Sample Spaces, Events, and Probability | 样本空间、事件与概率

📚 Sample Spaces, Events, and Probability | 样本空间、事件与概率

Probability theory begins with the fundamental concepts of sample spaces and events, which provide the framework for measuring uncertainty. In this article, we will explore the definitions of random experiments, sample spaces, events, and the basic rules that govern probability calculations, as required in the IB Mathematics curriculum.

概率论始于样本空间和事件的基本概念，它们为度量不确定性提供了框架。在本文中，我们将探讨随机试验、样本空间、事件的定义以及控制概率计算的基本规则，这些内容均属于 IB 数学课程的要求。

1. Random Experiments and Sample Spaces | 随机试验与样本空间

A random experiment is a process that leads to one of several possible outcomes, where the result cannot be predicted with certainty in advance. Tossing a coin, rolling a die, or drawing a card from a shuffled deck are all examples of random experiments.

随机试验是一个会产生若干可能结果之一的过程，其结果无法事先完全确定。抛硬币、掷骰子或从洗好的牌中抽一张牌都是随机试验的例子。

The sample space, denoted by S (or sometimes Ω), is the set of all possible outcomes of a random experiment. For a single coin toss, the sample space is S = {Heads, Tails}. For rolling a fair six-sided die, S = {1, 2, 3, 4, 5, 6}.

样本空间，记作 S（有时用 Ω），是随机试验所有可能结果的集合。对于一次抛硬币，样本空间为 S = {正面, 反面}。对于掷一枚公平的六面骰子，S = {1, 2, 3, 4, 5, 6}。

It is crucial to define the sample space correctly, as it forms the universal set for all probability calculations related to the experiment. A sample space can be discrete (finite or countably infinite) or continuous.

正确定义样本空间至关重要，因为它构成了与该试验相关的所有概率计算的通用集合。样本空间可以是离散的（有限或可数无限）或连续的。

2. Events as Subsets | 事件作为子集

An event is any subset of the sample space. It is a collection of outcomes that share a particular property or condition. Events are typically denoted by capital letters such as A, B, C. If the outcome of the experiment belongs to this subset, we say the event has occurred.

事件是样本空间的任意子集。它是具有特定属性或条件的结果的集合。事件通常用大写字母表示，如 A、B、C。如果试验的结果属于该子集，我们就说该事件发生了。

For example, when rolling a die, let A be the event of getting an even number. Then A = {2, 4, 6}. Rolling a 4 means event A occurs. An event consisting of a single outcome, such as E = {5}, is called a simple event (or elementary event).

例如，掷骰子时，令 A 为得到偶数点的事件。则 A = {2, 4, 6}。掷出 4 点意味着事件 A 发生。由单个结果组成的事件，如 E = {5}，称为简单事件（或基本事件）。

The sample space S itself is the sure event—it always occurs. The empty set ∅ is the impossible event—it never occurs.

样本空间 S 本身是必然事件——它总是发生。空集 ∅ 是不可能事件——它永远不会发生。

3. Types of Events: Simple, Compound, Sure, Impossible | 事件类型：简单、复合、必然、不可能

Events can be classified based on the number of outcomes they contain. A simple event contains exactly one outcome. A compound event contains two or more outcomes. The sure event S contains all outcomes, and the impossible event ∅ has no outcomes.

事件可以根据其所包含的结果数量进行分类。简单事件恰好包含一个结果。复合事件包含两个或两个以上的结果。必然事件 S 包含所有结果，不可能事件 ∅ 不包含任何结果。

In set notation, we use Venn diagrams or set listings to represent events. Understanding these types helps in constructing more complex probability statements.

用集合表示时，我们使用文氏图或集合列举来表示事件。理解这些类型有助于构建更复杂的概率陈述。

4. Mutually Exclusive and Exhaustive Events | 互斥事件与完备事件

Two events A and B are mutually exclusive (or disjoint) if they cannot occur at the same time. In set language, A ∩ B = ∅. For example, when drawing a single card from a deck, the events “drawing a heart” and “drawing a spade” are mutually exclusive.

如果两个事件 A 和 B 不能同时发生，则它们是互斥的（或不相交的）。用集合语言表示为 A ∩ B = ∅。例如，从一副牌中抽一张牌时，“抽到红心”和“抽到黑桃”的事件是互斥的。

Events are exhaustive if their union covers the entire sample space, i.e., A ∪ B ∪ … = S. A set of mutually exclusive and exhaustive events partitions the sample space. For a die roll, {1}, {2,3}, {4,5,6} is a partition if they are mutually exclusive and together include all outcomes.

如果事件的并集覆盖了整个样本空间，即 A ∪ B ∪ … = S，则这些事件是完备的。一组互斥且完备的事件构成了样本空间的一个划分。对于掷骰子，{1}, {2,3}, {4,5,6} 如果它们互斥且总和包含所有结果，则是一个划分。

5. The Probability of an Event: Classical Definition | 事件的概率：古典定义

The classical definition of probability applies when all outcomes in a finite sample space are equally likely. For an event A, the probability P(A) is defined as:

当有限样本空间中所有结果等可能时，

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