Generalizations of the M&M Game

Abstract: The M&M Game was created to help young kids explore probability by modeling a response to the question: \emph{If two people are born on the same day, will they die on the same day?} Each player starts with a fixed number of M&M's and a fair coin; a turn consists of players simultaneously tossing their coin and eating an M&M only if the toss is a head, with a person ``dying'' when they have eaten their stash. The probability of a tie can naturally be written as an infinite sum of binomial products, and can be reformulated into a finite calculation using memoryless processes, recursion theory, or graph-theoretic techniques, highlighting its value as an educational game. We analyze several extensions, such as tossing multiple coins with varying probabilities and evolving probability distributions for coin flips. We derive formulas for the expected length of the game and the probability of a tie by modeling the number of rounds as a sum of geometric waiting times.