The problem with that reasoning is that it still assumes past hashing somehow has an affect on the future. And that an average has anything to do with interval (there's a reason standard deviation exists, you might want to look it up before you keep throwing around the word "statistics" with any air of authority)

Do you believe that in any given second your odds of getting a share are better or worse than any other second? If not, why would it matter when you switch?

Now, someone answer me this time. If I'm throwing dice in a box and I get a point for every six I throw, does it matter if someone keeps swapping in different boxes? if not, how does this differ from hashing?