So the martingale has a better probability for turning 1000 into 1010, but how does the math change if you were to try to go from 1000 to 2000? I wonder if the martingale would still have a higher chance of winning than a single bet?
Yes. No matter what you're trying to do, two bets are better than one. I posted previously about how to turn 1 BTC into 2 BTC, and have modified the quote here, multiplying everything by 1000:
If you want to double your money, you have a higher chance if you place multiple bets than if you place a single bet. You have to pick the right multiple bets of course.
It's really quite easy to demonstrate:
If you place a single bet, then the chance of doubling your money is 49.5%.
Now consider this 2 bet sequence:
1. bet 414.21356 BTC with payout 3.41421356x and chance 28.99642866% to win 1414.21356 BTC for a profit of 1000 BTC
2. if you lose, bet 585.78644 BTC at the same payout and chance to win 2000 BTC for a net profit of 1000 BTC.
Your overall chance of success is 49.58492857%. That is higher than 49.5%.
(Note that those bets aren't exactly available on Just-Dice, since chance is only available to 4 significant figures, but that's just a nit-pick. It's still possible to double your money with a higher than 49.5% chance using 2 bets, and not using 1 bet).
In both cases you're going to run into the 'max profit' limit on Just-Dice, but we can ignore that for the purposes of this discussion.
If you're wondering where I got the numbers from, here's the calculation. "have" is how much we start with, and "gain" is how much we want to win. We calculate "stake", which is how much to bet on the first bet (we bet the rest on the 2nd bet if the 1st bet loses) and "payout" which is the payout multiplier for both bets:
>>> have = 1000
>>> gain = 1000
>>> stake = math.sqrt(gain*(gain+have)) - gain
>>> payout = (gain + stake) / stake
>>> stake
414.2135623730951
>>> payout
3.414213562373095