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.

Q: How do I approach Statistics & Math interview questions?

Statistics & Math questions require understanding of core concepts and practice. PracHub provides solutions with explanations to help you master statistics & math interviews.

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

Hints

Solution

Comments (0)