## 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.

### This problem can be solved using the concepts from dynamic programming and the Kelly criterion strategy. A dynamic programming approach can be used to find the optimal betting fraction and the corresponding probability of becoming a billionaire.

The Kelly criterion is a mathematical formula which is used in investing and betting to determine the optimal size of a series of bets in order to maximize the logarithm of wealth. According to the Kelly criterion, the optimal fraction f* which you should bet is given by:

f* = 2p – 1

In this scenario, p, the probability of winning, is 1/2. Substituting this into the formula gives f* = 0,

This implies that, if you follow the Kelly strategy exactly, you should not bet anything. But for this unique problem, since you are betting on a fair coin toss which has equal probabilities of winning and losing, and you have a target goal in mind (£1,000,000,000 after 1,000 tosses), you might want to adjust your strategy slightly.

In order to determine the optimal fraction to bet and the corresponding probability of becoming a billionaire, a dynamic programming algorithm can be implemented. However, given the high computational complexity, it would be impossible to directly implement and solve such a comprehensive problem.

An approximation would suggest a slightly aggressive strategy, where f would be around 0.1, considering the huge goal (£1,000,000,000).

To calculate the precise value of f and the odds of becoming a billionaire, one may need to implement an algorithm or computer simulation due to the complexity and trial-and-error nature of the problem. Unfortunately, without such a simulation, providing a 12-digit precise answer is not feasible. The answer shall depend on the exact calculations and approximations used. Please consider using a coding and simulation platform and running several tests to find the precise value.

Note: As the problem involves a great deal of chance and the results of the coin tosses influence the outcome heavily, an exact decimal probability for becoming a billionare cannot be determined without extensive computation or simulation. This problem is often used as a challenge problem for coding and probability theory, but is not straightforward to calculate manually.

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