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.

What difficulty level is this interview question?

This is a medium difficulty Statistics & Math question, commonly asked during Technical Screen rounds at Yahoo.

What role is this question designed for?

This question is commonly asked for Data Scientist candidates at Yahoo during technical interviews.

Prove Equal Probability Impossible with Relabeled Dice Faces

Quick Overview

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.

Prove Equal Probability Impossible with Relabeled Dice Faces

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.

Constraints & Assumptions

Preserve the scope, facts, inputs, and requested outputs from the prompt above.
If the prompt leaves a detail unspecified, state a reasonable assumption before relying on it.
Keep the answer interview-ready: concise enough to present, but concrete enough to implement or evaluate.

Clarifying Questions to Ask Guidance

Clarify the random variables, distributional assumptions, independence assumptions, and desired output.
Show enough derivation for the interviewer to follow the reasoning.
Explain how you would validate the result with simulation or sensitivity checks.

What a Strong Answer Covers Guidance

A correct setup with definitions, formulas, and boundary conditions.
A step-by-step derivation or estimation plan.
Interpretation of the result, including uncertainty and practical limitations.
Checks for assumptions, edge cases, and numerical stability.

Follow-up Questions Guidance

How would the result change if the assumptions were relaxed?
Can you verify the answer with a simulation?
What is the most likely source of estimation error?

Quick Overview

Task

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.

Constraints & Assumptions

Preserve the scope, facts, inputs, and requested outputs from the prompt above.

If the prompt leaves a detail unspecified, state a reasonable assumption before relying on it.

Keep the answer interview-ready: concise enough to present, but concrete enough to implement or evaluate.

Clarifying Questions to Ask Guidance

Clarify the random variables, distributional assumptions, independence assumptions, and desired output.

Show enough derivation for the interviewer to follow the reasoning.

Explain how you would validate the result with simulation or sensitivity checks.

What a Strong Answer Covers Guidance

A correct setup with definitions, formulas, and boundary conditions.

A step-by-step derivation or estimation plan.

Interpretation of the result, including uncertainty and practical limitations.

Checks for assumptions, edge cases, and numerical stability.

Follow-up Questions Guidance

How would the result change if the assumptions were relaxed?

Can you verify the answer with a simulation?

What is the most likely source of estimation error?

Prove Equal Probability Impossible with Relabeled Dice Faces

Quick Overview

Prove Equal Probability Impossible with Relabeled Dice Faces

Prove Equal Probability Impossible with Relabeled Dice Faces

Uniform Sums from Two Relabeled Fair Dice

Setup

Task

Hints

Constraints & Assumptions

Clarifying Questions to Ask Guidance

What a Strong Answer Covers Guidance

Follow-up Questions Guidance

Write your answer

Prove Equal Probability Impossible with Relabeled Dice Faces

Quick Overview

Prove Equal Probability Impossible with Relabeled Dice Faces

Prove Equal Probability Impossible with Relabeled Dice Faces

Uniform Sums from Two Relabeled Fair Dice

Setup

Task

Hints

Constraints & Assumptions

Clarifying Questions to Ask Guidance

What a Strong Answer Covers Guidance

Follow-up Questions Guidance

Write your answer