Trillionaire

Starting with 1 gram of gold you play a game. Each round you bet a certain amount of your gold: if you have $x$ grams you can bet $b$ grams for any $0 \le b \le x$. You then toss an unfair coin: with a probability of $0.6$ you double your bet (so you now have $x+b$), otherwise you lose your bet (so you now have $x-b$).

Choosing your bets to maximize your probability of having at least a trillion $(10^{12})$ grams of gold after $1000$ rounds, what is the probability that you become a trillionaire?

All computations are assumed to be exact (no rounding), but give your answer rounded to 10 digits behind the decimal point.

This problem is a variation of the Kelly Criterion problem in probability theory. The Kelly Criterion is a mathematical formulation used to find the optimal size of a series of bets. In principle, if we want to maximize the probability of reaching a certain wealth ($10^{12}$ grams) within a certain period (1000 rounds), we want to adapt our betting each round based on our current state.

Let $p(i,j)$ denote the probability of reaching at least $10^{12}$ grams of gold after $j$ rounds when we currently have $i$ grams of gold. Also, if we start with less than $10^{12}$, we would not be able to reach our goal within 0 rounds, so initially we have: $p(i,0) = 0$ for all $i<10^{12}$ and $p(i,0) = 1$ for all $i \geq 10^{12}$. For any $i$, $j$, we can look at the bet size $b$ which maximizes our expected value of $p(i,j)$ to update our probability from the outcomes of the next rounds: $p(x,j) = \max_{0\le b \le x} 0.6p(x+b, j-1) + 0.4p(x-b, j-1)$ However, as this is still an extensive computation to do manually or even by normal methods due to the large state space, typically simulation or approximation methods are used for estimating the result. Therefore, without using a simulation or computation, it's impossible to provide a numerical answer here. However, it should be pointed out that the answer would be computed by utilizing dynamic programming and deriving from the principle of Kelly Criterion as outlined above. If you can provide the use of a computer, approximation, or simulation method, a more specific numerical answer can be given. So in fact, probability must be calculated dynamically for each decision-tree branch (game round) and then calculated backwards to the root of the tree (start of the game) to get the exact answer with 10 digits behind the decimal point.

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