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EMSE 208 Lecture 1 - Introduction to Probability Theory

# EMSE 208 Lecture 1 - Introduction to Probability Theory - 1...

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1 Lecture 1: Introduction to Probability Theory Professor Thomas A. Mazzuchi

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2 Overview of Probability Terminology We Model The Outcomes of Random Experiments Outcome are unknown in advance List of possible outcomes is known Examples Flipping a coin Flipping a coin until a heads appears Rolling two die Selecting numbers in a lottery Observing the failure time for a group of components Observing the price of a stock at a given time Observing the integrity of a nuclear reactor at a given time Observing the weather tomorrow
3 Overview of Probability Terminology Consider a Nonempty Set, S , of Outcomes from a random Experiment, called the Sample Space and interpreted as the set of possible outcomes of a repeatable experiment (as in outcomes of a die roll) S = {1,2,3,4,5,6} Or observing the failure time of a component S = [0, ) the set of possible worlds or states of nature (as in the state of weather tomorrow) S = {Sunny, Cloudy, Rainy, Snowy}

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4 Examples Note The sample space for a given experiment is not necessarily unique. For example, throwing two coins can be capture by these two sample spaces: S 1 = {HH,HT,TH,TT}, S 2 = {0H, 1H, 2H} For the two coin experiment, the event A= “a head showing in coin 1”, is described by the subset A = {HH,HT} of S 1 . Notice that the second representation is not suitable to represent this event.
5 Set A which is a subset of S is called an event As Any Event is a Set, It Can Be Expressed by Roster or Rule S = {12,3,4,5,6} E = {1}, E = {2,4,6}, E is an even integer less than 4 Overview of Probability Terminology

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6 As events are subsets, they are subject to operations of union, intersection, and complementation, i.e. if A and B Overview of Probability Terminology
7 Visualizing Set Operations: Venn Diagrams Α Β C Α Β Α Β C Α Β A B A B (A B) C (A B) C c

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8 Basic Axioms ( 29 1 1 i i i i P A P A = = = U Probability is used to indicate the likeliness of an event occurring – the higher the probability the more likely the event Kolmogorov Axioms: Consider a Sample Space S, For Any Event A of the Sample Space 0 P(A) 1 P(S) = 1 for all A i which are events of S such that , A i A j = , i j where ... 2 1 1 = = A A A i i i These axioms can be used to prove all of the laws of probability
9 Using Basic Axioms ( 29 ( 29 ( 29 ( 29 1 1 1 1 1 but since 0 1 0 1 then 0 i i i i i i i i P A P A P P A P P = = = = = = = ∅ ≤ ∅ = U U Prove: P( ) = 0 Define A i = , i=1,…, (note that A i A j = for i j then

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10 Using Basic Axioms Prove: Define A 1 , …, A n such that A i A j = for i j and A k = for k=n+1, …, ( 29 1 1 n n i i i i P A P A = = = U ( 29 ( 29 ( 29 ( 29 1 1 1 1 1 1 n i i i i n i i i i i n n i i P A P A P A P A P P A = = = = = + = = = = + = U U
11 Visualizing Probability

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EMSE 208 Lecture 1 - Introduction to Probability Theory - 1...

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