Day 1: Foundations of Probability Theory

1. Basic Probability Concepts and Definitions

Sample Space and Events

• Sample Space (Ω): The set of all possible outcomes of an experiment
• Event (A): A subset of the sample space
• Elementary Event: A single outcome from the sample space

Example: Rolling a fair six-sided die

• Sample Space: Ω = {1, 2, 3, 4, 5, 6}
• Event A (rolling an even number): A = {2, 4, 6}
• Elementary Event: Rolling a 3

Probability Axioms (Kolmogorov Axioms)

1. Non-negativity: P(A) ≥ 0 for all events A
2. Normalization: P(Ω) = 1
3. Countable Additivity: For mutually exclusive events A₁, A₂, ...: $P(A₁ ∪ A₂ ∪ ...) = P(A₁) + P(A₂) + ...$

Some Properties

• Complement Rule: $P(Aᶜ) = 1 - P(A)$
• Addition Rule: $P(A ∪ B) = P(A) + P(B) - P(A ∩ B)$
• Empty Set: $P(∅) = 0$