Random Phenomena and Basic Concepts of Probability | 随机现象与概率基本概念

📚 Random Phenomena and Basic Concepts of Probability | 随机现象与概率基本概念

Probability is the mathematical language of uncertainty. It helps us describe random phenomena, measure how likely events are, and make reasoned decisions when outcomes cannot be predicted with certainty. This revision guide introduces the core definitions and rules you need for exams.

概率是描述不确定性的数学语言。它帮助我们解释随机现象、度量事件发生的可能性，并在无法确定结果的情况下做出理性决策。本复习指南将介绍考试所需的核心定义与运算法则。

1. Deterministic and Random Phenomena | 确定性现象与随机现象

A deterministic phenomenon is one in which the outcome is completely determined by the initial conditions. For example, if you drop an apple from a fixed height in a vacuum, its time of fall can be predicted exactly using physics.

确定性现象是指在给定初始条件下，结果是完全确定的现象。例如，在真空中从固定高度释放一个苹果，其下落时间可以通过物理学精确预测。

A random phenomenon, by contrast, can produce different outcomes when repeated under apparently identical conditions. Tossing a fair coin is a classic example: you know the possible outcomes are heads and tails, but you cannot predict which one will appear on the next toss.

相反，随机现象是指在表观相同的条件下重复进行，却可能产生不同结果的现象。掷一枚均匀硬币就是典型例子：你知道可能的结果是正面和反面，但无法预测下一次会出现哪一个。

Randomness does not mean “no rules”. In fact, the long-run behaviour of a random phenomenon often shows surprising regularity, which is exactly what probability theory studies.

随机并不等于“没有规律”。事实上，随机现象在长期重复中往往表现出惊人的规律性，这正是概率论所研究的内容。

2. Sample Space and Events | 样本空间与事件

For any random experiment, the sample space, often denoted by S or Ω, is the set of all possible outcomes. Each individual outcome is called a sample point.

对于任何随机试验，样本空间通常记作 S 或 Ω，是所有可能结果的集合。每一个单独的结果称为样本点。

Example: tossing a coin once gives S = {H, T}.

例：掷一枚硬币一次，样本空间 S = {H, T}。
Example: rolling a fair six-sided die gives S = {1, 2, 3, 4, 5, 6}.

例：掷一枚均匀六面骰子，样本空间 S = {1, 2, 3, 4, 5, 6}。

An event is any subset of the sample space. Events are usually denoted by capital letters such as A, B, and C. If the outcome of the experiment belongs to event A, we say that event A has occurred.

事件是样本空间的任意子集。事件通常用大写字母 A、B、C 表示。如果试验的结果属于事件 A，我们就说事件 A 发生了。

The empty set ∅ is called the impossible event, and the full sample space S is called the certain event. For example, rolling a 7 on a standard die is the impossible event, while rolling a number less than 7 is the certain event.

空集 ∅ 称为不可能事件，完整样本空间 S 称为必然事件。例如，掷一颗标准骰子得到 7 是不可能事件，而掷出的点数小于 7 是必然事件。

3. What Is Probability? | 概率是什么？

Probability is a numerical measure of how likely an event is to occur. It always lies between 0 and 1 inclusive. A probability of 0 means the event will never happen; a probability of 1 means the event is certain to happen.

概率是对事件发生可能性大小的数值度量。它的取值总是在 0 和 1 之间（含端点）。概率为 0 表示事件永远不会发生；概率为 1 表示事件必然发生。

Formally, a probability function P assigns a number P(A) to each event A, satisfying the following axioms:

严格来说，概率函数 P 为每个事件 A 赋予一个数 P(A)，并满足以下公理：

0 ≤ P(A) ≤ 1 for every event A.

对任意事件 A，有 0 ≤ P(A) ≤ 1。
P(S) = 1, where S is the sample space.

P(S) = 1，其中 S 是样本空间。
If A₁, A₂, A₃, … are mutually exclusive events (no two can occur at the same time), then P(A₁ ∪ A₂ ∪ A₃ ∪ …) = P(A₁) + P(A₂) + P(A₃) + …

如果 A₁、A₂、A₃、… 是两两互斥的事件（任意两个不能同时发生），则 P(A₁ ∪ A₂ ∪ A₃ ∪ …) = P(A₁) + P(A₂) + P(A₃) + …

These axioms are the foundation of all probability calculations.

这些公理是所有概率计算的基础。

4. Classical Probability | 古典概型

The classical definition of probability applies when all outcomes in a finite sample space are equally likely. If the sample space S contains n(S) equally likely outcomes and event A contains n(A) outcomes, then

古典概型适用于有限样本空间中所有结果等可能的情形。若样本空间 S 包含 n(S) 个等可能结果，事件 A 包含 n(A) 个结果，则

P(A) = n(A) / n(S)

For example, when rolling a fair die, each of the six outcomes is equally likely. The event A = “rolling an even number” contains three outcomes: 2, 4, and 6. Therefore P(A) = 3/6 = 1/2.

例如，掷一颗均匀骰子时，六个结果等可能。事件 A = “掷出偶数”包含三个结果：2、4、6。因此 P(A) = 3/6 = 1/2。

This definition works well for coins, dice, cards, and many other symmetric random experiments. However, it requires the assumption of equal likelihood, which must be checked carefully.

这一定义适用于硬币、骰子、扑克牌以及许多其他对称的随机试验。但前提是必须仔细检验“等可能”这一假设是否成立。

5. Relative Frequency and the Law of Large Numbers | 相对频率与大数定律

When outcomes are not equally likely, probability can be estimated by repeating the experiment many times. The relative frequency of an event A is

当结果并非等可能时，可以通过大量重复试验来估计概率。事件 A 的相对频率为

relative frequency of A = (number of times A occurs) / (total number of trials)

A 的相对频率 = A 发生的次数 / 试验总次数

As the number of trials increases, the relative frequency of A tends to stabilise and approach a fixed number, which we interpret as P(A). This is the idea behind the law of large numbers.

随着试验次数增加，A 的相对频率会趋于稳定并接近某个固定数值，我们将该数值理解为 P(A)。这就是大数定律的核心思想。

For example, if you toss a coin 10, 000 times, the proportion of heads will usually be close to 0.5, even though it may differ in the first few tosses.

例如，若你掷一枚硬币 10000 次，正面出现的比例通常会很接近 0.5，尽管最初几次的结果可能有所偏离。

6. The Complement Rule and the Addition Rule | 对立事件法则与加法法则

The complement of event A, written A′ or Aᶜ, is the set of all outcomes that are not in A. Since A and A′ together make up the whole sample space,

事件 A 的对立事件记作 A′ 或 Aᶜ，是不属于 A 的所有结果构成的集合。由于 A 与 A′ 合起来构成整个样本空间，因此

P(A′) = 1 − P(A)

This rule is extremely useful when an event has many outcomes but its complement has few. For example, to find the probability of rolling at least one 6 in two throws, it is easier to compute the probability of rolling no 6 at all and subtract from 1.

当事件包含的结果很多而对立事件包含的结果很少时，这个法则非常有用。例如，求两次抛掷中至少出现一次 6 的概率，更容易的做法是先计算两次都不出现 6 的概率，再用 1 减去它。

For any two events A and B, the probability of their union is given by the addition rule:

对于任意两个事件 A 和 B，其并事件的概率由加法法则给出：

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

If A and B are mutually exclusive, then P(A ∩ B) = 0 and the formula reduces to P(A ∪ B) = P(A) + P(B).

若 A 与 B 互斥，则 P(A ∩ B) = 0，公式简化为 P(A ∪ B) = P(A) + P(B)。

7. Conditional Probability | 条件概率

The conditional probability of B given A, written P(B | A), is the probability that B occurs given that A has already occurred. It is defined as

在事件 A 已发生的条件下，事件 B 发生的条件概率记作 P(B | A)，定义为

P(B | A) = P(A ∩ B) / P(A), provided P(A) > 0

P(B | A) = P(A ∩ B) / P(A)，其中 P(A) > 0

Example: Suppose a class contains 60 students, of whom 30 study mathematics, 20 study physics, and 10 study both. Let A be the event of studying mathematics and B the event of studying physics. Then P(B | A) = P(A ∩ B) / P(A) = (10/60) / (30/60) = 1/3. Given that a student studies mathematics, the chance they also study physics is 1/3.

例：假设一个班有 60 名学生，其中 30 人学数学，20 人学物理，10 人两科都学。设 A 表示学数学的事件，B 表示学物理的事件。则 P(B | A) = P(A ∩ B) / P(A) = (10/60) / (30/60) = 1/3。已知某学生学数学，则他同时学物理的概率为 1/3。

Conditional probability is not the same as unconditional probability. In exams, always check whether the condition restricts the sample space.

条件概率不同于无条件概率。考试中务必检查条件是否缩小了样本空间。

8. The Multiplication Rule and Independent Events | 乘法法则与独立事件

Rearranging the definition of conditional probability gives the multiplication rule:

将条件概率的定义变形，即可得到乘法法则：

P(A ∩ B) = P(A) × P(B | A) = P(B) × P(A | B)

This rule allows us to calculate the probability of two events both occurring by combining a marginal probability and a conditional probability.

该法则通过一个边缘概率和一个条件概率相乘，来计算两个事件同时发生的概率。

Two events A and B are said to be independent if the occurrence of one does not change the probability of the other. Mathematically,

若一个事件的发生不改变另一个事件发生的概率，则称 A 和 B 相互独立。数学表示为

P(A ∩ B) = P(A) × P(B)

Equivalently, A and B are independent if P(A | B) = P(A) and P(B | A) = P(B), provided the relevant conditional probabilities are defined.

等价地，如果 P(A | B) = P(A) 且 P(B | A) = P(B)，且相关条件概率有定义，则 A 与 B 独立。

Do not confuse independent events with mutually exclusive events. If two events are mutually exclusive, the occurrence of one means the other cannot occur, which is a very strong dependence.

不要将独立事件与互斥事件混淆。若两事件互斥，一个事件发生意味着另一个事件不可能发生，这实际上是极强的依赖关系。

9. Mutually Exclusive vs Independent Events | 互斥事件与独立事件的辨析

Mutually exclusive events cannot happen together: P(A ∩ B) = 0. Independent events satisfy P(A ∩ B) = P(A) × P(B). These two concepts are often tested together.

互斥事件不能同时发生：P(A ∩ B) = 0。独立事件满足 P(A ∩ B) = P(A) × P(B)。这两个概念经常放在一起考查。

If two events both have positive probability and are mutually exclusive, they cannot be independent. The reason is simple: P(A ∩ B) = 0, but P(A) × P(B) > 0, so the product rule fails.

如果两个事件都有正概率且互斥，那么它们不可能独立。原因很简单：P(A ∩ B) = 0，而 P(A) × P(B) > 0，因此乘法公式不成立。

Conversely, if two events are independent and both have positive probability, they cannot be mutually exclusive. Independence allows both to occur; mutual exclusion forbids it.

反过来，如果两个事件独立且都有正概率，那么它们不可能互斥。独立允许两事件同时发生；互斥则禁止它们同时发生。

Mutually exclusive: A and B cannot occur together.

互斥：A 与 B 不能同时发生。
Independent: knowing A occurs gives no information about whether B occurs.

独立：知道 A 发生不会提供关于 B 是否发生的信息。

10. Law of Total Probability and Bayes’ Rule | 全概率公式与贝叶斯公式

Suppose events B₁, B₂, …, Bₙ form a partition of the sample space: they are mutually exclusive and their union is S. Then for any event A,

假设事件 B₁、B₂、…、Bₙ 构成样本空间的一个划分：它们两两互斥且并集为 S。则对任意事件 A，

P(A) = P(B₁)P(A | B₁) + P(B₂)P(A | B₂) + … + P(Bₙ)P(A | Bₙ)

This is the law of total probability. It is useful when the probability of A is easier to compute separately in different scenarios.

这就是全概率公式。当事件 A 在不同场景下分别计算概率更为容易时，该公式非常有用。

Bayes’ rule revises our belief in a cause Bⱼ after observing an effect A. It states

贝叶斯公式用于在观察到结果 A 之后，修正对原因 Bⱼ 的初始判断。它给出

P(Bⱼ | A) = P(Bⱼ)P(A | Bⱼ) / P(A)

where the denominator can be expanded using the law of total probability.

其中分母可用全概率公式展开。

Example: A factory has two machines. Machine X produces 60% of the items and has a 2% defect rate; machine Y produces 40% of the items and has a 5% defect rate. The probability that a defective item came from machine Y is found by Bayes’ rule.

例：某工厂有两台机器。机器 X 生产 60% 的产品，次品率为 2%；机器 Y 生产 40% 的产品，次品率为 5%。求一件次品来自机器 Y 的概率，可用贝叶斯公式。

P(Y | D) = [0.4 × 0.05] / [0.6 × 0.02 + 0.4 × 0.05] = 0.02 / 0.032 = 0.625

Thus, given a defective item, there is a 62.5% chance it came from machine Y.

因此，已知一件产品是次品，它有 62.5% 的可能来自机器 Y。

11. Worked Examples and Exam Tips | 例题与考试提示

Worked Example 1: Two fair dice are rolled. Find the probability that the sum of the two numbers is 7.

例题 1：同时掷两枚均匀骰子，求点数之和为 7 的概率。

The sample space contains 6 × 6 = 36 equally likely ordered pairs. The outcomes with sum 7 are (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1), giving 6 favourable outcomes. Therefore P(sum = 7) = 6/36 = 1/6.

样本空间包含 6 × 6 = 36 个等可能的有序数对。和为 7 的结果有 (1,6)、(2,5)、(3,4)、(4,3)、(5,2)、(6,1)，共 6 个。因此 P(和为 7) = 6/36 = 1/6。

Worked Example 2: A bag contains 3 red balls and 2 blue balls. Two balls are drawn without replacement. Find the probability that both are red.

例题 2：一个袋子中有 3 个红球和 2 个蓝球。不放回地抽取两个球，求两个都是红球的概率。

Method 1: P(first red) = 3/5. After removing one red, there remain 2 red and 2 blue, so P(second red | first red) = 2/4 = 1/2. Thus P(both red) = 3/5 × 1/2 = 3/10.

方法 1：P(第一个为红球) = 3/5。取出一个红球后，剩下 2 个红球和 2 个蓝球，所以 P(第二个为红球 | 第一个为红球) = 2/4 = 1/2。因此 P(两个都是红球) = 3/5 × 1/2 = 3/10。

Method 2: Using combinations, P(both red) = C(3,2) / C(5,2) = 3/10.

方法 2：利用组合数，P(两个都是红球) = C(3,2) / C(5,2) = 3/10。

Exam Tips:

考试提示：

Always identify the sample space first and check whether outcomes are equally likely.

务必先确定样本空间，并检查各结果是否等可能。
Use the complement rule when the direct event is complicated: P(at least one) = 1 − P(none).

当所求事件较复杂时使用对立事件法则：P(至少一次) = 1 − P(没有一次)。
For “without replacement” problems, use conditional probabilities; for “with replacement”, use independence.

“不放回”问题应使用条件概率；“放回”问题可视为独立事件。
Do not round probabilities too early. Keep fractions or exact decimals until the final answer.

不要过早四舍五入。在得到最终答案前保留分数或精确小数。
Check that your final probability lies in the interval [0, 1].

检查最终概率是否落在区间 [0, 1] 内。

Understanding the basic concepts of probability is essential for solving more complex problems in statistics and mathematics. Master the definitions, practise the rules, and always reason clearly about the sample space.

理解概率基本概念对于解决统计与数学中更复杂的问题至关重要。掌握定义，熟练运用法则，并始终清晰地分析样本空间。

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Random Phenomena and Basic Concepts of Probability | 随机现象与概率基本概念