The M&M Game: From Morsels to Modern Mathematics

TL;DR

The paper introduces the M&M Game, a simple probabilistic model for children that illustrates randomness and leads to rich mathematical problems involving probability, combinatorics, and graph theory.

Contribution

It presents a novel, accessible game for teaching probability concepts and explores its mathematical properties and implications across various fields.

Findings

Derived formulas for the probability of a tie in the game

Connected the game to concepts in memoryless processes and hypergeometric functions

Provided educational insights for teaching probability to children

Abstract

To an adult, it's obvious that the day of someone's death is not precisely determined by the day of birth, but it's a very different story for a child. When the third named author was four years old he asked his father, the fifth named author: If two people are born on the same day, do they die on the same day? While this could easily be demonstrated through murder, such a proof would greatly diminish the possibility of teaching additional lessons, and thus a different approach was taken. With the help of the fourth named author they invented what we'll call \emph{the M\&M Game}: Given $k$ people, each simultaneously flips a fair coin, with each eating an M\&M on a head and not eating on a tail. The process then continues until all \mandms\ are consumed, and two people are deemed to die at the same time if they run out of \mandms\ together\footnote{Is one really living without \mandms?}. This led to a great concrete demonstration of randomness appropriate for little kids; it also led to a host of math problems which have been used in probability classes and math competitions. There are many ways to determine the probability of a tie, which allow us in this article to use this problem as a springboard to a lot of great mathematics, including memoryless process, combinatorics, statistical inference, graph theory, and hypergeometric functions.