Lets say I have 750 dollars and want at least 1250 dollars at the end of a 50/50 game where I can bet any possible value. Is there any way in which I can raise my chances of winning? And if there is, how can I define the best initial bet to be doubled?
(At first I thought problems like these were easy to solve and there was no possible way to "win", but at the same time I have some doubts about it and don't know the mathematical explanation for it.)
On
Your optimal probability of getting to \$1250 from \$750 with fair bets is 60% and you cannot do better. One strategy of doing so is given by @Robert's answer above, called "bold play" in the literature. If you follow his strategy, you find yourself along the following tree:
You take each path of this betting tree with probability $1/2$. If your initial probability of winning from \$750 is x, then one of three things happen on your way to victory:
So we have the equation $$x = 0.5 + 0.0625 + 0.0625x,$$ and when solved, we learn that $x = 9/16 + x/16$, or $15x/16 = 9/16$, whence $x = 9/15 = 0.6$. (The other $40\%$ probability is that you go broke.)
Why can't we do better than $60\%$? Consider your fortune $W_n$ on the $n$th bet, and imagine you make the bet $A_n$ subject to the constraints $A_n = 0$ if $W_n \geq 1250$ or $W_n = 0$, $0 \leq A_n \leq W_n$. Then $W_n$ is a uniformly integrable martingale, because your winning probability on each bet is $1/2$ and $$0 \leq W_n \leq 2*(1250-1) = 2498$$ is a bounded sequence above and below. This means that $W_n$ converges almost surely and in $L^1$ to a bounded random variable $W_\infty$, representing your limiting winnings. The $L^1$ convergence is crucial here, as it means $$\Bbb{E}[W_\infty] = 750.$$
You have "won" your betting sequence if $W_\infty \geq 1250$. What is the probability of that? By Markov's inequality, $$\Bbb{P}(W_\infty \geq 1250) \leq \frac{\Bbb{E}[W_\infty]}{1250} \leq \frac{750}{1250} = 60\%.$$
(If you started with $x$ dollars instead, the same argument shows you have an $x/1250$ chance of getting to \$1250, one way to do this being bold play.)
So bold play actually gives you the "best possible" chance of reaching \$1250--although it's also "worst possible" among strategies that achieve $60\%$, as the other $40\%$ of your time you lose all of your initial \$750.
Other betting strategies than bold play may leave you with a better probability of a nonzero limiting fortune $W_\infty$ (one example: bet $1/(n+1)$ of your current wealth $W_n$ on bet number $n$ or $1250 - W_n$, whichever is smaller). But no strategy, however clever or intricate, can do better than a $60\%$ chance of reaching \$1250 from an initial stake of \$750.
By "50/50 game" I assume you mean you can make any sequence of fair bets at $50/50$ odds, as long as you never bet more than you have at the time. Since these are fair bets, your expected value at the end is the same as your initial fortune, $750$. As long as you ensure that the only possible final outcomes are $0$ and $1250$, you maximize the probability of ending with $1250$. Thus one possible strategy is: whenever you have $x$ where $0 < x < 1250$, bet the minimum of $x$ and $1250 - x$. With probability $1$, you will eventually end up at either $1250$ or $0$.