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Probability: Terminology

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

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Probability: Terminology

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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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