So, I'll kind of second the observation in the comment above. It seems to me that, from the fact that reading the same story in the Washington Post does not make your epistemic situation better, it does not seem to follow that the Post story is not evidence that Bill win the lottery. That is: from the fact that a certain piece of evidence is swamped by another piece of evidence in a certain situation, it does not follow that the former is not evidence. We can see that it is evidence just following your steps: we conceive another situation where I didn't read the Times story but I read the Post story - and it is evidence that Bill win the lottery in this situation.

I agree that it seems just wrong to grant that strong evidence and weak evidence is determined by the access we have to evidence in order of time. But from the fact that one does not gain more justification to believe h by learning e it does not follow that e is evidence that h, all things considered.