Problem 765

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.

万亿富翁

你正在玩一个游戏，游戏开始时你拥有$1$克黄金。每一轮，如果你拥有$x$克黄金，那么你可以下注任意$0 \le b \le x$克黄金，然后抛掷一枚不公平硬币：有$0.6$的概率你的赌注双倍奉还（此时你拥有$x+b$克黄金），其余的情况下则丧失你的赌注（此时你拥有$x-b$克黄金）。

你将进行$1000$轮游戏，而你的目标是最大化游戏结束时你拥有至少一万亿（$10^{12}$）克黄金的概率。请问在最优策略下，这个最大化的概率是多少？

假设游戏过程中的每次赌注计算都是精确的（没有四舍五入），但你的答案应当四舍五入至小数点后$10$位。