Trillionaire
Starting with 1 gram of gold you play a game. Each round you bet a certain amount of your gold: if you have $x$ grams you can bet $b$ grams for any $0 \le b \le x$. You then toss an unfair coin: with a probability of $0.6$ you double your bet (so you now have $x+b$), otherwise you lose your bet (so you now have $x-b$).
Choosing your bets to maximize your probability of having at least a trillion $(10^{12})$ grams of gold after $1000$ rounds, what is the probability that you become a trillionaire?
All computations are assumed to be exact (no rounding), but give your answer rounded to 10 digits behind the decimal point.