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Probability-Basic Concepts and Approaches

Dr. Jerrell T. Stracener, SAE Fellow

EMIS 7370 STAT 5340

Probability and Statistics for Scientists and Engineers

Leadership in Engineering

Department of Engineering Management, Information and Systems

SMU BOBBY B. LYLESCHOOL OF ENGINEERING

EMIS - SYSTEMS ENGINEERING PROGRAM

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Probability-Basic Concepts and Approaches

•Basic Terminology & Notation•Basic Concepts•Approaches to Probability

oAxiomaticoClassical (A Priori)o Frequency or Empirical (A Posteriori)o Subjective

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• Definition – Experiment

Any well-defined action. It is any action or process that generates observations.

• Definition - Outcome

The result of performing an experiment

Basic Terminology

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• Definition - Sample SpaceThe set of all possible outcomes of a statistical experiment is called the sample space and is represented by S. Remark: Each outcome in a sample space is called an element or a member of the sample space or simply a samplepoint.

Basic Terminology

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An experiment consists of tossing a fair coin three times in sequence.

How many outcomes are in the sample space?

List all of the outcomes in the sample space.

Example

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An biased coin (likelihood of a head is 0.75) is tossed three times in sequence.

How many outcomes are in the sample space?

List all of the outcomes in the sample space.

Example

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• Definition - Event An event is the set of outcomes of the sample space each having a given characteristic or attribute

• Remark: An event, A, is a subset of a sample space, S, i.e.,

A S.

Basic Terminology

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• Definition - Types of EventsIf an event is a set containing only one element or outcome of the sample space, then it is called a simple event. A compound event is one that can be expressed as the union of simple events.

• Definition - Null EventThe null event or empty space is a subset of the sample space that contains no elements. We denote the event by the symbol .

Basic Terminology

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Certain operations with events will result in the formation of newevents. These new events will be subsets of the same sample space as the given events.

• Definition - The intersection of two events A and B, denoted by the symbol A B, or by AB is the event containing all elements that are common to A and B.

• Definition - Two events A and B are mutually exclusive if A B = .

• Definition - The union of two events A and B, denoted by the symbol A B, is the event containing all the elements that belong to A or to B or to both.

Operations With Events

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• Definition - The complement of an event A with respect to S is the set of all elements of S that are not in A. We denote thecomplement of A by the symbol A´.

Results that follow from the above definitions:• A = 0.• A = A.• A A´ = • A A´ = S.• S´ = .• ´ = S.• (A´) ´ = A.

A

A´

S

Venn Diagram

Operations With Events

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For any event A in S, the probability of A occurring isa number between 0 and 1, inclusive, i.e.,

where

and

where Ø is the null event

1)A(P0

0)( P

1)S(P

Basic Concept

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(1) First, there is a question of what we mean when we say that a probability is 0.82, or 0.25. - What is probability?

(2) Then, there is the question of how to obtain numerical values of probabilities, i.e., how do we determine that a certain probability is 0.82, or 0.25. - How is probability determined?

(3) Finally, there is the question of how probabilities can be combined to obtain other probabilities. - What are the rules of probability?

Probability-Basic Questions

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• Axiomatic

• Classical (A Priori)

• Frequency or Empirical (A Posteriori)

• Subjective

Approaches to Probability

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Given a finite sample space S and an event A in S, we define P(A), the probability of A, to be a value of an additive set function P, which must satisfy the following three conditions:

AXIOM 1. P(A) 0 for any event A in S.

AXIOM 2.P(S) = 1

Axiomatic Approach

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AXIOM 3.If A1, A2 …, Ak is a finite collection of mutually exclusive events in S, then

k

1ii

i

k

1i

APAP

Axiomatic Approach

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If an experiment can result in n equally likely and mutually exclusive ways, and if nA of these outcomes have the characteristic A, then the probability of the occurrence of A, denoted by P(A) is defined to be the fraction

n

n)A(P A

Probability - Classical Approach

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If an experiment is repeated or conducted n times, and if a particular attribute A occurred fA times, then an estimate ofthe probability of the event A is defined as:

Note that

Remark: Probability can be interpreted as relative frequency in the long run.

n

f)A(P A

^

asn

fAP A ,)( n

Frequency of Empirical Approach

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An experiment consists of tossing a fair coin three times in sequence.

What is the probability that 2 heads will occur?

Example

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An biased coin (likelihood of a head is 0.75) is tossed three times in sequence.

What is the probability that 2 heads will occur?

Example

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Relative Frequency vs. n

n

fA

n = number of experiments performed

1 2 3 . . . n

0

1

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•DefinitionThe probability P(A) is a measure of the degree of belief

one holds in a specified proposition A.

Note: Under this interpretation, probability may be directly relatedto the betting odds one would wager on the stated proposition.

•OddsThe relative chances for the event A and the event that A

does not occur, i.e.,

odds in favor of A

)(

)(

AP

AP

Probability - Subjective Approach