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 ∩ BA U B

A B A B

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. 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.

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 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, …

A. P(A1 U A2 U ... U Ak) = P(A1) + P(A2)+...+ P(Ak)

B. P(A1 U A2 U ...) = P(A1) + P(A2) + ...

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 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