We solve a version of gambler's ruin problem using the optional stopping theorem and effective resistances, drawing from theories of martingale and (probabilistic) electric network.
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Recommended References:
[Durrett]: "Probability: Theory and Examples" by Rick Durrett (particularly Chapter 4, Section 8) for the martingale approach.
[Levin & Peres]: "Markov Chain and Mixing Times" by David Levin and Yuval Peres (particularly Chapter 9) for the theory of electric networks and random walks.