Consider a gambler who starts with an initial fortune of $i$$and then on each successive gamble either wins $1$$ or loses $1$$independent of the past with probabilities $p$ and $q = 1−p$ respectively. The gambler’s objective is to reach a total fortune of $N$$, without first getting ruined (running out of money).

Let $P_i$ be the probability that the gambler wins when starting with $i$\$, we have

\begin{aligned} P_0 &= 0 \\ P_N &= 1 \\ P_i &= pP_{i+1} + qP_{i-1} \end{aligned}

Finally,

\begin{aligned} P_i = \begin{cases} \dfrac{1-\frac{q}{p}}{1-(\frac{q}{p})^N}, & \text{if } p \ne q; \\ \dfrac{1}{N} &\text{if }p=q=\frac{1}{2}. \end{cases} \end{aligned}

Note that, $1-P_i$ is the probability of ruin.

Another type of this question: Consider an ant walking along the positive integers. At position $i$, the ant moves to $i+1$ with probabilities $p$ and to $i-1$ with probabilities $q$. If the ant reach $0$, it stops walking. Starting from $i>0$, what is the probability that the ant reaches $i=N$ before reaching $0$?

Sometimes, we consider above problem as a random walk problem. This post is copied from this and we have a backup version here.

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