Probability Spaces

Discrete Math · Axiom Academy

The formal mathematical foundation of probability theory: sample spaces, events, and the axioms that make probability rigorous and consistent. Sample Space (Ω): The set of all possible outcomes of an experiment. Event (E): A subset of the sample space. An event "occurs" if the outcome is in that subset. Elementary Event: A single outcome, ω , where ω ∈ Ω. For example, when rolling a six-sided die, the sample space is Ω = 1, 2, 3, 4, 5, 6 . The event "rolling an even number" is the subset E = 2, 4, 6 . A probability function P assigns a number to each event. But not just any assignment works—it must satisfy three fundamental axioms proposed by Kolmogorov: Axiom 1 (Non-negativity): For any event E, P(E) ≥ 0 Axiom 2 (Normalization): The probability of the entire sample space is 1, i.e., P(Ω) = 1 Axiom 3 (Additivity): For disjoint events E₁ and E₂ (where E₁ ∩ E₂ = ∅), we have P(E₁ ∪ E₂) = P(E₁) + P(E₂) These axioms ensure consistency. For instance, they guarantee that P(∅) = 0 and that 0 ≤ P(E) ≤ 1 for all events E. Since events are sets, we can combine them using standard set operations. These correspond directly to logical operations in probability: 4. Probability Calculations with Venn Diagrams Venn diagrams provide a visual tool for understanding probability calculations. The most important formula combines union and intersection: P(A ∪ B) = P(A) + P(B) - P(A ∩ B) We subtract P(A ∩ B) because when we add P(A) and P(B), we count the intersection twice.

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