yaworsw/euler-manager

---
:id: 267
:name: Billionaire
:url: https://projecteuler.net/problem=267
:content: |+
  You are given a unique investment opportunity.

  Starting with £1 of capital, you can choose a fixed proportion, <var>f</var>, 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 <var>f</var> = 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 <var>f</var> 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.