Probability Theory

1. Types of Probability
   1. Classical Probability
      1. Definition and Overview
         1. Based on equally likely outcomes
         2. Used in situations where each outcome in a sample space is equally probable
      2. Key Characteristics
         1. Relies on symmetry and logical reasoning
         2. Best used in games of chance or theoretical scenarios
      3. Applications
         1. Tossing a fair coin
         2. Rolling a fair die
      4. Limitations
         1. Assumes equal likelihood, which is often not true in empirical situations
   2. Frequency (Empirical) Probability
      1. Definition and Overview
         1. Based on the frequency of occurrence of an event
         2. Derived from experimentation or historical data
      2. Key Characteristics
         1. Relies on the Law of Large Numbers for accuracy
         2. Observational and statistical in nature
      3. Applications
         1. Weather forecasting
         2. Quality control and defect analysis
      4. Strengths and Weaknesses
         1. More applicable to real-world and non-ideal conditions
         2. Requires a large amount of data for reliability
   3. Subjective Probability
      1. Definition and Overview
         1. Based on personal judgment or belief
         2. Influenced by individual opinions and experience
      2. Key Characteristics
         1. Not necessarily based on historical data or frequentist methods
         2. Reflects personal bias or intuition
      3. Applications
         1. Decision-making in business and economics
         2. Gambling and betting strategies
      4. Advantages and Concerns
         1. Can incorporate expert insights and nuances
         2. Susceptible to biases and less consistent
   4. Axiomatic Probability
      1. Definition and Overview
         1. Based on set theory and mathematical axioms
         2. Formal and rigorous approach to probability
      2. Key Axioms
         1. Non-negativity: Probability of any event is a non-negative number
         2. Normalization: Probability of the entire sample space is 1
         3. Additivity: Probability of the union of mutually exclusive events is the sum of their individual probabilities
      3. Applications
         1. Theoretical developments in statistics and probability
         2. Foundation for probabilistic reasoning in various fields
      4. Advantages
         1. Provides a consistent and universally applicable framework
         2. Supports advanced mathematical analysis and derivations