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chap2-3slidespt1

# chap2-3slidespt1 - Random experiment Definition A physical...

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Random experiment Definition A physical process whose outcome is not certain but all of its possible outcomes are known in advance. Sample space S : Set of all possible outcomes. Event A is a collection of some outcomes. Example: Roll of a die S = { 1 , 2 , 3 , 4 , 5 , 6 } Event “Even number”: A = { 2 , 4 , 6 } ECSE 305 - Winter 2012 1

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Random experiment Random experiment: sample space S Outcome s S A happens A does not happen s A s 6∈ A P ( A ) : Certainty that A happens Probability ECSE 305 - Winter 2012 2
Axiomatic approach Simplified version P ( A ) : Function that assigns a probability to the event A . It must satisfy: 1 For any event A , P ( A ) 0 2 P ( S ) = 1 3 If events A and B cannot occur simultaneously, then P ( A or B ) = P ( A ) + P ( B ) ECSE 305 - Winter 2012 3

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Axiomatic approach Remember... P ( A ) assigns a number to a set of outcomes. Anything that we want to calculate the probability of has to be written as a set of outcomes. ECSE 305 - Winter 2012 4
Set theory Definition A set is an unordered collection of objects, called elements. Sets are specified by: Enumeration of their elements, e.g., { 1 , 2 , 3 , 4 , 5 } Description of their elements, e.g., { a : a is a positive integer 5 } ECSE 305 - Winter 2012 5

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Set theory Notation To indicate that an object a is an element of a set A : a A If a is not an element of A : a 6∈ A ECSE 305 - Winter 2012 6
Set theory Equality Definition Two sets A and B are identical (or equal) if and only if (iff) they have the same elements. We write A = B In other words: A = B if and only if s A s B If A and B are not identical, we write A 6 = B . ECSE 305 - Winter 2012 7

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Set theory Subsets Definition A is a subset of B if every element of A is also an element of B . We write A B If A is not a subset of B , we write A 6⊆ B ECSE 305 - Winter 2012 8
Set theory Proper subsets Definition A is a proper subset of B when 1 A is a subset of B 2 but there exists b such that b B and b 6∈ A . We write A B If A is not a proper subset of B , we write A 6⊂ B ECSE 305 - Winter 2012 9

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Set theory Special sets Sample space S : In practical applications of set theory, all sets of interest in a given situation are usually subsets of a larger set called sample space, or universal set.
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chap2-3slidespt1 - Random experiment Definition A physical...

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