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I may be misreading your post, but it seems like there is some confusion between these two things:

1. Winning the lottery *twice*
2. Winning the lottery *a second time*

All of the remarks about the probability being the same are relevant to the second situation. Your calculations, however, address the *first* situation.

Perhaps it will be easier to avoid confusion if we use specifics. Let’s say that this is a lottery that will only ever be run twice, namely next Monday and Tuesday. Let’s say that both draws have a thousand entrants—in fact, the same thousand—and that all of them have an equal chance of winning ($p=\frac{1}{1000}$) in each draw. And let’s say that you, Chgg Clou, win Monday’s lottery.

Now, situation 2 is this: What is the probability that you win Tuesday’s lottery? It’s $\frac{1}{1000}$. The fact that you won Monday’s lottery doesn’t affect that (barring any rules forbidding a double win, or using your winnings to bribe the operators, or…).

Situation 1, meanwhile, is this: What is the probability that you win *both Monday’s and Tuesday’s lotteries*? Before Monday, we would have simply said that you (and everyone else) had a $\frac{1}{1\,000\,000}$ chance of doing so. But now?

Intuition’s a funny thing, but to me it seems very intuitive that this *is* improved by the fact that you won on Monday. You’re the *only* person who can possibly do it! Everyone else’s chances dropped to zero the moment Monday’s result came up. Your chances got better, and theirs got worse.

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More broadly, you can compare it to a sport or other contest. (For the best analogy, think of one like volleyball where you need to be the first to reach a certain score, rather than simply ending the game with a higher score. But if we’re after intuitive acceptance, that may not matter.)

Does it make intuitive sense that your chances of winning are higher when you’re in the lead? Even assuming your chances of scoring *at any given point* are always the same (no psychological advantage to being in the lead, no pressure not to blow it), being in front makes you more likely to win.