Detailed problem statement can be found here Gambler's Ruin

General solution to the problem is;

\(1- a_1 = a_1 \sum\limits_{i = 1}^{k-1} \left(\frac{1 - p}{p}\right)^i\)

For a fair game of \(p = \frac{1}{2}\), \(a_1 = \frac{1}{k}\)

and, for an unfair game of \(p …