Three-Roll Dice Betting Math

This question presents a simple multi-stage dice game where the player can stop early for a guaranteed payout or continue rolling in search of a higher value. It captures the trade-off between locking in a known outcome and gambling on future randomness under a finite time horizon. The candidate must recognize that the decision points are sequential, that information arrives gradually with each roll, and that the forced final roll creates a natural terminal condition. This style of optimal-stopping game is common in quant interviews for trading and structuring roles, where evaluating when to "exercise" or "walk away" under uncertainty is central.

Solving it leans on backward induction, conditional expectations, and a clear understanding of state-dependent decision rules. The interviewer is looking for someone who can formalize the problem, compute continuation values correctly, and translate them into a simple threshold-based strategy. They will watch for the ability to articulate assumptions, handle symmetry and discrete distributions without overcomplication, and remain consistent about risk-neutral, expectation-maximizing behavior. Clean reasoning, rather than heavy computation, is the main signal.

What it tests

The core structure of this problem class is optimal stopping, where at each stage you must decide whether to accept a known reward or risk it for a potentially better but uncertain future outcome. The principle is that the value of continuing is the expected value of all possible future paths, and you should only continue if this expected value exceeds your current offer. Backward induction is the tool that lets you solve these problems: by starting at the last possible decision point (where you have no choice), you can recursively determine the optimal action at each earlier stage. This works because the future is uncertain, but its expected value is calculable, so you can always compare the certain present to the probabilistic future. The pattern holds because, at every step, the rational choice is to maximize expected value by comparing immediate payoff to the expected value of future decisions, given optimal play from that point onward.