Matching adjacent cards

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.