On the Asymptotic Solution of a Card-Matching Problem

Introduction Recently S. G. Penrice [8] has shown how an asymptotic result concerning the number of derangements in a classical card-matching problem can be obtained by using two "hard" celebrated inequalities, known as Minc's conjecture and Van der Waerden's conjecture. Both were verified years after they were proposed. It will be demonstrated here that this result in fact follows from easy inequalities and a straightforward application of the principle of inclusion and exclusion. We show this by proving a more general result. However, as will be explained, the inequalities for the derangement probability established by Penrice yield an asymptotic result that otherwise seems difficult to verify. For a standard deck of cards, with k = 4 suits and n = 13 denominations in each suit, the problem is equivalent to a card game played in Las Vegas: The cards of a shuffled deck are dealt one at a time and face up. At the same time the player calls the denominations in the order ace, two, three,... queen, king; ace, two,... . A match occurs when the player calls the same denomination as the card dealt; the suits need not match. The player wins if no match occurs. If one continues the comparison through the whole deck, the number of matches X is a random variable on the sample space S13x4, which consists of all possible shufflings of the cards. What can we say, to begin, about the derangement probability po = P(X = 0), which is the probability that the player wins? Using the Penrice inequalities (6) below,