Gambling on a dicing site with a 1% house edge gives you a 49.5% chance to turn your bankroll to 200% of your original bankroll, while giving you a 50.5% chance to turn your bankroll into 0% of your original bankroll - assuming you only stop when you reach 200% or 0% of your original bankroll.
That's not true - your chance of doubling your bankroll depends a lot on your strategy.
For an example of a very bad strategy, suppose you go all-in at 98% over and over until you lose, or double your bankroll:
each time you win, you multiply your bankroll by 99/98 = 1.0102040816
so to double your bankroll, you need to win math.log(2, 99.0/98) = 68.27441 times
the chance of winning 68 times in a row is (98.0/100) ** 68 = 0.25314 = 25.314% - much lower than 49.5%
For an example of a better strategy, try:
bet 0.5 at 33%
if you win, you get 99/33 = 3x, so you get 1.5 back, a profit of 1 - you've doubled your money
if you lose, bet the other 0.5 at 24.75%
if you win, you get 99/24.75 = 4x, so you get 2 back - you've doubled your money
the only way it goes wrong is if you lose both the 33% and the 24.75% - the chance of that is (1-0.33) * (1-0.2475) = 0.504175
so the chance of success is 1 - 0.504175 = 0.495825 = 49.5825% - a fraction higher than 49.5
So it's possible to find strategies that are better or worse than 49.5% chance of doubling up.
I also seen something about a single martingale sequence being better than going all in on a certain % How does this exactly happened, if your chances are 49.5% how are you actually able to improve them, slightly. Is the math on casinos not perfect when calculating odds and results?
