📚 How to Calculate Theoretical Probability in IB Maths | IB数学：理论概率的计算方法
Theoretical probability is one of the most intuitive yet powerful tools in mathematics. In this article, we will explore the key formulas, logical structures, and common pitfalls that IB students face when calculating probabilities in pure and applied contexts.
理论概率是数学中最直观却最强大的工具之一。本文将系统讲解IB学生在纯数学与应用情境中计算概率时需要掌握的核心公式、逻辑结构与常见误区。
1. Definition of Theoretical Probability | 理论概率的定义
Theoretical probability assumes that all outcomes in a sample space are equally likely. It is defined as the ratio of the number of favourable outcomes to the total number of possible outcomes:
理论概率假设样本空间中的所有结果等可能发生。它定义为有利结果的数量与可能结果总数之比：
P(A) = n(A) / n(U)
where n(A) is the number of outcomes in event A and n(U) is the total number of outcomes in the sample space. For example, when rolling a fair die, the probability of rolling a 4 is 1/6 because there is one favourable outcome and six possible outcomes.
其中 n(A) 是事件 A 中包含的结果数，n(U) 是样本空间中的结果总数。例如，掷一枚公平骰子时，掷出 4 的概率是 1/6，因为有利结果只有一个，而可能结果共有六个。
2. Sample Space and Events | 样本空间与事件
The sample space U is the set of all possible outcomes of a random experiment. An event is any subset of the sample space. IB questions often ask students to list the sample space before calculating probabilities.
样本空间 U 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。IB考试题经常要求学生先列出样本空间，再计算概率。
For tossing two coins, the sample space is {HH, HT, TH, TT}. The event “at least one head” corresponds to {HH, HT, TH}, so its probability is 3/4. Clearly identifying the sample space avoids double-counting and incomplete enumeration.
以抛两枚硬币为例，样本空间为 {HH, HT, TH, TT}。“至少一个正面”对应 {HH, HT, TH}，所以概率为 3/4。清晰列出样本空间可以避免重复计数和枚举不全。
3. The Addition Rule | 加法规则
The addition rule is used to find the probability of the union of two events. In general:
加法规则用于求两个事件并集的概率。一般形式为：
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
The subtraction of P(A ∩ B) corrects for double-counting the outcomes that lie in both events. If A and B are mutually exclusive, meaning they cannot occur together, then P(A ∩ B) = 0 and the formula simplifies to P(A ∪ B) = P(A) + P(B).
减去 P(A ∩ B) 是为了修正两个事件公共部分被重复计算的情况。如果 A 与 B 互斥，即它们不可能同时发生，那么 P(A ∩ B) = 0，公式简化为 P(A ∪ B) = P(A) + P(B)。
For example, in a class of 30 students, 12 play football, 15 play basketball, and 5 play both. The probability that a randomly chosen student plays football or basketball is P(F ∪ B) = 12/30 + 15/30 − 5/30 = 22/30 = 11/15.
例如，某班有30名学生，12人踢足球，15人打篮球，5人两项都参加。随机选一名学生，他参加足球或篮球的概率为 P(F ∪ B) = 12/30 + 15/30 − 5/30 = 22/30 = 11/15。
4. The Multiplication Rule | 乘法规则
The multiplication rule determines the probability of the intersection of two events. For all events A and B:
乘法规则用于求两个事件交集的概率。对于任意事件 A 与 B：
P(A ∩ B) = P(A) × P(B|A)
If A and B are independent, the occurrence of one does not affect the probability of the other, so P(B|A) = P(B), and the formula becomes P(A ∩ B) = P(A) × P(B). For example, rolling a 5 on a die and flipping heads on a coin are independent events.
如果 A 与 B 独立，一个事件的发生不影响另一个事件的概率，则 P(B|A) = P(B)，公式变为 P(A ∩ B) = P(A) × P(B)。例如，掷骰子得到 5 和抛硬币得到正面是两个独立事件。
For dependent events, the multiplication rule remains valid but requires the conditional probability. Drawing two cards from a deck without replacement is a classic dependent-event scenario.
对于相依事件，乘法规则仍然成立，但需要使用条件概率。从一副牌中不放回地抽取两张牌是典型的相依事件场景。
5. Conditional Probability | 条件概率
Conditional probability measures the probability of event A given that event B has already occurred. It is defined as:
条件概率衡量在事件 B 已发生的条件下，事件 A 发生的概率。其定义为：
P(A|B) = P(A ∩ B) / P(B), where P(B) > 0
This formula is fundamental in IB Paper 2 and Paper 3 questions. A typical question: a bag contains 4 red and 6 blue marbles. Two marbles are drawn without replacement. What is the probability that the second marble is red given that the first is blue?
该公式是IB Paper 2 和 Paper 3 中的基础内容。一个典型问题：袋中有 4 个红球和 6 个蓝球，不放回地抽取两个球。已知第一个球是蓝球，求第二个球是红球的概率。
Since the first marble was blue, only 4 red and 5 blue marbles remain. Thus P(second red | first blue) = 4/9. Note that conditional probability reflects a reduced sample space.
因为第一个球是蓝球，袋中剩余 4 个红球和 5 个蓝球。因此 P(第二个红 | 第一个蓝) = 4/9。注意，条件概率体现了样本空间的缩减。
6. Tree Diagrams | 树形图
Tree diagrams visually organise multi-stage experiments. Each branch is labelled with a probability, and every path from root to leaf represents a combined outcome. The probability of each path is the product of the probabilities along the branches.
树形图以可视化方式组织多阶段试验。每条分支标注概率，从根到叶的每条路径代表一个复合结果。每条路径的概率等于路径上各分支概率的乘积。
Consider an experiment where a fair coin is tossed twice. The tree diagram yields four paths: HH, HT, TH, TT, each with probability 1/4. For conditional situations, the second-stage branch probabilities change after the first outcome.
以抛一枚公平硬币两次为例。树形图给出四条路径：HH、HT、TH、TT，每条概率为 1/4。在条件情境中，第二阶段的分支概率会随第一次结果而变化。
IB questions frequently combine tree diagrams with conditional probability, particularly when the first-stage outcomes have non-uniform probabilities. Always sum the probabilities of all paths to check that they total 1.
IB试题经常将树形图与条件概率结合，尤其当第一阶段结果的概率不相等时。务必累加所有路径的概率，检验总和是否等于 1。
7. Venn Diagrams in Probability | 维恩图与概率
Venn diagrams are powerful tools to represent events and their intersections visually. Regions in a Venn diagram correspond exactly to the terms in the addition and multiplication rules.
维恩图是直观展现事件及其交集的有力工具。维恩图中的区域与加法规则、乘法规则中的各项精确对应。
For two events A and B, the diagram splits the sample space into four disjoint regions: A only, B only, both, and neither. Labelling these regions with counts or probabilities allows direct computation of complex expressions.
对于两个事件 A 和 B，维恩图将样本空间划分为四个互不相交的区域：仅 A、仅 B、二者都有、两者皆无。用计数或概率标记这些区域，可以直接计算复杂表达式。
| Region | 区域 | Meaning | 含义 | Probability | 概率 |
| A only | occurs but not B | P(A ∩ B′) |
| B only | occurs but not A | P(A′ ∩ B) |
| both | A and B occur | P(A ∩ B) |
| neither | neither A nor B | P(A′ ∩ B′) |
8. Permutations and Combinations in Probability | 排列组合与概率
When the sample space is large, counting outcomes efficiently becomes essential. Permutations count ordered arrangements, while combinations count unordered selections. In probability, combinations are more frequently used because the order of selected items rarely matters.
当样本空间很大时，高效计数变得至关重要。排列计数有序排列，组合计数无序选择。在概率计算中，组合更为常用，因为所选项目的顺序通常不重要。
P(n, r) = n! / (n − r)!, C(n, r) = n! / [r!(n − r)!]
A standard IB question: from a group of 8 students, 3 are selected randomly to form a committee. What is the probability that the two specified students are both selected? The number of ways to choose 3 students is C(8,3) = 56. The number of favourable ways is C(2,2) × C(6,1) = 6. Thus the probability is 6/56 = 3/28.
一道标准IB题：从8名学生中随机选出3人组成委员会。指定的两名学生都被选中的概率是多少？选择3名学生的总数为 C(8,3) = 56。有利方式数为 C(2,2) × C(6,1) = 6。因此概率为 6/56 = 3/28。
9. Bayes’ Theorem | 贝叶斯定理
Bayes’ theorem allows us to reverse conditional probabilities. Given a partition of the sample space into mutually exclusive events B₁, B₂, …, Bₙ, the posterior probability P(Bⱼ|A) is:
贝叶斯定理允许我们逆转条件概率。设有互斥事件 B₁, B₂, …, Bₙ 构成样本空间的一个分割，后验概率 P(Bⱼ|A) 为：
P(Bⱼ|A) = P(A|Bⱼ) × P(Bⱼ) / [Σ P(A|Bᵢ) × P(Bᵢ)]
This formula is particularly useful in diagnostic testing and decision theory. For instance, if 2% of a population has a disease, a test is 95% accurate, and the false positive rate is 5%, the probability that a positive test truly indicates the disease is not simply 95%.
该公式在诊断测试和决策理论中特别有用。例如，某人口中2%的人患有某种疾病，检测准确率为95%，假阳性率为5%，那么检测为阳性时真正患病的概率并非简单的95%。
Here the sample space must be partitioned by disease status. Bayes’ theorem correctly combines the prior probability P(disease) with the test characteristics to yield the posterior probability. This counter-intuitive result highlights why IB examiners frequently test it in Paper 3.
此处样本空间必须按患病状态分割。贝叶斯定理将先验概率 P(患病) 与检测特征结合，正确得到后验概率。这一违背直觉的结果解释了为何IB考官常在 Paper 3 中考查它。
10. The Binomial Distribution as a Probability Model | 二项分布作为概率模型
The binomial distribution models the number of successes in n fixed independent trials, each with the same probability of success p. The probability of exactly k successes is:
二项分布对 n 次固定独立试验中的成功次数建模，每次试验的成功概率均为 p。恰好 k 次成功的概率为：
P(X = k) = C(n, k) × pᵏ × (1 − p)ⁿ⁻ᵏ
Valid conditions for a binomial setting are: fixed number of trials, each trial has two possible outcomes, trials are independent, and p remains constant. For example, shooting free throws 10 times with a 70% success rate follows a binomial distribution with n = 10, p = 0.7.
二项分布的适用条件为：固定试验次数、每次试验只有两种结果、各次试验相互独立、概率 p 恒定。例如，以70%的命中率投篮10次，符合 n = 10, p = 0.7 的二项分布。
IB students should know how to compute individual binomial probabilities, cumulative probabilities, and the conditions under which a normal approximation may be used for large n.
IB学生应掌握如何计算单个二项概率、累积概率，以及在 n 较大时正态近似的适用条件。
11. Common Mistakes and Exam Strategy | 常见错误与考试策略
One frequent error is confusing independence and mutual exclusivity. Independent events can occur together P(A ∩ B) = P(A)P(B); mutually exclusive events cannot. Another common mistake is forgetting to subtract P(A ∩ B) in the addition rule.
一个常见错误是混淆独立与互斥。独立事件可以同时发生 P(A ∩ B) = P(A)P(B)；而互斥事件不能同时发生。另一个常见错误是在加法规则中忘记减去 P(A ∩ B)。
To avoid losing marks in exams, first write down the given events and the target probability in symbols. Determine whether the events are independent, mutually exclusive, or neither. Then select the appropriate formula and verify the final probability lies between 0 and 1.
为避免考试失分，请先写出给定事件与目标概率的符号表示。判断事件是独立、互斥，还是均不属于这两类。然后选择合适的公式，最后验证最终概率在0和1之间。
Always show intermediate steps: listing the sample space, substituting into formulas clearly, and simplifying fractions. IB marking schemes reward method marks even when the final answer is incorrect.
务必要展示中间步骤：列出样本空间、清晰代入公式、化简分数。IB评分标准即便最终答案有误也会给方法分。
12. Summary | 小结
Theoretical probability in IB Mathematics is built on a compact set of definitions and rules: the basic ratio, addition and multiplication rules, conditional probability, tree diagrams, Venn diagrams, combinatorics, Bayes’ theorem and the binomial model. Mastery of these tools equips students to approach any probability question systematically.
IB数学中的理论概率建立在一组紧凑的定义与规则之上：基本比例、加法规则、乘法规则、条件概率、树形图、维恩图、组合数学、贝叶斯定理和二项分布模型。熟练运用这些工具，学生将能够系统地解出任何概率题。
Rather than memorising formulas mechanically, focus on understanding the logical structure of each problem: what is the sample space, what conditions are given, and which rule connects the known quantities to the unknown probability.
与其机械记忆公式，不如专注于理解每道题的逻辑结构：样本空间是什么、给定了哪些条件、哪条规则能将已知量与未知概率联系起来。
With deliberate practice and careful attention to the assumptions behind each formula, theoretical probability becomes one of the highest-scoring topics on the IB examination.
通过刻意练习，并仔细关注每条公式背后的假设条件，理论概率将成为IB考试中得分率最高的考点之一。
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