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

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 U A U B AB AB

 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 Probability: Set Notation A S

Probability: Terminology  Mutually Exclusive Events  Events with no outcomes in common.  A 1, A 2, …, A k such that A i ∩ A j = {} for all i≠j.  Exhaustive Events  Events which collectively include all distinct outcomes in sample space  A 1, A 2, …, A k such thatA 1 U A 2 U … U A k = S.

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

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

Some Properties of Probability 1. For any event A, P(A) = 1 – P(A’) 2. P({}) = 0 3. If A is a subset of B, then P(A) ≤ P(B) 4. For all events A, P(A) ≤ P(S) = 1 0 = P({}) ≤ P(A) ≤ P(S) = 1

Some Properties of Probability 5. For any events A and B, P(A U B) = P(A) + P(B) – P(A ∩ B) 6. 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)

Classical Definition  Suppose that an experiment consists of N equally likely distinct outcomes.  Each distinct outcome o i has probability P(o i ) = 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