Probability (概率) — Lecture 8
1. Introduction (简介)
Definition (定义)
- Probability is a numerical measure of the likelihood of an event.
- 概率是事件发生可能性的数值度量。
Scale (范围)
- Ranges from 0 (impossible) to 1 (certain).
- 取值范围为 0（不可能）到 1（必然）。
Purpose (目的)
- To quantify uncertainty in decision-making.
- 用于在决策中量化不确定性。
2. Experiments and Sample Space (实验与样本空间)
Experiment (实验)
- A process leading to well-defined outcomes.
- 导致确定结果的过程。
Sample Space (样本空间)
- The set of all possible outcomes.
- 所有可能结果的集合。
Example (例子)
- Investment outcomes for Markley Oil & Collins Mining.
- Markley Oil 与 Collins Mining 的投资结果。
3. Counting Rules (计数规则)
Multiplication Rule (乘法规则)
- Total outcomes = n1 × n2 × … × nk.
- 总可能结果数 = n1 × n2 × … × nk。
Tree Diagram (树形图)
- Visual representation of multi-step experiments.
- 多步骤实验的直观表示方法。
Example (例子)
- 8 outcomes for two stocks (4 × 2).
- 两只股票的投资结果，共 8 种可能。
4. Assigning Probabilities (分配概率的方法)
Classical Method (古典法)
- Equal likelihood: each outcome has probability 1/n.
- 等可能性：每个结果概率均为 1/n。
Relative Frequency Method (相对频率法)
- Probability = frequency ÷ total trials.
- 概率 = 事件出现次数 ÷ 总次数。
Subjective Method (主观法)
- Based on judgment and estimation.
- 基于判断与估计。
Example (例子)
- Dice rolling (classical).
- 工具租赁记录 (relative frequency).
- Expert forecasts (subjective).
5. Events and Probabilities (事件与概率)
Event (事件)
- A collection of sample points.
- 样本点的集合。
Event Probability (事件的概率)
- Sum of probabilities of all included outcomes.
- 包含样本点概率之和。
Example (例子)
- Event M (Markley profitable), P(M) = 0.70.
- 事件 M (Markley 盈利)，P(M) = 0.70。
6. Basic Relationships (基本关系)
Complement (补集)
- Outcomes not in event A.
- 不在事件 A 中的结果。
- Formula: P(Aᶜ) = 1 − P(A).
Union (并集)
- Outcomes in A or B or both.
- 属于 A 或 B 或两者的结果。
- Example: P(M ∪ C) = 0.82.
Intersection (交集)
- Outcomes in both A and B.
- 同时属于 A 和 B 的结果。
- Example: P(M ∩ C) = 0.36.
Mutually Exclusive (互斥事件)
- No outcomes in common, P(A ∩ B) = 0.
- 没有共同结果，P(A ∩ B) = 0。
7. Probability Laws (概率法则)
Addition Law (加法法则)
- P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
- 避免重复计算交集。
Example (例子)
- P(M ∪ C) = 0.70 + 0.48 − 0.36 = 0.82.
- 至少一家公司盈利的概率 = 0.82。
8. Applications (应用)
Business Context (商业背景)
- Investment risk evaluation.
- 投资风险评估。
Decision Making (决策制定)
- Supports forecasting and risk analysis.
- 支持预测与风险分析。
Portfolio Analysis (投资组合分析)
- Evaluates combined profitability.
- 评估组合的整体盈利可能性。
9. Summary (总结)
Key Points (关键点)
- Probability definition and range.
- 概率的定义与范围。
- Methods of assigning probabilities.
- 分配概率的方法。
- Events and relationships (complement, union, intersection, exclusivity).
- 事件及其关系（补集、并集、交集、互斥）。
- Addition law and its applications.
- 加法法则及应用。
Importance (重要性)
- Probability provides the foundation for advanced statistics, risk management, and decision-making.
- 概率为高级统计、风险管理和决策制定奠定基础。