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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