The following is an example problem very common in a Computational Statistics course I have. I'm asked to comment the result of the following experiment:
A person has certain amount of money $C$ and participates in the following game: in each time unit, with probability $p$ wins a unit, with probability $q$, looses a unit and with probability $r$ remains equal. We assume that $p+q+r = 1$ and $C \ge 1$. The game continues until the player's ruin ($C \le 0$) and we are interested in knowing the time $t$ in which he ruins.
I'm asked to do a simulation of the process but my doubt here is a claim by my professor:
If there is probability to be ruined, then eventually one will ruin.
I googled Gambler's ruin problem instinctively and thought that my situation fits into that framework. The difference is that here I have just a conventional player and I think that the second player could be a player with an infinite number of coins so that he never ruins. In that sense, the Huygens' formulas that Wikipedia offer would confirm the claim of my professor.
So my question is, is my view correct or I should refer to another model for it? Also, the simulation we did was with $p = q$ (fair coin flipping), so, is the claim still valid with unfair flipping?