课堂笔记 （Probability (概率)）

Slide 1 — MGS 2150 Business Statistics

第1页——MGS 2150 商业统计学

Knowledge Points (知识点)

1. Course Title: Business Statistics (课程名称：商业统计学)
2. Instructor & Context (教师与背景)：Prof. Rongjuan Chen, Fall 2025, Wenzhou-Kean University.

Business Statistics (商业统计学)

• Explanation (解释):
  Business Statistics is the study of applying statistical tools and concepts in the context of business and economics.
  商业统计学是在商业和经济学中应用统计工具与概念的一门学科。

• Example (例子):
  Companies use statistical analysis to identify sales patterns, forecast demand, and manage risks.
  企业利用统计分析来识别销售模式、预测需求和管理风险。

• Extension (拓展):
  Business statistics not only supports decision-making but also improves operational efficiency, reduces uncertainty, and provides evidence-based insights for managers.
  商业统计学不仅支持决策，还能提升运营效率、减少不确定性，并为管理者提供基于数据的洞察。

Summary (总结)

本页主要介绍课程名称、背景信息，说明了课程的定位：统计学在商业与经济中的实际应用价值。

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Slide 2 — Lecture 8: Probability

第2页——讲座8：概率

Knowledge Points (知识点)

1. Chapter 4: Probability (第4章：概率)
2. Focus Topics (重点内容)：Experiments, counting rules, assigning probabilities, events, probability relationships.

• Explanation (解释):
  This lecture introduces the foundation of probability, covering experiments, methods of assigning probabilities, and relationships between events.
  本讲座介绍概率的基础，包括实验、概率分配方法以及事件之间的关系。

• Example (例子):
  In business, probability helps firms assess the risk of investments, customer behavior, or market trends.
  在商业中，概率帮助公司评估投资风险、客户行为或市场趋势。

• Extension (拓展):
  Understanding probability is critical for quantitative decision-making, financial modeling, and business forecasting.
  掌握概率对于定量决策、金融建模和商业预测至关重要。

Summary (总结)

本页介绍了讲座的总体框架，强调学习概率的核心意义。

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Slide 3 — What is Probability?

第3页——什么是概率？

Knowledge Points (知识点)

1. Definition (定义)：A numerical measure of likelihood of an event’s occurrence.
2. Scale (范围)：Values range from 0 to 1.

• Explanation (解释):
  Probability quantifies uncertainty by measuring how likely an event is to occur, with 0 meaning impossible and 1 meaning certain.
  概率是衡量事件发生可能性的一种数值度量，0 表示不可能，1 表示必然发生。

• Example (例子):
  The probability of flipping a fair coin and getting heads is 0.5.
  抛一枚公平硬币得到正面的概率是 0.5。

• Extension (拓展):
  Probability values help businesses measure risk. For example, the likelihood of a product defect, or the chance of stock price increase.
  概率值帮助企业衡量风险，例如产品缺陷的概率，或股票价格上涨的可能性。

Summary (总结)

本页定义了概率，说明它是介于0到1之间的数值，用于表示事件发生的可能性。

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Slide 4 — Example: Bradley Investments

第4页——示例：布拉德利投资公司

Knowledge Points (知识点)

1. Context (背景)：Bradley invested in two stocks: Markley Oil and Collins Mining.
2. Goal (目标)：Determine possible investment outcomes in 3 months.

• Explanation (解释):
  By listing possible gains or losses from two stocks, we can form a sample space for probability analysis.
  通过列出两只股票的可能盈亏结果，可以构建一个概率分析的样本空间。

• Example (例子):

  • Markley Oil possible outcomes: +10, +5, 0, −20 (in $000).
  • Collins Mining possible outcomes: +8, −2 (in $000).Exampleoutcome:(10,8)meansMarkleyearns$10,000 and Collins earns $8,000.
  • Markley Oil 可能结果：+10、+5、0、−20 (单位：千美元)
  • Collins Mining 可能结果：+8、−2 (单位：千美元)
    示例结果：(10, 8) 表示 Markley 获利 $10,000，Collins获利$8,000。

• Extension (拓展):
  This illustrates how probability can be applied to portfolio analysis, evaluating combined risks and returns.
  这个例子说明了如何用概率进行投资组合分析，评估综合风险与回报。

Summary (总结)

本页通过投资案例引出概率的实际应用，展示如何定义样本空间以进行后续分析。

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Slide 5 — Counting Rules (计数规则)

第5页——计数规则

Knowledge Points (知识点)

1. Multiple-step sequence (多步骤序列)：n1, n2, …, nk.
2. Total outcomes = n1 × n2 × … × nk (总可能结果数 = 各步骤可能数的乘积).

• Explanation (解释):
  When an experiment consists of several steps, the total number of possible results is the product of possibilities at each step.
  当一个实验包含多个步骤时，总的可能结果数等于各步骤可能结果数的乘积。

• Example (例子):

  • Step 1 (Markley Oil): 4 possible outcomes.
  • Step 2 (Collins Mining): 2 possible outcomes.
    Total = 4 × 2 = 8 outcomes.
  • 第1步 (Markley Oil)：4种可能结果。
  • 第2步 (Collins Mining)：2种可能结果。
    总数 = 4 × 2 = 8 种结果。

• Extension (拓展):
  Counting rules are fundamental in probability theory and are often visualized using tree diagrams.
  计数规则是概率论的基础，常通过树形图来直观表示。

Summary (总结)

本页介绍了多步骤实验的计数规则，强调通过乘法原理计算总结果数。

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Slide 6 — Counting Rules with Tree Diagram (树形图示例)

第6页——树形图计数规则

Knowledge Points (知识点)

1. Tree diagrams (树形图)：直观展示多步骤实验的结果。
2. Bradley Investment example: outcomes represented as (Markley, Collins).

• Explanation (解释):
  Tree diagrams help visualize multi-step experiments by branching each possible outcome at each stage.
  树形图通过在每个阶段分支所有可能结果，帮助直观展示多步骤实验。

• Example (例子):

  • (10, 8): Gain $18,000
  • (10, -2): Gain $8,000
  • (0, -2): Lose $2,000
  • (-20, -2): Lose $22,000
  • (10, 8)：盈利 $18,000
  • (10, -2)：盈利 $8,000
  • (0, -2)：亏损 $2,000
  • (-20, -2)：亏损 $22,000

• Extension (拓展):
  Tree diagrams are widely used in decision analysis, showing not only probabilities but also payoffs and risks.
  树形图广泛应用于决策分析，不仅展示概率，还能显示收益与风险。

Summary (总结)

本页通过树形图展示了投资案例的所有可能结果，直观体现计数与结果对应。

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Slide 7 — Assigning Probabilities (概率的分配方法)

第7页——如何分配概率

Knowledge Points (知识点)

1. Classical method (古典法)：等可能性。
2. Relative frequency method (相对频率法)：基于历史数据。
3. Subjective method (主观法)：基于判断。

• Explanation (解释):
  There are three main methods to assign probabilities:

  • Classical: equal likelihood when outcomes are equally possible.
  • Relative frequency: based on observed data.
  • Subjective: based on intuition or judgment.
    分配概率的三种主要方法：
  • 古典法：在所有结果等可能时，概率相等。
  • 相对频率法：基于历史数据或经验。
  • 主观法：基于个人直觉或判断。

• Example (例子):

  • Rolling a die (classical).
  • Rental records of tools (relative frequency).
  • Predicting election results (subjective).
  • 掷骰子（古典法）。
  • 工具租赁记录（相对频率法）。
  • 预测选举结果（主观法）。

• Extension (拓展):
  These methods can be combined. For example, when data is incomplete, firms may mix historical data with expert judgment.
  三种方法可以结合使用，例如数据不完整时，企业会将历史数据与专家判断相结合。

Summary (总结)

本页介绍了三种常见的概率分配方法，分别是古典法、相对频率法和主观法。

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Slide 8 — Classical Method (古典方法)

第8页——古典概率方法

Knowledge Points (知识点)

1. Formula (公式)：If n possible outcomes, each has probability 1/n.
2. Sample space (样本空间)：所有可能结果的集合。

• Explanation (解释):
  When all outcomes are equally likely, each outcome has probability 1/n.
  当所有结果等可能时，每个结果的概率是 1/n。

• Example (例子):

  • Rolling a die with 8 sides: sample space S = {1,2,3,4,5,6,7,8}.
  • Each outcome has probability 1/8.
  • 掷一个八面骰子：样本空间 S = {1,2,3,4,5,6,7,8}。
  • 每个结果的概率为 1/8。

• Extension (拓展):
  Classical probability assumes fairness. In practice, real-world events may deviate, requiring empirical or subjective adjustments.
  古典概率假设绝对公平，但在现实中可能会偏离，需要通过经验或主观判断修正。

Summary (总结)

本页介绍了古典概率分配方法，强调等可能情况下概率均为 1/n。

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Slide 9 — Relative Frequency Method (相对频率法)

第9页——相对频率方法

Knowledge Points (知识点)

1. Probability based on historical records (基于历史记录的概率).
2. Formula: Probability = Frequency of event ÷ Total trials (概率 = 事件频数 ÷ 总实验次数).

• Explanation (解释):
  Relative frequency assigns probabilities based on how often an event occurs in repeated trials or historical data.
  相对频率法根据事件在重复试验或历史数据中出现的频率来分配概率。

• Example (例子):
  Lucas Tool Rental recorded the number of polishers rented each day over 40 days.

  • 0 polishers: 4 days
  • 1 polisher: 6 days
  • 2 polishers: 18 days
  • 3 polishers: 10 days
  • 4 polishers: 2 days
    40 total days.
    Lucas 工具租赁公司记录了40天的抛光机租赁数量：
  • 0 台：4天
  • 1 台：6天
  • 2 台：18天
  • 3 台：10天
  • 4 台：2天
    总计：40天。

• Extension (拓展):
  Relative frequency is widely used in business forecasting, quality control, and insurance risk estimation.
  相对频率广泛应用于商业预测、质量控制和保险风险评估。

Summary (总结)

本页展示了相对频率法，强调通过历史数据计算事件发生的概率。

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Slide 10 — Relative Frequency Method: Example (例子)

第10页——相对频率法：示例

Knowledge Points (知识点)

1. Convert frequency into probability (频数转化为概率).
2. Probabilities must sum to 1 (概率之和等于1).

• Explanation (解释):
  By dividing each frequency by the total number of trials, we obtain the probability distribution of the event.
  通过将每个频数除以总实验次数，可以得到事件的概率分布。

• Example (例子):
  Lucas Tool Rental example (40 days):

  • 0 polishers: 4/40 = 0.10
  • 1 polisher: 6/40 = 0.15
  • 2 polishers: 18/40 = 0.45
  • 3 polishers: 10/40 = 0.25
  • 4 polishers: 2/40 = 0.05
    Sum = 1.00
    例子（40天）：
  • 0 台：0.10
  • 1 台：0.15
  • 2 台：0.45
  • 3 台：0.25
  • 4 台：0.05
    总和 = 1.00

• Extension (拓展):
  This method is useful for demand forecasting, inventory management, and service planning.
  这种方法对需求预测、库存管理和服务规划非常有用。

Summary (总结)

本页通过计算展示了如何将频数转化为概率，并形成完整的概率分布。

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Slide 11 — Subjective Method (主观方法)

第11页——主观概率方法

Knowledge Points (知识点)

1. Used when data is limited or incomplete (用于数据有限或不完整时).
2. Based on judgment, experience, or intuition (基于判断、经验或直觉).
3. Can be combined with other methods (可与其他方法结合使用).

• Explanation (解释):
  Subjective probability relies on human judgment rather than precise data, often applied when historical data is insufficient.
  主观概率依赖于人的判断，而非精确数据，常用于历史数据不足的情况。

• Example (例子):
  An investor estimates the chance of stock price rising based on market trends and expert opinion.
  投资者根据市场趋势和专家意见来估计股票价格上涨的可能性。

• Extension (拓展):
  In business, subjective probability is common in forecasting new product success, political risks, or rare events.
  在商业中，主观概率常用于预测新产品成功率、政治风险或罕见事件。

Summary (总结)

本页介绍了主观概率法，强调它在缺乏充分数据时的重要性。

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Slide 12 — Bradley Investments: Assigning Probabilities (概率分配案例)

第12页——布拉德利投资：分配概率

Knowledge Points (知识点)

1. Assigning probabilities to each outcome (为每个结果分配概率).
2. Probabilities based on analysis or judgment (基于分析或判断的概率).

• Explanation (解释):
  For Bradley’s two-stock investment, probabilities can be assigned to each outcome, reflecting likelihood of combined gains or losses.
  对于布拉德利的双股票投资，可以为每个结果分配概率，反映综合盈亏的可能性。

• Example (例子):
  Assigned probabilities for outcomes:

  • (10, 8): 0.20
  • (10, -2): 0.08
  • (5, 8): 0.16
  • (5, -2): 0.26
  • (0, 8): 0.10
  • (0, -2): 0.12
  • (-20, 8): 0.02
  • (-20, -2): 0.06
    投资结果的分配概率：
  • (10, 8)：0.20
  • (10, -2)：0.08
  • (5, 8)：0.16
  • (5, -2)：0.26
  • (0, 8)：0.10
  • (0, -2)：0.12
  • (-20, 8)：0.02
  • (-20, -2)：0.06

• Extension (拓展):
  Probability distributions like this are useful in risk analysis, allowing investors to evaluate expected returns and losses.
  这种概率分布在风险分析中非常有用，能帮助投资者评估预期收益与亏损。

Summary (总结)

本页通过投资案例展示了如何为不同结果分配概率，建立概率分布模型。

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Slide 13 — Events and Probabilities (事件与概率)

第13页——事件与概率

Knowledge Points (知识点)

1. Event (事件)：由一组样本点组成。
2. Event probability (事件的概率)：等于该事件包含的所有样本点概率之和。

• Explanation (解释):
  An event is any collection of sample points from the sample space. Its probability is the sum of the probabilities of all included outcomes.
  事件是样本空间中的一个集合，其概率等于所有包含样本点的概率之和。

• Example (例子):

  • Event M = “Markley Oil is profitable.”
  • M = {(10,8), (10,-2), (5,8), (5,-2)}.
  • P(M) = 0.20 + 0.08 + 0.16 + 0.26 = 0.70.
  • 事件 M = “Markley Oil 盈利。”
  • M = {(10,8), (10,-2), (5,8), (5,-2)}
  • P(M) = 0.70

• Extension (拓展):
  Event probability calculation is the basis of probability theory, used in reliability analysis, quality control, and finance.
  事件概率的计算是概率论的基础，用于可靠性分析、质量控制和金融研究。

Summary (总结)

本页定义了事件，并展示了如何通过样本点的概率求和得到事件的概率。

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Slide 14 — Example: Event C (Collins Mining Profitable)

第14页——示例：事件 C（Collins Mining 盈利）

Knowledge Points (知识点)

1. Event C = Collins Mining is profitable.
2. Probability P(C) = sum of included sample points.

• Explanation (解释):
  To evaluate if Collins Mining is profitable, we sum the probabilities of outcomes where Collins Mining’s gain > 0.
  判断 Collins Mining 是否盈利，需要将其盈利情况下的概率相加。

• Example (例子):

  • C = {(10,8), (5,8), (0,8), (-20,8)}
  • P(C) = 0.20 + 0.16 + 0.10 + 0.02 = 0.48
  • C = {(10,8), (5,8), (0,8), (-20,8)}
  • P(C) = 0.48

• Extension (拓展):
  This type of calculation is widely used in portfolio analysis to determine the probability of profits from specific assets.
  这种计算常用于投资组合分析，评估某个特定资产盈利的可能性。

Summary (总结)

本页通过 Collins Mining 的盈利情况说明如何计算特定事件的概率。

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Slide 15 — Basic Relationships of Probability (概率的基本关系)

第15页——概率的基本关系

Knowledge Points (知识点)

1. Complement of an event (事件的补集).
2. Union of two events (两个事件的并集).
3. Intersection of two events (两个事件的交集).
4. Mutually exclusive events (互斥事件).

• Explanation (解释):

  • Complement: all outcomes not in event A.
  • Union: outcomes in A, or B, or both.
  • Intersection: outcomes in both A and B.
  • Mutually exclusive: events cannot occur together.
  • 补集：所有不在事件 A 中的结果。
  • 并集：在事件 A 或事件 B 或两者中。
  • 交集：在 A 和 B 中同时出现的结果。
  • 互斥事件：两个事件不能同时发生。

• Example (例子):

  • Tossing a coin: Event A = heads, Event B = tails. A and B are mutually exclusive.
  • 抛硬币：事件 A = 正面，事件 B = 反面。A 和 B 互斥。

• Extension (拓展):
  These relationships are the foundation for probability laws like the addition law and complement rule.
  这些关系是概率法则（如加法法则和补集法则）的基础。

Summary (总结)

本页总结了概率论中四个最基本的事件关系：补集、并集、交集和互斥。

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Slide 16 — Complement of an Event (事件的补集)

第16页——事件的补集

Knowledge Points (知识点)

1. Complement notation: Aᶜ.
2. Includes all outcomes not in event A.

• Explanation (解释):
  The complement of an event A consists of all sample points not in A.
  事件 A 的补集包含所有不在 A 中的样本点。

• Example (例子):

  • Tossing a die: Event A = rolling an even number {2,4,6}.
  • Aᶜ = rolling an odd number {1,3,5}.
  • 掷骰子：事件 A = 出现偶数 {2,4,6}。
  • Aᶜ = 出现奇数 {1,3,5}。

• Extension (拓展):
  Complement probabilities are useful in simplifying calculations:
  P(Aᶜ) = 1 − P(A).
  补集概率常用于简化计算：P(Aᶜ) = 1 − P(A)。

Summary (总结)

本页介绍了补集的概念，强调了 P(Aᶜ) = 1 − P(A) 的基本关系。

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Slide 17 — Union of Two Events (两个事件的并集)

第17页——两个事件的并集

Knowledge Points (知识点)

1. Union notation: A ∪ B.
2. Definition: all sample points in A, B, or both.

• Explanation (解释):
  The union of events A and B contains all outcomes that belong to A, or B, or both.
  事件 A 与 B 的并集包含所有属于 A 或 B 或两者的结果。

• Example (例子):

  • Event A = student passes math.
  • Event B = student passes English.
  • A ∪ B = student passes math, or English, or both.
  • 事件 A = 学生通过数学。
  • 事件 B = 学生通过英语。
  • A ∪ B = 学生通过数学或英语，或两者都通过。

• Extension (拓展):
  Union events are key in evaluating combined probabilities, such as “profitable in at least one investment.”
  并集事件在计算组合概率时非常重要，比如“至少一项投资盈利”。

Summary (总结)

本页介绍了并集的定义及其在概率计算中的作用。

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Slide 18 — Example: Bradley Investments (Union)

第18页——示例：布拉德利投资（并集）

Knowledge Points (知识点)

1. Event M = Markley Oil profitable.
2. Event C = Collins Mining profitable.
3. Union: M ∪ C = M profitable OR C profitable.

• Explanation (解释):
  The union of events M and C includes outcomes where at least one company is profitable.
  M 和 C 的并集包含至少一家公司盈利的结果。

• Example (例子):
  M ∪ C = {(10,8), (10,-2), (5,8), (5,-2), (0,8), (-20,8)}
  P(M ∪ C) = 0.20 + 0.08 + 0.16 + 0.26 + 0.10 + 0.02 = 0.82
  M ∪ C = {(10,8), (10,-2), (5,8), (5,-2), (0,8), (-20,8)}
  P(M ∪ C) = 0.82

• Extension (拓展):
  Union probabilities are widely applied in finance, e.g., estimating the chance of at least one project being successful.
  并集概率常用于金融分析，如估算至少一个项目成功的可能性。

Summary (总结)

本页用投资案例展示了如何计算两个事件并集的概率。

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Slide 19 — Intersection of Two Events (两个事件的交集)

第19页——两个事件的交集

Knowledge Points (知识点)

1. Intersection notation: A ∩ B.
2. Definition: outcomes belonging to both A and B.

• Explanation (解释):
  The intersection of events A and B contains all outcomes that belong to both A and B simultaneously.
  事件 A 与 B 的交集包含同时属于 A 和 B 的所有结果。

• Example (例子):

  • Event A = student passes math.
  • Event B = student passes English.
  • A ∩ B = student passes both subjects.
  • 事件 A = 学生通过数学。
  • 事件 B = 学生通过英语。
  • A ∩ B = 学生同时通过数学和英语。

• Extension (拓展):
  Intersection events are crucial in joint probability analysis, e.g., probability of both investments being profitable.
  交集事件在联合概率分析中非常重要，例如计算两项投资都盈利的概率。

Summary (总结)

本页介绍了交集的定义，强调其在联合事件分析中的应用。

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Slide 20 — Example: Bradley Investments (Intersection)

第20页——示例：布拉德利投资（交集）

Knowledge Points (知识点)

1. Event M = Markley Oil profitable.
2. Event C = Collins Mining profitable.
3. Intersection: M ∩ C = M AND C profitable.

• Explanation (解释):
  The intersection includes outcomes where both Markley Oil and Collins Mining are profitable.
  M 和 C 的交集包括两家公司同时盈利的结果。

• Example (例子):
  M ∩ C = {(10,8), (5,8)}
  P(M ∩ C) = 0.20 + 0.16 = 0.36
  M ∩ C = {(10,8), (5,8)}
  P(M ∩ C) = 0.36

• Extension (拓展):
  This analysis is vital for portfolio diversification, measuring the chance of both investments performing well.
  这种分析对投资组合多样化非常重要，用于衡量两项投资同时表现良好的可能性。

Summary (总结)

本页通过投资案例展示了如何计算两个事件交集的概率。

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Slide 21 — Addition Law (加法法则)

第21页——加法法则

Knowledge Points (知识点)

1. Formula: P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
2. Prevents double-counting the intersection.

• Explanation (解释):
  The addition law is used to calculate the probability of the union of two events, subtracting the overlap (intersection) to avoid double-counting.
  加法法则用于计算两个事件并集的概率，要减去交集部分，避免重复计算。

• Example (例子):
  P(M ∪ C) = P(M) + P(C) − P(M ∩ C).
  = 0.70 + 0.48 − 0.36 = 0.82
  P(M ∪ C) = 0.82

• Extension (拓展):
  Addition law is fundamental in probability theory, especially for evaluating overlapping risks or events.
  加法法则是概率论的基础，尤其在评估重叠风险或事件时非常重要。

Summary (总结)

本页介绍了加法法则及其在并集计算中的应用。

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Slide 22 — Example: Bradley Investments (Addition Law)

第22页——示例：布拉德利投资（加法法则）

Knowledge Points (知识点)

1. Event M = Markley Oil profitable (P(M) = 0.70).
2. Event C = Collins Mining profitable (P(C) = 0.48).
3. Intersection: P(M ∩ C) = 0.36.
4. Union: P(M ∪ C) = 0.82.

• Explanation (解释):
  Applying the addition law, the probability that at least one of the two investments is profitable is 0.82.
  使用加法法则计算，至少有一项投资盈利的概率为 0.82。

• Example (例子):
  P(M ∪ C) = 0.70 + 0.48 − 0.36 = 0.82
  P(M ∪ C) = 0.82

• Extension (拓展):
  In finance, this helps measure the likelihood that a portfolio achieves gains from at least one asset.
  在金融中，这有助于衡量投资组合中至少一项资产获得收益的可能性。

Summary (总结)

本页通过投资案例展示了加法法则的实际应用。

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Slide 23 — Mutually Exclusive Events (互斥事件)

第23页——互斥事件

Knowledge Points (知识点)

1. Definition: Two events have no sample points in common.
2. If A and B are mutually exclusive, P(A ∩ B) = 0.
3. Formula: P(A ∪ B) = P(A) + P(B).

• Explanation (解释):
  Mutually exclusive events cannot occur simultaneously. Their intersection probability is zero.
  互斥事件不能同时发生，其交集概率为零。

• Example (例子):

  • Tossing a die: Event A = {1,2}, Event B = {3,4}.
  • A and B are mutually exclusive.
  • P(A ∪ B) = P(A) + P(B).
  • 掷骰子：事件 A = {1,2}，事件 B = {3,4}。
  • A 与 B 互斥。
  • P(A ∪ B) = P(A) + P(B)。

• Extension (拓展):
  This concept is essential in risk management, e.g., identifying independent failure modes in engineering.
  这一概念在风险管理中很重要，例如识别工程中的独立故障模式。

Summary (总结)

本页介绍了互斥事件，强调其特点是交集概率为零。

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Slide 24 — End of Lecture 8 (讲座总结)

第24页——第8讲总结

Knowledge Points (知识点)

1. Probability definition (概率的定义).
2. Methods of assigning probability (概率分配方法：古典、相对频率、主观).
3. Event probability and relationships (事件概率及其关系：补集、并集、交集、互斥).
4. Addition law (加法法则).

• Explanation (解释):
  Lecture 8 provided the foundation of probability, covering definitions, methods of assigning probabilities, and relationships among events.
  本讲座系统讲解了概率的基础，包括定义、分配方法以及事件之间的关系。

• Example (例子):
  Bradley Investments example illustrated how to compute probabilities for different scenarios in business.
  布拉德利投资的案例展示了如何在商业场景中计算不同结果的概率。

• Extension (拓展):
  These concepts are widely used in business statistics, risk analysis, financial modeling, and decision-making.
  这些概念广泛应用于商业统计、风险分析、金融建模和决策制定。

Summary (总结)

本页总结了概率论的基本知识点，为后续学习更复杂的统计与概率模型奠定基础。

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