📚 7A Introduction to Probability | 7A 概率论入门
Probability is the branch of mathematics that deals with the likelihood of events occurring. In IB Mathematics, understanding probability is essential for analysing real-world situations, from games of chance to statistical predictions. This chapter introduces the core concepts: sample spaces, events, probability rules, Venn diagrams, and tree diagrams, providing the foundation for further work in statistics.
概率论是研究事件发生可能性的数学分支。在IB数学中，理解概率对于分析从博弈到统计预测的现实情境至关重要。本章将介绍核心概念：样本空间、事件、概率法则、维恩图和树形图，为后续统计学学习奠定基础。
1. Key Definitions | 关键定义
An experiment is a repeatable process that produces an outcome, such as rolling a die or tossing a coin. A single performance of an experiment is called a trial. The set of all possible outcomes is the sample space, often denoted by U (or S). An event is any subset of the sample space, representing a particular collection of outcomes we are interested in.
试验是一个会产生结果的可重复过程，例如掷骰子或抛硬币。试验的单次执行称为一次尝试。所有可能结果的集合称为样本空间，通常用 U（或 S）表示。事件是样本空间的任意子集，代表我们感兴趣的特定结果集合。
For example, when rolling a fair six‑sided die, the sample space is U = {1, 2, 3, 4, 5, 6}. The event 'rolling an even number' y likely if each has the same chance of occurring, which is the assumption for a fair die.
例如，掷一枚均匀的六面骰子时，样本空间为 U = {1, 2, 3, 4, 5, 6}。事件"掷出偶数"可写作 A = {2, 4, 6}。如果每个结果出现的可能性相同，则称结果等可能，这也是公平骰子的前提。
2. Probability of an Event | 事件的概率
If all outcomes in the sample space are equally likely, the probability of an event A is given by the classical probability formula:
若样本空间中所有结果等可能出现，则事件 A 的概率由古典概率公式给出：
P(A) = n(A) / n(U)
where n(A) is the number of favourable outcomes in event A and n(U) is the total number of outcomes in the sample space. Probability is always a number between 0 and 1. A probability of 0 means the event is impossible, while a probability of 1 means the event is certain.
其中 n(A) 是事件 A 中有利结果的数量，n(U) 是样本空间中结果的总数。概率始终是一个介于 0 和 1 之间的数。概率为 0 表示事件不可能发生，概率为 1 表示事件必然发生。
the die example, for A = {2, 4, 6}, n(A) = 3 and n(U) = 6, so P(A) = 3/6 = 1/2. The event 'rolling a 7' is the empty set with probability 0.
以上述骰子为例，对于 A = {2, 4, 6}，n(A) = 3，n(U) = 6，因此 P(A) = 3/6 = 1/2。事件"掷出 7 点"为空集，概率为 0。
3. Probability Scale and Complement | 概率尺度与补集
Probabilities can be expressed as fractions, decimals or percentages. The complement of an event A, written as A', is the event that A does not occur. Since A and A' together cover the entire sample space, their probabilities add up to 1:
概率可用分数、小数或百分比表示。事件 A 的补集，记作 A'，是 A 不发生的事件。由于 A 和 A' 共同覆盖了整个样本空间，它们的概率之和为 1：
P(A') = 1 − P(A)
or instance, the probability of not rolling a 6 on a fair die is P(not 6) = 1 − 1/6 = 5/6.
当计算补集的概率更容易时，这个关系式非常有用。例如，在公平骰子上不掷出 6 的概率为 P(非 6) = 1 − 1/6 = 5/6。
4. Expected Frequency | 期望频率
If an experiment is repeated n times, the expected frequency of event A is the number of times we would expect A to occur in the long run. It is calculated as:
若试验重复 n 次，事件 A 的期望频率是我们预期 A 在长期中发生的次数。计算公式为：
Expected frequency = n × P(A)
ample, if you roll a fair die 300 times, the expected number of sixes is 300 × (1/6) = 50.
这并不保证在特定的一组试验中 A 恰好发生那么多次，但它给出了多次重复下的平均结果。例如，若将一枚公平骰子掷 300 次，出现 6 的期望次数为 300 × (1/6) = 50。
5. Venn Diagrams and Set Notation | 维恩图与集合表示
Venn diagrams are visual tools that represent the sample space as a rectangle and events as circles inside it. The intersection of two events A and B, written A ∩ B, contains outcomes that belong to both A and B. The union A ∪ B contains outcomes that belong to A or B or both.
维恩图是以矩形表示样本空间、以内部圆形表示事件的可视化工具。两个事件 A 和 B 的交集，记作 A ∩ B，包含同时属于 A 和 B 的结果。并集 A ∪ B 包含属于 A 或 B 或两者的结果。
In a Venn diagram, overlapping areas show A ∩ B, while the total shaded area of both circles shows A ∪ B. The region outside circle A inside the rectangle represents A'. These diagrams help to visualise relationships between events and to count outcomes correctly.
在维恩图中，重叠区域表示 A ∩ B，而两个圆形的全部阴影区域表示 A ∪ B。矩形内圆形 A 之外的区域表示 A'。这些图形有助于直观理解事件间的关系并正确计算结果的数目。
6. Mutually Exclusive Events | 互斥事件
joint) if they cannot occur at the same time. In set terms, their intersection is empty: A ∩ B = ∅, so P(A ∩ B) = 0. For example, when rolling a die, the events 'rolling a 3' and 'rolling an even number' are mutually exclusive.
若两个事件 A 和 B 不能同时发生，则称它们为互斥事件（或不相交事件）。用集合语言表示，即它们的交集为空：A ∩ B = ∅，故 P(A ∩ B) = 0。例如，掷骰子时，"掷出 3"与"掷出偶数"是互斥事件。
For mutually exclusive events, the addition rule simplifies to:
对于互斥事件，加法法则简化为：
P(A ∪ B) = P(A) + P(B)
This means the probability that either A or B occurs is simply the sum of their individual probabilities. However, this rule only applies when A and B share no outcomes.
这意味着 A 或 B 发生的概率就是它们各自概率之和。但该规则仅当 A 与 B 无共同结果时适用。
7. General Addition Rule | 一般加法法则
When events are not mutually exclusive, the intersection A ∩ B is non‑empty, and simply adding P(A) and P(B) would count the overlap twice. The general addition rule for any two events is:
当事件不互斥时，交集 A ∩ B 非空，简单地将 P(A) 和 P(B) 相加会重复计算重叠部分。对于任意两个事件的一般加法法则为：
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
For instance, when drawing a card from a standard deck, let A be 'a red card' and B be 'a king'. There are 26 red cards, 4 kings, and 2 red kings (A ∩ B). Thus P(A ∪ B) = 26/52 + 4/52 − 2/52 = 28/52 = 7/13.
例如，从一副标准扑克牌中抽一张牌，设 A 为"红牌"，B 为"K"。有 26 张红牌，4 张 K，2 张红桃/方片 K（A ∩ B）。因此 P(A ∪ B) = 26/52 + 4/52 − 2/52 = 28/52 = 7/13。
8. Independent Events | 独立事件
Two events A and B are independent if the occurrence of one does not affect the probability of the other. Mathematically, independence is defined by:
若一个事件的发生不影响另一个事件的概率，则称 A 和 B 为独立事件。数学上，独立性定义为：
P(A ∩ B) = P(A) × P(B)
a fair coin twice, the outcome of the first flip does not affect the second, so the events 'heads on first flip' and 'heads on second flip' are independent: P(H₁ ∩ H₂) = ½ × ½ = ¼.
必须使用该等式来检验独立性；不能仅凭假设认为事件独立。例如，将一枚均匀硬币抛两次，第一次的结果不影响第二次，因此事件"第一次正面"和"第二次正面"独立：P(H₁ ∩ H₂) = ½ × ½ = ¼。
9. Conditional Probability | 条件概率
Conditional probability measures the probability of event A occurring given that event B has already occurred. It is written as P(A | B) and defined by:
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。记作 P(A | B)，定义为：
P(A | B) = P(A ∩ B) / P(B), provided P(B) > 0
deck of cards, the probability that a card is a king given that it is a red card is P(king | tional probability is closely linked to independence: if A and B are independent, then P(A | B) = P(A).
此公式实质上是将样本空间限制为 B。例如，从一副牌中，已知抽到一张红牌，则该牌为 K 的概率为 P(K | 红) = (2/52) / (26/52) = 2/26 = 1/13。条件概率与独立性密切相关：若 A 与 B 独立，则 P(A | B) = P(A)。
10. Probability Tree Diagrams | 概率树形图
Tree diagrams are branching diagrams that show all possible outcomes of a multi‑stage experiment. Each branch represents an outcome and is labelled with its probability. The probabilities on branches from any node must sum to 1. To find the probability of a path, multiply the probabilities along the branches. To find the probability of an event combining several paths, add the path probabilities.
树形图是展示多阶段试验所有可能结果的分支图。每个分支代表一个结果并标注其概率。从任一节点出发的各分支概率之和必须为 1。要求出某条路径的概率，将沿路分支的概率相乘；要求出组合多个路径的事件的概率，则将各路径概率相加。
ee diagrams are especially useful for conditional probabilities and for problems involving 'with replacement' and 'without replacement' /4) = 6/20 = 3/10, while the probability of getting at least one blue is 1 − P(RR) = 7/10.
树形图对于条件概率以及涉及"有放回"和"无放回"的问题尤为有用。例如，袋中有 3 个红球和 2 个蓝球。无放回地抽取两个球，得到两个红球的概率为 (3/5) × (2/4) = 6/20 = 3/10，而至少得到一个蓝球的概率为 1 − P(RR) = 7/10。
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