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  #31  
Old 12-29-2017, 06:19 PM
stefanos stefanos is offline
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Originally Posted by Z3ta View Post
Yes just apply the following:



This, in a sense, is true One can decompose A_1 in countably many pieces, move them around, and then reassemble them as A_2, regardless of their areas.

(A special case of the Banach-Tarski paradox)
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  #32  
Old 12-29-2017, 07:27 PM
Academic Actuary Academic Actuary is online now
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Originally Posted by Tsin View Post
I found other solutions and they are waaay more complicated. Seems like this problem is super hard.

http://mathforum.org/library/drmath/view/56637.html
I did try the formula given and got .85975 so it seems quite accurate for a large n.
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  #33  
Old 12-29-2017, 07:41 PM
Z3ta Z3ta is offline
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Originally Posted by stefanos View Post
This, in a sense, is true One can decompose A_1 in countably many pieces, move them around, and then reassemble them as A_2, regardless of their areas.

(A special case of the Banach-Tarski paradox)
Edit: I should say it is not true because it requires going outside of the sigma algebra the probability measure is defined on... so it is not true in any sense in our framework.

Last edited by Z3ta; 12-29-2017 at 07:56 PM..
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  #34  
Old 12-29-2017, 08:25 PM
stefanos stefanos is offline
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Originally Posted by Z3ta View Post
Edit: I should say it is not true because it requires going outside of the sigma algebra the probability measure is defined on... so it is not true in any sense in our framework.
If it wasn't clear before: my intention was to alter the framework by using paradoxical (non-measurable) decompositions, which represent very legit math, but half-jokingly in this context.
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  #35  
Old 12-29-2017, 08:37 PM
Z3ta Z3ta is offline
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Originally Posted by stefanos View Post
If it wasn't clear before: my intention was to alter the framework by using paradoxical (non-measurable) decompositions, which represent very legit math, but half-jokingly in this context.
You mean it’s weird to respond seriously to a post with a wink emoji that was a response to something I wrote in jest? You may have a point
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