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Understanding Probability and Set Notation: Definitions and Axioms

Explore the fundamental concepts of probability, set notation, and the axioms that govern them. Learn about sample spaces, events, unions, intersections, complements, and DeMorgan's Law. Understand terms like mutually exclusive and exhaustive events, theoretical and empirical probabilities, and the Axioms of Probability. Dive into the properties of probability, Kolmogorov's work, and the Classical Definition. Enhance your knowledge of probability theory and its foundational principles.

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Understanding Probability and Set Notation: Definitions and Axioms

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  1. Probability: Terminology • Sample Space • Set of all possible outcomes of a random experiment. • Random Experiment • Any activity resulting in uncertain outcome • Event • Any subset of outcomes in the sample space • An event is said to occur if and only if the outcome of a random experiment is an element of the event • Simple Event has only one outcome

  2. A ∩ B A U B Probability: Set Notation • A U B – Union of A and B (OR) • set containing all elements in A or B • A ∩B –Intersection of A and B (AND) • set containing elements in both A and B • Venn Diagrams A B A B

  3. S A Probability: Set Notation • A’ – Complement of A (NOT) • set containing all elements not in A • { } – Null or Empty Set • Set which contains no elements • A U B = (A' ∩ B')' - DeMorgan’s Law

  4. Probability: Terminology • Mutually Exclusive Events • Events with no outcomes in common. • A1, A2, … , Ak such that Ai ∩Aj = {} for all i≠j. • Exhaustive Events • Events which collectively include all distinct outcomes in sample space • A1, A2, … , Ak such that A1 U A2 U … U Ak = S.

  5. Probability: Terminology • Mutually Exclusive & Exhaustive Events • Events with no outcomes in common that collectively include all distinct outcomes in the sample space. • P(A) Denotes the Probability of Event A • Theoretical – exact, not always calculable • Empirical – relative frequency of occurrence • Converges to theoretical as number of repetitions gets large

  6. Axioms of Probability • 6th of Hilbert's 23 Math Problems in 1900 • Kolmogorov found in 1933 • Axiom 1: P(A) ≥ 0 • Axiom 2: P(S) = 1 • Axiom 3: For mutually exclusive events A1, A2, A3, … • P(A1 U A2 U ... U Ak) = P(A1) + P(A2)+...+ P(Ak) • P(A1 U A2 U ...) = P(A1) + P(A2) + ...

  7. Some Properties of Probability • For any event A, P(A) = 1 – P(A’) • P({}) = 0 • If A is a subset of B, then P(A) ≤ P(B) • For all events A, P(A) ≤ P(S) = 1 0 = P({}) ≤ P(A) ≤ P(S) = 1

  8. Some Properties of Probability • For any events A and B, P(A U B) = P(A) + P(B) – P(A ∩ B) • For any events A, B and C, P(A U B U C) = P(A) + P(B) + P(C) – P(A ∩ B) – P(A ∩ C) – P(B ∩ C) + P(A ∩ B ∩ C)

  9. Classical Definition • Suppose that an experiment consists of N equally likely distinct outcomes. • Each distinct outcome oi has probability P(oi) = 1/N • An event A consisting of m distinct outcomes has probability P(A) = m / N • If an experiment has finite sample space with equally likely outcomes, then an event A has probability P(A) = N(A) / N(S) • where N() is the counting function, so N(A) is the number of distinct outcomes in A

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