It has been a long while since I solved any Project Euler problem. For some reason I read an article about it and someone references Problem 267, so I decided to take a look at it, and it sucked me in. The problem reads

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

fto 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.