5-Sided Die Sum Chase

This quant interview question is about understanding how a random process evolves as you repeatedly add independent outcomes until a target is reached. It sits at the intersection of expected value, stopping times, and Markov chains, all central ideas in serious quant prep. You need to think in terms of states that summarize progress so far, and how random increments move you between them on each roll.

It trains your ability to formalize a stochastic process, isolate the relevant state variables, and express expectations in a consistent framework. You practice conditioning on the next step, handling multiple future branches cleanly, and keeping track of how uncertainty compounds over time. It also reinforces comfort with discrete distributions and structured expectations under a stopping rule.

This matters for quant interviews because many real trading and risk problems involve cumulative quantities crossing thresholds: PnL hitting limits, inventory breaching bounds, or risk metrics reaching triggers. Interviewers use this style of question to see whether you can translate a verbal description into a precise stochastic model, reason about expected times to events, and keep your logic organized under pressure. It is a compact test of core probability instincts that underlie market-making, execution algorithms, and risk management.

What it tests

When dealing with processes where progress accumulates in random increments until reaching a threshold, the core structure is a Markov process with states representing the current progress. The expected time to reach the threshold from any state can be expressed recursively in terms of the expected times from subsequent states, weighted by the transition probabilities. This principle holds because, at each step, the process 'forgets' its past and only the current state matters for future evolution, making the system memoryless. The recursive equations arise from conditioning on the outcome of the next random event, which partitions the future into mutually exclusive possibilities. Solving these equations, starting from the absorbing (terminal) state, allows you to compute the expected time to absorption from any initial state.