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Prove Equal Probability Impossible with Relabeled Dice Faces

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This interview question evaluates statistical assumptions, formulas, estimation strategy, uncertainty, edge cases, and interpretation in a realistic interview setting. A strong answer for Prove Equal Probability Impossible with Relabeled Dice Faces states assumptions, handles edge cases, explains trade-offs, and shows how to validate the result clearly.

Scenario

Interview assessing candidate's understanding of probability distributions with a dice-configuration puzzle used by a gaming analytics team.

Question

You have two fair six-sided dice. Can you relabel or configure the faces so that when both dice are rolled, the sums 1 through 12 each appear with equal probability? If yes, provide the new face labels; if no, give a rigorous proof of impossibility.

Hints

Equal probability requires each of the 12 sums to occur exactly 3 times out of 36. Analyze face-pair counts and parity constraints.