denoted A' (or Aᶜ), consists of all outcomes in the sample space that are not in A. Because A and A' together cover the entire sample space with no overlap, we have the complementary rule:
事件 A 的补事件，记为 A'（或 Aᶜ），包含样本空间中所有不在 A 中的结果。由于 A 和 A' 共同覆盖整个样本空间且无重叠，我们得到互补规则：
P(A') = 1 – P(A)
This relationship is extremely useful in IB problems where counting 'not A' le, the probability of getting at least one head in three coin tosses is easier found as 1 – P(no heads) = 1 – (1/2)³ = 7/8.
这一关系在计算“非A”比计算A更容易的IB问题中极为有用。例如，抛三枚硬币至少得到一次正面的概率，可以通过 1 – P(没有正面) = 1 – (1/2)³ = 7/8 更方便地求得。
Students must remember that complementary events are mutually exclusive and exhaustive. The rule only works if A' is truly the complement relative to the defined sample space.
学生必须记住，互补事件是互斥且穷尽的。只有当 A' 确实是相对于所定义样本空间的补事件时，该规则才有效。
6. Experimental versus Theoretical Probability | 实验概率与理论概率
of times A occurred) / (total trials). IB questions often ask you to compare these two and discuss the effect of trial size.
理论概率给出基于模型我们所期望发生的结果，而实验概率（或经验概率）则是通过实际进行试验计算得出：P(A) ≈（A发生的次数）/（总试验次数）。IB题目常要求你比较两者，并讨论试验次数的影响。
As the number of trials increases, the experimental probability tends to stabilise and approach the theoretical probability – a phenomenon known as the Law of Large Numbers. A coin flipped 10 times might give 7 heads (experimental 0.7), but over 10,000 flips the relative frequency will converge towards 0.5.
随着试验次数的增加，实验概率趋于稳定并接近理论概率——这一现象称为大数定律。抛硬币10次可能出现7次正面（实验概率0.7），但经过10,000次抛掷，相对频率将收敛于0.5。
In IB assessments, this distinction is especially relevant in the Applications & Interpretation course, where simulation and data-based probability play a larger role.
在IB测评中，这一区别在应用与解释课程中尤为相关，因为该课程更侧重模拟和基于数据的概率。
7. Compound Events: Union and Intersection | 复合事件：并集与交集
When dealing with two events A and B, the union (A ∪ B) represents outcomes in A or B or both, while the intersection (A ∩ B) represents outcomes common to both. The addition rule for theoretical probability is:
当处理两个事件 A 和 B 时，并集 (A ∪ B) 表示在 A 或 B 中或两者皆有的结果，而交集 (A ∩ B) 表示同时属于两者的结果。理论概率的加法规则为：
P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
This prevents double‑counting outcomes that appear in both events. If A and B are mutually exclusive – meaning they cannot occur together, so A ∩ B = ∅ – then P(A ∩ B) = 0 and the formula simplifies to P(A ∪ B) = P(A) + P(B).
这避免了重复计算同时出现在两个事件中的结果。如果 A 和 B 互斥——即它们不能同时发生，所以 A ∩ B = ∅——那么 P(A ∩ B) = 0，公式简化为 P(A ∪ B) = P(A) + P(B)。
a deck: P(red) = 26/52, P(king) = 4/52, but the two red kings are counted twice, so P(red ∪ king) = 26/52 + 4/52 – 2/52 = 28/52 = 7/13.
例如，从一副牌中抽出一张红色牌或一张K：P(红色) = 26/52，P(K) = 4/52，但两张红色K被重复计算，因此 P(红色 ∪ K) = 26/52 + 4/52 – 2/52 = 28/52 = 7/13。
8. Independent Events | 独立事件
Two events are independent if the occurrence of one does not affect the probability of the other. In theoretical probability, we check independence using the multiplication rule:
如果两个事件中一个的发生不影响另一个的概率，则它们相互独立。在理论概率中，我们使用乘法规则来检验独立性：
P(A ∩ B) = P(A) × P(B)
die and flipping a coin are independent: P(rolling a 6 and heads) = (1/6) × (1/2) = 1/12.
如果该等式成立，则 A 与 B 相互独立。例如，掷骰子和抛硬币是独立事件：P(掷出6且正面) = (1/6) × (1/2) = 1/12。
IB problems often require you to test for independence by comparing P(A ∩ r to calculate combined probabilities assuming independence. Be careful: mutually exclusive events are definitely not independent unless one has zero probability.
IB题目常要求你通过比较 P(A ∩ B) 与 P(A)×P(B) 来检验独立性，或在假设独立下计算联合概率。注意：互斥事件除非其中一个概率为零，否则绝不独立。
9. Dependent Events and Conditional Probability | 相关事件与条件概率
When events are not independent, we use conditional probability. The probability of A given B, written P(A|B), measures how likely A is if we know B has occurred. The formula is:
当事件不独立时，我们使用条件概率。已知 B 发生时 A 的概率，记为 P(A|B)，衡量在知道 B 已发生的情况下 A 的可能性。公式为：
P(A|B) = P(A ∩ B) / P(B), provided P(B) > 0
replacement. For example, drawing two hearts from a deck without replacement: P(2nd heart | 1st heart) = 12/51, because after one heart is removed, 12 hearts remain among 51 cards.
这在涉及不放回抽样的 IB 题目中特别有用。例如，不放回地从一副牌中抽出两张红心：P(第二张红心 | 第一张红心) = 12/51，因为移去一张红心后，剩下51张牌中有12张红心。
Rearranging the formula gives the general multiplication rule for dependent events: P(A ∩ B) = P(A) × P(B|A). This forms the basis for tree diagram calculations in IB exams.
将公式变形即可得到相关事件的一般乘法规则：P(A ∩ B) = P(A) × P(B|A)。这构成了IB考试中树状图计算的基础。
10. Tree Diagrams and Probability | 树状图与概率
Tree diagrams are a powerful visual tool for multistage experiments where events are conditional. Each branch represents a possible outcome at that stage, labelled with its probability. To find the probability of a final combined outcome, multiply along the branches; to find the probability of an event encompassing multiple outcomes, add the relevant end‑node probabilities.
树状图是多阶段实验（事件具有条件性）的一种强大可视化工具。每条分枝代表该阶段的一个可能结果，并标注其概率。要找出最终组合结果的概率，沿分枝相乘；要找出包含多个结果的事件的概率，则将相关的末端节点概率相加。
For example, a bag with 3 red and 5 blue balls, drawing twice without replacement. The first branch probabilities are 3/8 (red) and 5/8 (blue). Second‑stage probabilities change depending on the first draw. IB students must label all branches unambiguously and often find P(at least one red) by summing relevant end‑to‑end products.
例如，一个袋中有3个红球和5个蓝球，不放回地抽取两次。第一层分枝概率为 3/8（红）和 5/8（蓝）。第二层概率根据第一次抽取而变化。IB学生必须明确标注所有分枝，并经常通过求和相关端到端乘积来求得 P(至少一个红球)。
Tree diagrams also elegantly handle 'with replacement' scenarios, where second‑stage probabilities remain constant, indicating independence.
树状图还可以优雅地处理“有放回”情景，此时第二层概率保持不变，表明事件独立。
11. Venn Diagrams and Probability | 维恩图与概率
e circles represent events A and B. Overlapping regions show intersections, and regions outside circles show complements.
维恩图有助于在样本空间中组织事件及其重叠部分。矩形代表整个样本空间，P(S)=1，圆圈代表事件A和B。重叠区域表示交集，圆圈之外的区域表示补集。
IB questions frequently present Venn diagrams with numbers representing counts or probabilities. From these, you can deduce P(A), P(B), P(A ∩ B) and apply the addition rule. They are particularly useful for solving problems where you are given P(A ∪ B) and need to find missing probabilities.
IB题目常以数字（代表计数或概率）的形式给出维恩图。从中你可以推导出 P(A)、P(B)、P(A ∩ B) 并应用加法规则。在给出 P(A ∪ B) 并需要找出缺失概率的问题中，维恩图尤其有用。
A correctly drawn Venn diagram also makes it easy to test for mutual exclusivity (circles do not overlap) or to calculate P(A ∪ B)' directly.
一幅正确绘制的维恩图还能方便地检验互斥性（圆圈不重叠）或直接计算 P(A ∪ B)'。
12. Expected Value and Fair Games | 期望值与公平游戏
Theoretical probability extends to calculating the expected value E(X) of a discrete random variable, which is the long‑run average if an experiment is repeated many times. The formula is:
理论概率还延伸到计算离散随机变量的期望值 E(X)，即若实验重复多次的长期平均值。公式为：
E(X) = Σ xᵢ × P(X = xᵢ)
where xᵢ are the possible values of X. In IB Mathematics, this concept appears in games of chance, insurance policies, and investment decisions. A game is considered fair if the expected gain (or net payoff) is zero.
其中 xᵢ 是 X 的可能取值。在IB数学中，这一概念出现在机会游戏、保险政策和投资决策中。如果期望收益（或净回报）为零，则该游戏被认为是公平的。
For example, a game costs $2 to play: you roll a die and win $10 if you roll a 6, otherwise you win nothing. Your net gain X is $8 with probability 1/6 and –$2 with probability 5/6. E(X) = 8×(1/6) + (–2)×(5/6) = 8/6 – 10/6 = –2/6 ≈ –$0.33. The negative expectation means the game is unfavourable in the long run.
例如，一个游戏花费2美元参与：掷骰子，若掷出6则赢10美元，否则无奖金。你的净收益 X 以概率1/6为8美元，以概率5/6为–2美元。E(X) = 8×(1/6) + (–2)×(5/6) = 8/6 – 10/6 = –2/6 ≈ –0.33美元。负期望值意味着长期来看该游戏不利。
Understanding expected value connects theoretical probability to real‑world decision‑making, a key theme in IB Applications & Interpretation.
理解期望值将理论概率与现实决策联系起来，这是IB应用与解释课程中的一个关键主题。
Published by TutorHao | IB 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