Prove Equal Probability Impossible with Relabeled Dice Faces

Q: Prove Equal Probability Impossible with Relabeled Dice Faces

This question evaluates discrete probability and combinatorial reasoning, including parity and modular arithmetic, along with formal proof-writing skills in the context of relabeling dice faces.

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Question

Uniform Sums from Two Relabeled Fair Dice

Setup

You have two fair six-sided dice. You may relabel each die's faces with integers; duplicates are allowed. Each face on a given die is equally likely to appear on a roll.

Task

Can you relabel the faces so that when both dice are rolled, each sum from 1 through 12 occurs with equal probability? If yes, give one valid labeling (for both dice). If no, provide a rigorous proof of impossibility.

Equal probability over 36 outcomes means each sum 1–12 must occur exactly 3 times.
You may use parity and face-pair counting arguments.

Hints

Each of the 12 sums must occur exactly 3 out of 36 outcomes.
Think about extreme sums (1 and 12) and parity of face labels.
A generating-function or modular (mod 3) argument can greatly constrain possible face multiplicities.

Prove Equal Probability Impossible with Relabeled Dice Faces

Uniform Sums from Two Relabeled Fair Dice

Setup

Task

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Solution

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Prove Equal Probability Impossible with Relabeled Dice Faces

Overview

Uniform Sums from Two Relabeled Fair Dice

Setup

Task

Hints

Solution

Comments (0)