Matching adjacent cards
Kent E. Morrison
TL;DR
This paper analyzes the distribution of adjacent-rank matches in a shuffled deck with $k$ suits and $n$ ranks, focusing on the exact distribution of the random variable $M$ counting adjacent equal-rank pairs. It develops an exact combinatorial framework using partitions of $k$ across ranks and generating functions $A(x)$ and $B(x)$, with the core relation $A(x)=B(x-1)$, to enumerate the distribution of $M$, and specializes to both the standard deck ($k=4$) and general $k$. The authors show that a binomial model with $N=kn-1$ trials and $p=(k-1)/(kn-1)$ and a Poisson limit with parameter $\lambda=k-1$ both approximate $M$, proving variance equality and a Poisson limit as $n\to\infty$ (via Soon's bound), and validating the theory with numerical comparisons for the standard deck. They also provide exact results for the case $n=2$ and establish connections to Carlitz permutations and lattice-path interpretations, offering practical guidance for estimating the distribution of adjacent matches in decks.
Abstract
In a well-shuffled deck of cards, what is the probability that somewhere in the deck there are adjacent cards of the same rank? What is the average number of adjacent matches? What is the probability distribution for the number of matches? We answer these and related questions for both the standard $52$-card deck with four suits and $13$ ranks and for generalized decks with $k$ suits and $n$ ranks. We also determine the limiting distribution as $n$ goes to infinity with $k$ fixed.