Matching adjacent cards

TL;DR

This paper analyzes the probability and distribution of adjacent matching cards in well-shuffled decks, providing exact and limiting results for standard and generalized decks with various suits and ranks.

Contribution

It offers new formulas and asymptotic analysis for the probability and distribution of adjacent card matches in both standard and generalized decks.

Findings

Probability of at least one adjacent match in standard deck

Expected number of adjacent matches in various deck configurations

Limiting distribution as number of ranks grows large

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.