But we're not talking about the deduction in the article, but to the deduction I gave above for the iterated experiment! (I thought I was being quite clear about that.) I'll break my argument down for you in detail is as follows. A is an arbitrary constant. n is large enough that we can use the law of large numbers. 
 # For every n times we are given an envelope, ½n times we have the smaller envelope and ½n times we have the larger envelope (because we choose at random). 
 # If A is the larger amount the other envelope contains ½A. 
 # If A is the smaller amount the other envelope contains 2A. 
 # So for every n times we find A in our envelope, ½n times there is 2A in the other envelope and ½n times there is ½A. 
 # So for every n times we find A in our envelope, if we keep our envelope we will have nA (on those occasions on which we found A).  
 # But if we switch each time, we have 1¼An (on those occasions). 
 # So we do better if we swap whenever we find A in the envelope. 
 So once again, is this argument correct? If not, where is the mistake and what is it?