Bob's 30-20 Die Split Win Trick
Probability Alice beats Bob in dice game is a hard quant interview question on Conditional Probability, reported to have been seen at Citadel, DRW and Jane Street.
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This quant interview question is about understanding how extra options change winning chances in a head-to-head random game. You have asymmetric dice, a special privilege for one player, and a winner-takes-all outcome, all wrapped into a clean conditional probability setting. It's a classic quant prep scenario where intuition often fails unless you carefully formalize who really has the edge.
It trains your ability to model threshold behavior, conditional probability, and expected values under partial information. You must translate a verbal description of a re-roll privilege into a precise probabilistic framework and identify an optimal decision rule. This is the sort of probabilistic thinking that comes up repeatedly in quant interviews and advanced quant prep.
It matters because trading, market making, and risk decisions often resemble this structure: asymmetry, options to act again, and noisy information about the other side. Interviewers use such problems to see whether you can reason rigorously about conditional probabilities, think strategically under uncertainty, and justify an optimal policy clearly and quickly during a quant interview.
What it tests
This problem class is governed by the principle of optimal stopping and threshold strategies in competitive games with asymmetric information and options. When one player can take an extra action (such as a re-roll) without knowing the opponent's result, their best move is to compare the expected value of keeping their current result versus the expected value of re-rolling, given the distribution of possible outcomes. The threshold is set where the expected gain from re-rolling equals the expected gain from keeping the current roll. This principle holds because, under uncertainty and with fair dice, the only way to improve one's odds is to use the available option (re-roll) precisely when it is statistically advantageous, which is determined by comparing conditional probabilities. The structure is always about maximizing or minimizing expected outcomes under partial information, and the solution is found by equating these expectations to find the optimal cutoff.
Practise this question with written feedback, or hear it in a spoken mock interview.
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