Bob's Rebuy Gamble

This quant interview question is about a simple stochastic game where wealth moves back and forth between two players, but with an asymmetric rebuy rule. It embeds a finite-state Markov chain with absorbing states and a restart mechanism for one player, making it a clean example of conditional expectation and conditional probability in a gambling setup. The scenario mimics a stylized trading PnL process with top-ups and drawdown limits.

It trains your ability to model random walks with absorbing barriers, to recognize when a rebuy effectively changes the initial capital, and to compute conditional expectations across stages of a process. It also reinforces comfort with stopping events, symmetry arguments, and translating a word problem into a small, tractable state space that can be analyzed rigorously.

This matters for quant interviews and quant prep because interviewers want to see whether you can turn an informal gambling story into a precise probabilistic model and reason correctly about outcomes and risk. The same mindset underlies pricing, risk limits, and capital allocation problems in quantitative finance interviews, where rebalance rules or margin calls change the evolution of a process.

What it tests

Problems of this class are governed by the principle of state transitions and absorbing barriers in random walks, specifically the Gambler's Ruin framework. The key is that the probability of ruin (or victory) depends only on the initial relative stakes and the rules for absorbing (ending) states, not on the path taken. When a player is allowed to rebuy or re-enter with additional resources, the process is equivalent to starting with a larger initial stake, as the random walk simply continues from the new state. This equivalence arises because the memoryless property of the fair coin and the linearity of expectation mean that the probability of eventual ruin is determined by the total resources available before the final absorbing barrier is reached. Thus, the overall probability can be computed by summing the probabilities of sequential ruin events or by considering a single walk with the combined resources.