Given that one starts at $i$ dollars, how does one find the probability of getting infinitely rich/getting ruined (the complement)?
Let's say that $p = \text{probability of winning an extra dollar}$, and $q = \text{probability of losing a dollar}$. The gambler is ruined once he has $0$ dollars (I think the assumption is that the opposing party has unlimited dollars).
I found the formula online, which is that if $p \leq \frac{1}{2}$, then the probability of winning becomes $0$. However, if $p \geq \frac{1}{2}$, then the probability of winning is not $1$, but follows this formula:
$P(\text{becoming infinitely rich, given that you start with $i$ dollars}) = 1 - (\frac{q}{p})^i$.
What's the intuition behind this?