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https://www.youtube.com/watch?v=zTsRGQj6VT4
Start with 1,000. If you bet everything, you have a 50% chance of winning.
But using this method
Starting at 5, how many bets before we become bankrupt (5, 10, 20, 40, 80, 160, 320, this adds up to 635 with 365 remaining coins) lets round up to 8. Whats the chance we lose 8 times
1/2 * 1/2 ... (8 times) = 0.39%
And we need to win 200 times to double our initial investment, so whats the chance we win 200 times in a row without going bankrupt once?
Chance of us winning is 1 - 1/256 = 255/256 = 99.61%
Winning 200 times in a row is 255/256 * 255/256 ... (200 times) = 45.7%
In reality, since we rounded up to 8 earlier, our odds are worse, if you take this to the extreme, starting at 1,048,575 and an initial bet of 1, your odds are:
20 bets, 1/1048576 chance to bankrupt, 99.9999046% to win once, means 36.788% chance to double your money