Gambling device: What's my probability to win at 5 dollars before going bankrupt?

Just a setup.

So definitely not a full answer, but too much for a comment.

For $n\in\mathbb{Z}$ let $p_{n}$ denote the probability that the game ends in winning if your credit is $n$.

Then $p_{n}=0$ if $n\leq0$ and $p_{n}=1$ if $n\geq15$.

For $n\in\left\{ 1,\dots,14\right\} $ we have the relation:

$$p_{n}=\frac{1}{5}p_{n+3}+\frac{4}{5}p_{n-1}$$

You are looking for $p_{10}$.