Edit: It affects my chance of being one of the winners, not my actual chance of winning, though the distinction is slight. If my chance of being one of the winners increases so does my chances of winning directly.
I think your math is correct, right up to the part where you try to express it in English. It seems you agree with my the odds of the ticket winning do not change, no matter how many other people enter. That's correct, and there's no disagreement there. But then you leave math, and extrapolate colloquially, and make a statement that is blatantly false. Since my chance of winning never changes, my chance of being one of the winners also does not change. Those two are mathematically distinct things, but you've calculated one and then simply asserted the two are the same. Try calculating directly. What are the odds not of winning, but of being one of the winners, if I am the only ticket entered. Now calculate directly, not indirectly, what are the odds of being one of the winners, if a million other people enter.
The odds of a winner existing at all increases with more entries, but the odds I am one of them do not. I'm beginning to wonder if you're making the same error I made earlier in the thread. When I first calculated the odd of winning, I made the incorrect assumption that if there was no winner, all previous entries were still alive: that you could enter once, and if there was no winner you still had a chance to win until there was a winner. That's not true for powerball. If there is no winner, all previous entries lose and in effect there is a brand new drawing held *only* with the tickets purchased since the last drawing, with the prize rolling over into the new lottery. That's the only explanation I have for why you would think the number of entries affects the odds of winning for a particular person. If I'm the *only* entry and tickets actually rolled over with the prize, then my odds of winning are 100% - because there has to be a winner eventually and I'm the only entry. It'll take about 40,000 years for me to win, but I will eventually win. For more entries to *increase* the chances for me to be one of the winners requires a different set of lottery rules that don't match Powerball.