📚 Classical Probability & Simple Probability Calculations | 古典概型与简单概率计算
Probability is a branch of mathematics that quantifies uncertainty. Classical probability, also called theoretical probability, is the simplest and most widely taught approach: it assumes that all outcomes of a random experiment are equally likely. In this article we explore the definition, key rules, counting techniques, and worked examples of classical probability and simple probability calculations.
概率是数学中量化不确定性的分支。古典概型，又称理论概率，是最基础且最常教授的方法：它假定随机实验的所有可能结果等可能发生。本文将探讨古典概型的定义、核心法则、计数技巧以及简单概率计算的例题。
1. What is Classical Probability? | 什么是古典概型？
Classical probability was formalised in the 17th century by mathematicians such as Pascal and Fermat while studying games of chance. The classical definition states: if a random experiment has N equally likely outcomes, and event A occurs in n(A) of those outcomes, then the probability of A is
古典概型在17世纪由帕斯卡和费马等数学家在研究机会游戏时正式确立。古典定义如下：若某个随机实验具有N个等可能的结果，且事件A包含其中n(A)个结果，那么事件A的概率为
P(A) = n(A)/N = n(A)/n(S)
where S is the sample space, so N = n(S). This formula is valid only when every single outcome is equally likely, such as a fair coin, a fair die, or a well-shuffled deck of cards.
其中S为样本空间，即N = n(S)。此公式仅在每个单独结果都等可能时才成立，例如一枚公平的硬币、一粒公平的骰子或一副充分洗过的扑克牌。
2. Sample Spaces and Events | 样本空间与事件
The sample space S is the set of all possible outcomes of an experiment. An event A is any subset of S. For a single toss of a fair coin, S = {H, T}; for a roll of a fair die, S = {1, 2, 3, 4, 5, 6}. An event may be simple, containing one outcome, or compound, containing two or more outcomes.
样本空间S是实验所有可能结果的集合。事件A是S的任意子集。掷一枚公平硬币时，S = {H, T}；掷一粒公平骰子时，S = {1, 2, 3, 4, 5, 6}。事件可以是简单事件（只含一个结果），也可以是复合事件（含有两个或更多结果）。
For example, when rolling a die, the event “an odd number” is A = {1, 3, 5}. Because there are 3 favourable outcomes out of 6, P(odd) = 3/6 = 1/2. The impossible event is the empty set ∅ with probability 0, and the certain event S has probability 1.
例如，掷骰子时，事件“掷出奇数”为 A = {1, 3, 5}。由于6个结果中有3个有利结果，P(奇数) = 3/6 = 1/2。不可能事件是空集∅，其概率为0；必然事件S的概率为1。
3. Basic Probability Rules | 概率的基本性质
Three fundamental rules govern all probabilities. First, for any event A, 0 ≤ P(A) ≤ 1. Second, the sum of the probabilities of all outcomes in the sample space is 1. Third, the complement rule: the probability that A does not occur is
三条基本法则支配着所有概率。第一，对任意事件A，有 0 ≤ P(A) ≤ 1。第二，样本空间中所有结果的概率之和为1。第三，互补法则：事件A不发生的概率为
P(A′) = 1 − P(A)
Complement notation is A′, Aᶜ, or Ā. For example, if the probability of rain is 0.3, then the probability of no rain is 0.7.
互补事件的记法有 A′、Aᶜ 或 Ā。例如，若下雨的概率为0.3，则不下雨的概率为0.7。
4. Counting Outcomes: The Multiplication Principle | 计数结果：乘法原理
Many probability problems require counting outcomes. The multiplication principle states: if one task can be done in m ways and a second independent task can be done in n ways, then the two tasks together can be done in m × n ways.
许多概率问题需要计数结果。乘法原理指出：若完成第一件事有m种方式，完成第二件独立的事有n种方式，那么两件事依次完成共有 m × n 种方式。
Example: A restaurant offers 3 kinds of soup and 4 kinds of main dish. A customer chooses one soup and one main dish. The total number of meals is 3 × 4 = 12. If we also choose one dessert from 2 options, the total becomes 3 × 4 × 2 = 24.
例如：一家餐厅提供3种汤和4种主菜。顾客选择一份汤和一份主菜，总共有 3 × 4 = 12 种套餐。若再选一种甜品（有2种选择），则总数为 3 × 4 × 2 = 24。
5. Permutations | 排列
A permutation is an ordered arrangement of distinct objects. The number of ways to arrange r objects selected from n distinct objects is
排列是对不同对象的有序安排。从n个不同对象中选取r个并按顺序排列的方法数记为
ⁿPᵣ = n! / (n − r)!
where n! = n × (n − 1) × ⋯ × 1. For example, the number of ways to arrange 3 books chosen from 5 books on a shelf is ⁵P₃ = 5! / (5 − 3)! = 5 × 4 × 3 = 60.
其中 n! = n × (n − 1) × ⋯ × 1。例如，从5本书中选3本排在书架上的方法数为 ⁵P₃ = 5! / (5 − 3)! = 5 × 4 × 3 = 60。
Permutations are used when order matters: pin codes, race rankings, seating arrangements.
排列用于顺序重要的场景：密码、比赛名次、座位安排等。
6. Combinations | 组合
A combination is an unordered selection of objects. The number of ways to choose r objects from n distinct objects without regard to order is
组合是无视顺序的对象选取。从n个不同对象中不考虑顺序地选取r个的方法数记为
ⁿCᵣ = n! / [r! (n − r)!]
This expression is also written C(n,r) or nCr. For example, the number of ways to choose a committee of 3 people from 8 candidates is ⁸C₃ = 8! / (3! × 5!) = 56.
该式也记为 C(n,r) 或 nCr。例如，从8名候选人中选出3人组成委员会的方法数为 ⁸C₃ = 8! / (3! × 5!) = 56。
Combinations are used when order does not matter: lottery numbers, handshake counts, selecting teams.
组合用于顺序无关的场景：彩票号码、握手次数、选择队伍等。
7. Probability Using Combinations | 用组合计算概率
When selecting objects from a finite set, classical probability often uses combinations to count favourable and total outcomes. Suppose a bag contains 6 red and 4 blue marbles. We draw 3 marbles without replacement. The total number of possible outcomes is ¹⁰C₃ = 120. The number of ways to draw exactly 2 red and 1 blue is ⁶C₂ × ⁴C₁ = 15 × 4 = 60. Thus
当从有限集合中选取对象时，古典概型常使用组合来计数有利结果与总结果。假设袋中有6个红球和4个蓝球，不放回地抽取3个球。总的可能结果数为 ¹⁰C₃ = 120。恰好抽到2个红球和1个蓝球的方法数为 ⁶C₂ × ⁴C₁ = 15 × 4 = 60。因此
P(exactly 2 red) = 60/120 = 1/2
This illustrates the core idea: probability as the ratio of favourable combinations to total combinations.
这体现了核心思想：概率等于有利组合数与总组合数之比。
8. Mutually Exclusive Events and the Addition Rule | 互斥事件与加法法则
Two events are mutually exclusive if they cannot occur at the same time. For mutually exclusive events A and B, the addition rule gives
如果两个事件不能同时发生，则称它们互斥。对于互斥事件A和B，加法法则为
P(A ∪ B) = P(A) + P(B)
For example, rolling a die: the events “score 2” and “score 5” are mutually exclusive, so P(2 or 5) = 1/6 + 1/6 = 1/3.
例如，掷骰子时，“点数为2”和“点数为5”互斥，因此 P(2或5) = 1/6 + 1/6 = 1/3。
If A and B are not mutually exclusive, the general addition rule subtracts the overlap:
若A和B不互斥，则一般加法法则需要减去重叠部分：
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
For a single card drawn from a deck, let A = “heart” and B = “king”. Then P(A) = 13/52, P(B) = 4/52, and P(A ∩ B) = 1/52. Hence P(A ∪ B) = 13/52 + 4/52 − 1/52 = 16/52 = 4/13.
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