Dice, (De)Convolution and Generating Functions

The blog post discusses the mathematical exploration of customizing dice to achieve the same probability distribution as standard dice when rolled. It highlights the concept of Sicherman Dice, which are labeled differently yet yield the same outcomes as traditional dice. The author explains the probability distribution of rolling two standard dice, noting that certain sums are more likely than others due to the number of combinations that can produce them. The post details how to represent dice rolls using polynomials, allowing for the multiplication of these polynomials to simulate the addition of dice. The author also introduces the idea of convolution, explaining how it relates to the probability distributions of the dice. Additionally, the post touches on generating functions, which serve as a powerful mathematical tool for solving combinatorial problems. Through examples, the author illustrates how generating functions can be used to determine the number of ways to achieve specific outcomes, such as summing numbers or selecting jelly beans under certain constraints. The overall theme emphasizes the intersection of probability, combinatorics, and polynomial mathematics in the context of dice and random distributions.

- Sicherman Dice provide an alternative labeling to achieve the same probability distribution as standard dice.

- The probability distribution of rolling two dice is not uniform, with some sums being more likely than others.

- Polynomials can represent dice rolls, allowing for the multiplication of polynomials to simulate the addition of dice.

- Convolution is a key concept in understanding the probability distributions of random variables.

- Generating functions are useful for solving combinatorial problems and determining the number of ways to achieve specific outcomes.