The Game of Band or Bump

TL;DR

This paper generalizes the game of Book or Band to arbitrary decks with multiple ranks and cards, deriving probabilistic expressions for game outcomes and stopping times using hypergeometric distributions.

Contribution

It introduces a generalized framework for the game with arbitrary deck configurations and provides analytical formulas for outcome probabilities and stopping times.

Findings

Derived joint stopping time distribution formulas.

Expressed outcomes in terms of hypergeometric probabilities.

Extended the game analysis to arbitrary deck sizes and configurations.

Abstract

In this report we generalize the game of Book or Band described in Levin (2024) to an arbitrary playing deck with $m$ ranks and $s$ cards in each rank, for a total of $t=ms$ cards. Two events (a band or a bump) are defined in terms of given non-negative integers $0\le l\le u \le s$, not necessarily with $l+u=s$. We derive expressions for the joint stopping time distribution and outcome band or bump in terms of rectangular event probabilities for central multiple hyper-geometric random variables.