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elemprob-fall2010-page1

# elemprob-fall2010-page1 - A is 1 36 times the number of...

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Math 3160, Lecture 3 1 Framework We start with a sample space Ω, a class of events, and a probability. The sample space can be any set whatsoever. An event is a subset of the sample space. Occasionally we restrict the collection of subsets to what is called a σ -field, but most often we allow any subset of Ω to be an event. To each event A we assign a number P ( A ), the probability of A , which is a number between 0 and 1. As an example, suppose we roll a die. (“Die” is the singular form of the plural “dice.”) We set Ω = { 1 , 2 , 3 , 4 , 5 , 6 } and the set of events is the collection of all subsets of Ω. We assume the die is fair, and we set P ( A ) to be equal to the number of elements in A divided by 6. Thus P ( { 1 , 3 , 5 } ) = 3 6 = 1 2 . For another example, suppose we roll two dice, one green and one red. Then we take Ω to be all pairs (1 , 1) , (1 , 2) , . . . , (6 , 6) and the probability of
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Unformatted text preview: A is 1 / 36 times the number of elements in A . In both these cases, we assume that each number on each die is equally likely. We might have a “loaded” die, where P ( { 1 } ) = . 10, P ( { 2 } ) = . 15, etc. We do want something to happen, so P (Ω) = 1 and P ( ∅ ) = 0. The collection of events have to be what is called a σ-ﬁeld. A collection F subsets of Ω is a σ-ﬁeld if Ω , ∅ ∈ F , A c ∈ F whenever A ∈ F , and ∪ ∞ i =1 A i and ∩ ∞ i =1 A i are in F whenever all the A i are in F . A probability is a function on the events such that (1) 0 ≤ P ( A ) ≤ 1 for each event A . (2) P (Ω) = 1 and P ( ∅ ) = 0. (3) If A and B are disjoint, which means A ∩ B = ∅ , then P ( A ∪ B ) = P ( A ) + P ( B ) . 1...
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