A Random Card Shuffling Process

TL;DR

This paper analyzes a stochastic process for rearranging a deck of 2n cards to achieve an alternating color pattern, using combinatorics, probability, and linear algebra to model the process as a Markov chain.

Contribution

It introduces a novel Markov chain model for a specific card shuffling process and analyzes its properties to determine the average number of moves needed.

Findings

Derived the expected number of moves for the process

Modeled the shuffling as a finite Markov chain

Applied combinatorics and linear algebra techniques

Abstract

Consider a randomly shuffled deck of $2n$ cards with $n$ red cards and $n$ black cards. We study the average number of moves it takes to go from a randomly shuffled deck to a deck that alternates in color by performing the following move: If the top card and the bottom card of the deck differ in color place the top card at the bottom of the deck, otherwise, insert the top card randomly in the deck. We use tools from combinatorics, probability, and linear algebra to model this process as a finite Markov chain.