Probability Theory

Probability Theory is a branch of mathematics that deals with the analysis and interpretation of random phenomena. It provides a framework for quantifying uncertainty, allowing for the calculation of the likelihood of various outcomes based on known information. Probability Theory is foundational for various fields, including statistics, finance, science, and engineering, and encompasses concepts such as random variables, probability distributions, expectation, and statistical inference. Through its principles, it enables the modeling and understanding of complex systems where uncertainty exists.

1. Fundamental Concepts of Probability Theory
   1. Random Experiments
      1. Definition and Characteristics
         1. Unpredictability
         2. Repeatability
         3. Outcome Set
      2. Examples
         1. Tossing a coin
         2. Rolling a die
         3. Drawing a card
   2. Sample Space
      1. Definition
         1. Set of all possible outcomes
      2. Types of Sample Spaces
         1. Discrete Sample Space
            1. Finite number of outcomes
            2. Countable outcomes
         2. Continuous Sample Space
            1. Uncountable outcomes
            2. Example: Range of real numbers between 0 and 1
      3. Representation
         1. List notation
         2. Set builder notation
         3. Venn diagrams
   3. Events
      1. Definition
         1. Subset of the sample space
      2. Types of Events
         1. Simple Events
            1. Events with a single outcome
            2. Example: Getting a head when a coin is tossed
         2. Compound Events
            1. Events formed by combining two or more simple events
            2. Use of logical operations (union, intersection)
         3. Mutually Exclusive Events
            1. Events that cannot occur simultaneously
            2. Example: Rolling an odd and an even number in a single die throw
         4. Exhaustive Events
            1. A set of events covering the entire sample space
            2. Every possible outcome is included
      3. Operations on Events
         1. Union of Events
            1. Definition and notation
            2. Example using Venn diagrams
         2. Intersection of Events
            1. Definition and notation
            2. Example using Venn diagrams
         3. Complement of an Event
            1. Definition and notation
            2. Example using Venn diagrams
   4. Probability Axioms
      1. Non-negativity
         1. Probability of any event is a non-negative number
      2. Additivity
         1. For mutually exclusive events, probability of their union is the sum of their individual probabilities
      3. Normalization
         1. Total probability of the entire sample space equals one
   5. Properties Derived from Axioms
      1. Monotonicity
         1. Probability of any event is less than or equal to one
      2. Complement Rule
         1. Probability of an event plus the probability of its complement is one
      3. Inclusion-Exclusion Principle
         1. Relationship to calculate probability of the union of two events
   6. Real-world Applications
      1. Weather Forecasting
         1. Modeling uncertainty in predictions
      2. Quality Control
         1. Assessing probabilities of defects in manufacturing
      3. Epidemiology
         1. Estimating the likelihood of disease outbreaks
   7. Limitations and Paradigms
      1. Classical Interpretation
         1. Based on symmetry and equally likely outcomes
      2. Frequentist Interpretation
         1. Long-run relative frequency of an event
      3. Subjective Probability
         1. Based on personal belief or opinion