Difference between Power Sets and Sample Spaceby thrillhouse86 Tags: power set, probability space, sample space 

#1
Jul2710, 11:26 PM

P: 80

Hey All,
In my probability theory class we have just started learning about how a probability space is defined by a sample space (which contains all possible events), events and a measure. We briefly went over the idea of the Power Set, which seems to be the set of all subsets in your sample space. My question is what is the difference between the Power Set and the Sample Space ? aren't they both just a collection of all possible outcomes ? Thanks 



#2
Jul2810, 12:51 AM

P: 42

The powerset is the sample space of the experiment:
'Of all subsets of X, what is the chance that this subset is one specific subset Y. provided an even distribution' Well, then your chance happens to be 1/P(X).. This is of course nothing special. This is just because the powerset is the defined as the set of all subsets. If you ask, 'what is the chance that you have one specific subset which meets property Q' Then your chance is also simply 1/{x sub X : Q(x)} ... The powerset can be defined with the identity property (the property which is always true), which is x = x basically. So we get P(x) = {x sub X : x = x}. Edit: not sure you know 'set builder notation', but if Phi(x) and Theta(x) are formulae in x, such as x > 3, or x ^ 2 = 3 et cetera. See {Phi(x) : Theta(x)} as the set of all x for which both formulae are true. 



#3
Jul2810, 04:24 PM

Sci Advisor
P: 5,939

Simple example:
Sample space {1,2,3} Power set {(),(1),(2),(3),(1,2),(1,3),(2,3),(1,2,3)} 



#4
Jul2810, 04:29 PM

P: 42

Difference between Power Sets and Sample SpaceIf A is a sample space, then B is the powerset of a sample space. A sample space is with respect to some experiment. 



#5
Jul2910, 04:28 PM

Sci Advisor
P: 5,939





#6
Jul2910, 05:12 PM

P: 42

You just gave a sample space, and the powerset of that space as an illustration that powerspaces and sample sets are somehow different concepts, which they are, but not via this logic, because a powerset is of course always different to its set. 



#7
Jul3010, 06:39 AM

P: 608

Perhaps he is talking about the set of all events, and asking why that is not just the power set ...




#8
Aug110, 04:35 AM

P: 534




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