Graduate Math Seminar

Title: Gambler's Ruin and Simple Random Walk

Abstract: The gambler's ruin problem is one of the classical problems in probability theory. A gambler starts with a finite amount of capital and repeatedly wins or loses one unit according to a fixed probability. The game ends when the gambler either becomes bankrupt or reaches a prescribed target.

In this seminar we formulate the problem using a simple random walk, derive the probability of winning using first-step analysis, determine

the expected duration of the game, and discuss the important distinction between fair and biased games. We conclude with connections to absorbing Markov chains, martingales, and random walks.