Might be my definition/example of the gambler's fallacy is not accurate, cause otherwise I still don't understand how it's a fallacy:

Say I buy a lottery ticket with a win probability of 1/10^9 everyday for the next 1000 days and I lose on all 1000. I understand if I play on the 1001th day, my odds are still 1/10^9 for that ticket, but doesn't the probability of losing on every ticket I buy decrease? i.e. (9.999999999*10^8/10^9)^1000 < (9.999999999*10^8/10^9)^1001 ? Isn't this pretty straightforward? As I continue to buy tickets and continue to lose them, doesn't the probability of at least winning one continue to increase? (Practically speaking it'd be better to shotgun them all into one roll, but for the sake of my question.) How does this factor in to the 1/10^9 chance?