Expected Payout in a Die-Rolling Game

This interview question describes a simple die-rolling game with a random stopping time and a path-dependent payoff. The candidate must reason about how many times the game tends to continue before ending, how the score accumulates along the way, and how the final outcome can either realize or wipe out that accumulated score. The setup is representative of problems used in quant interviews to test understanding of expected value when the payoff depends both on the length of play and on the nature of the terminating event.

To solve it, the candidate needs a solid grasp of geometric stopping times, conditional expectation, and linearity of expectation. The structure invites modeling the number of continuation steps as a geometric random variable and then conditioning on the terminal face of the die. Interviewers watch for whether the candidate can decompose the process into "accumulation" and "payout event," handle independence and the memoryless property correctly, and express the expectation cleanly without getting lost in infinite sums or unnecessary algebra.

What it tests

This problem class is governed by the interplay between geometric stopping times and conditional payouts. Whenever a process continues with fixed probability and terminates with a set of possible outcomes, the number of steps before stopping follows a geometric distribution. The expected value of any cumulative reward up to the stopping point can be separated from the probability of receiving the reward, especially if the final outcome determines whether the accumulated value is paid out. The principle is to decompose the process: first, compute the expected accumulation up to the random stopping time, then multiply by the probability that the terminal event results in a payout. This works because expectation is linear and the terminal event is independent of the path taken to reach it, given the memoryless property of geometric processes.