# Problem 267: Billionaire

![graphic](img267.gif)

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.