Gambler's Ruin Target Fortune

This classic gambler's ruin question considers a one-dimensional random walk for a gambler's fortune, with absorbing boundaries at zero and at a target wealth level. The candidate is asked to compute the probability of eventual success, starting from an arbitrary initial fortune, under a fixed win/loss probability on each play. It is a canonical model of risk of ruin versus hitting a profit target, and often appears in interviews for quantitative finance and probability-focused roles because it compresses many core ideas of discrete-time stochastic processes into a very compact setup.

The solution leans on Markov chain reasoning, conditioning on the next step to derive a recurrence for the success probability. Candidates are expected to translate the verbal setup into a recursive relation with appropriate boundary conditions, and then solve that recurrence systematically rather than by ad hoc guessing. Interviewers watch for comfort with difference equations, recognition of symmetry and special cases, and the ability to handle both fair and biased random walks cleanly. Clarity of notation, correct use of conditional probability, and logical structuring of the argument are all important signals.

What it tests

This problem class is governed by the idea of Markov processes with absorbing barriers, where the future evolution depends only on the current state and not the path taken to reach it. The key structure is that the process is memoryless and transitions between states are governed by fixed probabilities, leading to a recurrence relation for the probability of reaching a boundary before another. The recurrence arises because, at each step, the probability of eventual success from a given state is a weighted average of the probabilities from the neighboring states, reflecting the possible outcomes of the next move. The boundary conditions (absorbing states) anchor the solution, and the recurrence relation, together with these boundaries, uniquely determines the probability of reaching one barrier before the other. The geometric nature of the solution comes from the constant ratio of transition probabilities, which leads to a geometric series when solving the recurrence.