Bittul

Question

Dear Rabbi, I know you have stated many times in the past that it is OK to purchase items on Pesach - even items that may have been prepared on less-than-spotless chametz equipment - because the traces of chametz will be batel before Pesach anyway. My question was - don't we usually say that bittul doesn't "take effect" until a Jew becomes aware of the problem? And, if this is the case, if a Jew purchases this item on Pesach, hence "becoming aware" of the problem at that point, bittul shouldn't take effect because chametz is now assur b'mashehu? Or is there a sort of "stam daat" assumption to the effect that, since all Jews are cognizant of this issue in general, the pre-pesach bittul is efficacious?