As an independent event, yes. Otherwise, no. On your principal the probability of a roulette wheel being red for 1,000 goes is the same as it being 500 black and 500 red.
Spins of a roulette wheel are independent of each other and the probability of getting red 1000 times in a row is the same as getting black 500 times in a row and then red 500 times in a row. If you are not considering that particular order then the probability would not be the same but that's because of the amount of combinations, not any dependency between events.