Chapter2 - Chapter 2 lecture notes Math 431, Spring 2011...

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Chapter 2 lecture notes Math 431, Spring 2011 Instructor: David F. Anderson Chapter 2: Axioms of Probability Section 2.2: Sample Space and Events We consider an experiment whose outcome is not predictable with certainty. However, we suppose that the set of all possible outcomes is known. Definition 1. The set of all possible outcomes of an experiment is the sample space of an experiment, and is denoted by S . Subsets of S are called events . Elements of S , x S , are called outcomes . In this course, we will care about the probability of both events and specific outcomes. That is, P ( E ) and P ( { x } ) = P ( x ). Example 1. A coin is tossed twice and the outcome of each is recorded. Then, S = { ( H,H ) , ( H,T ) , ( T,H ) , ( T,T ) } . The event that the second toss was a Head is the subset E = { ( H,H ) , ( T,H ) } . Example 2. Consider 3 light-bulbs. Our experiment consists of finding out which light-bulb burns out first, and how long (in hours) it takes for this to happen. S = { ( i,t ) : i ∈ { 1 , 2 , 3 } ,t 0 } , i tells you which one burns out, and t gives how long it lasted, in hours. The event that the 2nd bulb burns out first, and it lasts less than 3 hours is the set E = { (2 ,t ) : t < 3 } . Example 3. You roll a four sided die until a 4 comes up. The event you are interested in is getting a three with the first two rolls. S = { ( a 1 ,a 2 , ··· ,a n ) | n 1 ,a n = 4 ,a i ∈ { 1 , 2 , 3 } for i 6 = n } E = { (3 , 3 ,a 3 ,a 4 , ··· ,a n ) | n 3 ,a n = 4 ,a i ∈ { 1 , 2 , 3 } for i 6 = n } 1
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We care about the probability of events, which are sets. We want to know how to manipulate sets so that we can say things like P ( E or F ) or P ( E and F ). Notation: We write x E to denote that “ x is in E ”. Definition 2. For two sets E,F S , the union of E and F , denoted E F , consists of those outcomes that are in either E or F (or both). Formal definition: E F = { x S : x E or x F } . Example: If E = { ( H,H ) , ( H,T ) } (Head on first toss) and F = { ( T,T ) , ( H,T ) } (tail on the second toss). Then, E F = { ( H,H ) , ( H,T ) , ( T,T ) } , only missing ( T,H ). Definition 3. The intersection of E,F S , denoted EF or E F , consists of those events that are in both E and F . Formal def: E F = { x S : x E and x F } . Previous Example: If E = { ( H,H ) , ( H,T ) } (Head on first toss) and F = { ( T,T ) , ( H,T ) } (tail on the second toss). Then, EF = { ( H,T ) } , The empty set , {} = is the set consisting of nothing. Definition 4. Two sets are mutually exclusive if EF = . A set of sets, { E 1 ,E 2 ,..., } are mutually exclusive if E i E j = for all i 6 = j .
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Chapter2 - Chapter 2 lecture notes Math 431, Spring 2011...

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