Dice rolls with re-rolls expected payoff

This question presents a simple stopping-time game built from repeated rolls of a fair die. The candidate must understand that the game's payoff is determined only by the final roll, even though there may be several preliminary rolls that do not directly contribute to the payout. The twist is that some outcomes allow the process to continue, while others force it to stop and pay out, so the game behaves like a resettable random experiment. Variants of this setup often appear in junior quant and data science interviews to test comfort with expected values in processes that can, in principle, run indefinitely.

To solve it cleanly, a candidate needs to recognize the stationarity of the game after each qualifying roll and to model the expectation using a self-referential equation. The key tools are conditional expectation, geometric-like stopping behavior, and manipulating a simple linear equation in the unknown expected payoff. Interviewers are watching for an organized derivation, correct conditioning on the different branches of the roll, and comfort treating the expectation as an unknown variable rather than trying to enumerate long paths.

What it tests

This problem class is governed by the law of total expectation and the recursive structure of stopping-time processes. When a process can either terminate with a known outcome or continue with a probabilistic reset, the overall expectation is a weighted sum of the immediate outcomes and the expected value of the process itself. The key is recognizing that the expectation can be expressed in terms of itself, leading to a solvable equation. This recursive self-reference arises because, at each stage, the process 'restarts' with the same rules, so the expected value from that point onward is unchanged. The principle holds because the future of the process, conditioned on not stopping, is statistically identical to the original process, making recursion both valid and necessary.