Stat 310 Chapter 1

Front 1

PROBABILITY

Back 1

The mathematical model of chance phenomena concerned with random outcomes

Front 2

SAMPLE SPACE

Back 2

The set of all possible outcomes corresponding to an experiment, denoted by the Greek letter Omega.

Front 3

EVENT

Back 3

A particular subset of Omega, denoted by a capital letter. The components of the event are listed in brackets.

Front 4

UNION

Back 4

The union of two events A and B is event C, where either A or B or A and B could occur. Denoted by C=A U B.

Front 5

INTERSECTION

Back 5

The intersection of two events A and B is event C, where A and B both occur. Denoted by C=A n B.

Front 6

COMPLEMENT

Back 6

The complement is the event that A does not occur, or all elements in the sample space not in A. Denoted by Acomplement.

Front 7

EMPTY SET

Back 7

An empty set is a set with no elements, denoted by 0 with a line through it.

Front 8

DISJOINT

Back 8

If two or more events have no elements in common, they are disjoint.

Front 9

COMMUTATIVE LAW OF SET THEORY

Back 9

A U B = B U A
A n B = B n A

Front 10

ASSOCIATIVE LAW OF SET THEORY

Back 10

(A U B) U C = A U (B U C)
(A n B) n C = A n (B n C)

Front 11

DISTRIBUTIVE LAW OF SET THEORY

Back 11

(A U B) n C = (A n C) U (B n C)
(A n B) U C = (A U C) n (B U C)

Front 12

PROBABILITY MEASURE

Back 12

A probability measure on Omega is a function P from subsets of Omega to the real numbers that satisfies:

1. P(Omega) = 1
2. If A < Omega, then P(A) >= 0 (no negative probability)
3. If A1 and A2 are disjoint, then P(A1 U A2) = P(A1) + P(A2)

Front 13

PROPERTIES OF PROBABILITY MEASURES

Back 13

Property A: P(Acomplement) = 1 - P(A)

Proof: Because A and Acomplement are disjoin, P(A U Acomplement) = P(A) + P(Acomplement); also, A U Acomplement = Omega and P(Omega) - P(A U Acomplement) = 1 therefore 1 = P(A) + P(Acomplement), or P(Acomplement) = 1 - P(A)

Property B: P(empty set) = 0

Property C: If A < B, then P(A) <= P(B)

Property D: P(A U B) = P(A) + P(B) - P(A n B)