Probability Theory

1. Probability Rules
   1. Addition Rule
      1. Definition
      2. Application in events
         1. Mutually exclusive events
         2. Non-mutually exclusive events
      3. Formula and Examples
         1. \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \)
         2. Application in real-life scenarios
   2. Multiplication Rule
      1. Definition
      2. Application in independent events
         1. Independent event criteria
         2. Example of independent events
      3. Application in dependent events
         1. Dependent event criteria
         2. Calculating joint probability
      4. Formula and Examples
         1. \( P(A \cap B) = P(A) \times P(B|A) \)
         2. Practical scenarios illustrating the rule
   3. Complementary Rule
      1. Definition
      2. Concept of complement events
      3. Calculation of complementary probability
         1. Formula: \( P(A') = 1 - P(A) \)
      4. Use cases
         1. Simplifying calculations
         2. Applying where outcomes are easier to count
   4. Conditional Probability
      1. Definition
      2. Formula: \( P(B|A) = \frac{P(A \cap B)}{P(A)} \)
      3. Concept of conditioning
      4. Real-life applications
         1. Risk analysis
         2. Decision making in uncertainty
      5. Conditional probability table
   5. Bayes’ Theorem
      1. Definition and Importance
      2. Formula: \( P(A|B) = \frac{P(B|A)P(A)}{P(B)} \)
      3. Components
         1. Prior Probability
            1. Definition and examples
            2. Role in Bayesian framework
         2. Likelihood
            1. Definition and interpretation
            2. Example scenarios
         3. Posterior Probability
            1. Definition and calculation
            2. Importance in updating beliefs
      4. Applications
         1. Medical diagnosis
         2. Machine learning algorithms
         3. Fraud detection
      5. Bayesian Decision Theory
         1. Overview
         2. Use in decision-making processes
         3. Examples and case studies