The M&M Game: From Morsels to Modern Mathematics

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.