Puzzle/Paradox..

Easy question for everyone here I am sure.

I saw this somewhere and I have a question, but it is different from the one asked in the puzzle I saw.

Box A
$100 + $100

Box B
$100 + $1

Ok. So you are presented with two identical boxes and told that you can reach in and grab one of the two bills from a box. You are told that in Box A there are two $100 bills and in Box B there is a $100 bill and a $1 bill. You are not told which is Box A or B.

You are going to get two turns to pick, so we won’t even go through the scenario where you pick the $1 bill first, as no matter which box you pick from you will get a $100 bill with your second pick.

But, how about if you pick a $100 bill on your first pick. Is there a strategy for picking a box that will give you a greater chance of getting another $100?

-If you picked from box A there is a 100 % chance of getting another $100 if you pick box A again.
-If you picked from box B there is a 0% chance to get $100 if you pick from box B again.

That said, the very fact that you pulled a $100 bill on your first pick makes it twice as likely that you picked box A, and thus you should pick from that same box again. Or does it?

Should you pick from the same box again? It seems like a yes to me.