Problem 267

Billionaire

You are given a unique investment opportunity.

Starting with £1 of capital, you can choose a fixed proportion, f, of your capital to bet on a fair coin toss repeatedly for 1000 tosses.

Your return is double your bet for heads and you lose your bet for tails.

For example, if f = 1/4, for the first toss you bet £0.25, and if heads comes up you win £0.5 and so then have £1.5. You then bet £0.375 and if the second toss is tails, you have £1.125.

Choosing f to maximize your chances of having at least £1,000,000,000 after 1,000 flips, what is the chance that you become a billionaire?

All computations are assumed to be exact (no rounding), but give your answer rounded to 12 digits behind the decimal point in the form 0.abcdefghijkl.

百万富翁

现在你面前有一个仅此一次的投资机会。

一开始，你有£1的资本。你可以选择一个固定的比例f，每一轮投入这个比例的资本，并抛掷一枚标准硬币，这样的过程重复一千轮。

如果硬币是正面朝上，你的回报是投入资本的两倍，如果是反面朝上则没有回报。

例如，如果选择f = 1/4，第一轮投入的资本是£0.25，假定正面朝上，那么你获得的回报是£0.5，因此总资本变成£1.5。下一轮你投入的资本是£0.375，假定反面朝上，那么你现在的总资本是£1.125。

经过1,000轮后你有一定几率拥有至少£1,000,000,000的资本。选择最大化这个几率的f时，你有多大的几率成为百万富翁？

计算过程中必须是精确值（不四舍五入），但将你的答案四舍五入到小数点后12位小数，即形式为0.abcdefghijkl。