For a giveaway with 200 entries (i.e. you have a 1/200 chance of winning), you have to enter 138 giveaways to have a 50% chance of winning a game.

With a 1/500 chance, you have to enter 346 giveaways for a 50% chance of winning.

With a 1/1000 chance, you have to enter 693 giveaways for a 50% chance of winning.

With a 1/2000 chance, you have to enter 1386 giveaways for a 50% chance of winning.