Can you play a fair game of craps with a loaded pair of dice?

TL;DR

This paper investigates the conditions under which the probability distributions of individual components in a product space are uniquely determined by the overall joint distribution, with a focus on special cases involving loaded dice.

Contribution

It provides new insights into the relationship between joint distributions and marginal distributions in finite probability spaces, especially for loaded dice scenarios.

Findings

Identifies conditions where marginal distributions are uniquely determined by the joint.

Analyzes special cases of loaded dice to understand distribution dependencies.

Provides theoretical results on distribution determination in finite product spaces.

Abstract

We study, in various special cases, total distributions on the product of a finite collection of finite probability spaces and, in particular, the question of when the probability distribution of each factor space is determined by the total distribution.