Die Roll Game Expected Gain

This interview question introduces a simple die-rolling game that continues until a "valid" outcome appears, then asks for the expected value of that final roll. The setup highlights a stopping rule based on avoiding a particular subset of outcomes, while focusing attention on the distribution and average of the terminal value. It is a classic expected value problem that can show up in entry-level quant, trading, and risk interviews, because it tests whether candidates understand how conditioning and stopping rules interact with basic probability.

To answer it well, a candidate must recognize that the repeated trials form a memoryless process and that conditioning on "not stopping yet" does not distort the relative likelihoods among the eventual allowed outcomes. The key tools are conditional probability, geometric-type stopping arguments, and computing an expectation via a simple average. Interviewers pay close attention to whether the candidate can justify the distribution of the terminal roll clearly, avoid overcomplicating the infinite process, and articulate why the stopping time does not bias which valid face is ultimately observed.

What it tests

When a process repeats until a certain condition is met—such as rolling until a result outside a forbidden set appears—the final outcome is distributed uniformly over the allowed set, regardless of the number of trials. This is because each trial is independent and identically distributed, and the process is memoryless: the probability of ending on a specific allowed outcome is the same as if you had rolled just once, conditioned on not getting a forbidden result. The expectation is then simply the average of the allowed outcomes. This principle holds because the geometric waiting time until success does not bias which allowed outcome appears; it only delays the inevitable selection from the allowed set.