Unit 1196: Probability Theory and Its Applications

1. Introduction to Probability Theory

• Definition of probability
• Sample spaces and events
• Types of events: mutually exclusive, exhaustive
• Fundamental rules of probability
  • Addition rule
  • Multiplication rule
  • Complement rule

2. Probability Distributions

2.1 Discrete Probability Distributions

• Definition and examples (Bernoulli, Binomial, Poisson)
• Probability mass function (PMF)
• Properties and characteristics

2.2 Continuous Probability Distributions

• Definition and examples (Uniform, Normal, Exponential)
• Probability density function (PDF)
• Properties and characteristics

3. Conditional Probability

• Definition and formula
• Relationship between events
• Independent vs dependent events
• Law of total probability

4. Bayes' Theorem

• Statement and formula
• Intuition behind Bayes' Theorem
• Applications in updating probabilities with new evidence

5. Combinatorics and Probability

• Fundamental counting principle
• Permutations: definition and formulas
• Combinations: definition and formulas
• Applying combinatorics to probability problems

6. Expected Value and Variance

• Definition of expected value (mean) for discrete and continuous variables
• Calculation methods
• Variance and standard deviation: definitions and formulas
• Interpretation and importance in decision-making

7. Central Limit Theorem

• Statement of the theorem
• Importance in sampling distributions
• Relationship with the law of large numbers
• Practical implications for normal approximation

8. Hypothesis Testing

• Introduction to hypothesis testing
• Null and alternative hypotheses
• Significance level and p-values
• Type I and Type II errors
• Basic test procedures

9. Applications of Probability Theory

• Use cases in statistics (e.g., inferential statistics)
• Applications in finance (risk assessment, portfolio theory)
• Engineering applications (reliability, quality control)
• Healthcare applications (diagnostic testing, epidemiology)
• Decision-making under uncertainty