Probability Theory: Basic Definitions, Rules, and Applications in Economics (ECON 223), Lecture notes of Economics

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Mathematics for Economics II Mathematics for Economics II ECON 223 ECON 223

By By Vishal ChoudhuryVishal Choudhury

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Chapter 2 Chapter 2

Probability Probability

 Define probability, sample space, and event.

 Distinguish between subjective and objective probability.

 Describe the complement of an event, the intersection, and the union of two

events.



Compute probabilities of various types of events.

 Explain the concept of conditional probability and how to compute it.

 Describe permutation and combination and their use in certain probability

computations.

 Explain Bayes’ theorem and its applications.

LEARNING OBJECTIVES LEARNING OBJECTIVES

After studying this chapter, you should be able to: After studying this chapter, you should be able to:

(^22)

2-1 2-1 Probability is:Probability is:

A quantitative measure of uncertainty

A measure of the strength of belief in the occurrence of an

uncertain event

A measure of the degree of chance or likelihood of

occurrence of an uncertain event

Measured by a number between 0 and 1 (or between 0% and

Types of Probability (Continued) Types of Probability (Continued)

Subjective Probability

 (^) based on personal beliefs, experiences, prejudices, intuition - personal judgment  (^) different for all observers (subjective)  (^) examples: Super Bowl, elections, new product introduction, snowfall

Set - a collection of elements or objects of interest

 (^) Empty set (denoted by ) 

a set containing no elements

 (^) Universal set (denoted by S) 

a set containing all possible elements

 (^) Complement (Not). The complement of A is 

a set containing all elements of S not in A

 A  2-2 2-2 Basic DefinitionsBasic Definitions

Intersection (And)

• a set containing all elements in both A and B

Union (Or)

• a set containing all elements in A or B or both  A^  B   A^  B   A^  B   A^  B  Basic Definitions (Continued) Basic Definitions (Continued)

A A  BB Sets: A Intersecting with B Sets: A Intersecting with B

A

B

S

• Mutually exclusive or disjoint sets
  • sets having no elements in common, having no intersection, whose intersection is the empty set
• Partition
  • a collection of mutually exclusive sets which together include all possible elements, whose union is the universal set Basic Definitions (Continued) Basic Definitions (Continued)

Mutually Exclusive or Disjoint Sets Mutually Exclusive or Disjoint Sets

A

B

S

Sets have nothing in common

• Process that leads to one of several possible outcomes *, e.g.:

 Coin toss

• (^) Heads, Tails

 Throw die

• (^) 1, 2, 3, 4, 5, 6

 Pick a card

 AH, KH, QH, ...

 Introduce a new product

• Each trial of an experiment has a single observed outcome.
• (^) The precise outcome of a random experiment is unknown before a trial.
  • Also called a basic outcome, elementary event, or simple event

• Also called a basic outcome, elementary event, or simple event Experiment Experiment

Sample Space or Event Set

 (^) Set of all possible outcomes (universal set) for a given experiment 

E.g.: Roll a regular six-sided die

 (^) S = {1,2,3,4,5,6} 

Event

 (^) Collection of outcomes having a common characteristic

 E.g.: Even number

 (^) A = {2,4,6}  (^) Event A occurs if an outcome in the set A occurs 

Probability of an event

 (^) Sum of the probabilities of the outcomes of which it consists 

P(A) = P(2) + P(4) + P(6)

Events : Definition Events : Definition

Pick a Card: Sample Space Pick a Card: Sample Space

Union of Event ‘Ace’

Events ‘Heart’

and ‘Ace’

Event ‘Heart’

The intersection of the

events ‘Heart’ and ‘Ace’

comprises the single point

circled twice: the ace of hearts

P Heart Ace n Heart Ace n S

P Heart n Heart n S

P Ace n Ace n S

P Heart Ace n Heart Ace n S

Hearts Diamonds Clubs Spades

A A A A

K K K K

Q Q Q Q

J J J J

Range of Values for P(A):

Complements - Probability of not A

Intersection - Probability of both A and B

 (^) Mutually exclusive events (A and C) :

Range of Values for P(A):

Complements - Probability of not A

Intersection - Probability of both A and B

 (^) Mutually exclusive events (A and C) : 0  P ( A ) 1 P ( A )  1  P ( A ) P A B n A B n S ( ) ( ) ( )    P ( A  C ) 0 2-3 Basic Rules for Probability 2-3 Basic Rules for Probability