On duality of probabilities for card dealing

Abstract

We simplify and generalize a result of Sobel and Frankowski concerning the equality in distribution of certain random quantities associated with dealing cards Sobel and Frankowski [1] recently described a notion of for carddealing probabilities. They prove the equality of certain probabilities for different card-dealing schemes, and then discuss applications of their work. Their proof is quite computationally involved. The purpose of this short note is to present a much simpler and more conceptual proof of their result, which also generalizes it substantially. Consider a deck of N cards, called Deck #1, which contains b different (e.g., suits), of sizes mI, ... , Mb . Consider dealing (uniformly at random) j different hands from this deck, of sizes n, ...I , nj . Without loss of generality we assume that mI +.. +mb=n1 + ++nj=N (since if not, we could always include an extra category or an extra hand to make this so). For clarity, the case N = 52, b = j = 4, imI = = m. 4 = n = ...= n4 = 13 corresponds to dealing an ordinary bridge game. For 1 < x < b and 1 < y < j, we let Cxy by the random variable representing the number of cards from category x dealt to hand y. Let C record these numbers as a random b x j matrix. To explain the duality notion, consider a second deck of N cards, Deck #2, which has the sizes of categories and hands reversed. Thus, this second deck has j different categories of sizes nI, ... , nj, and we deal out b different hands of sizes mI, ... , Mbb. We let Dxy be the random variable representing the number of cards from category x in hand y2. Let D record these numbers as a random j x b matrix. Received by the editors December 15, 1992 and, in revised form, March 31, 1993. 1991 Mathematics Subject Classification. Primary 60C05. ? 1994 American Mathematical Society 0002-9939/94 $1.00 + $.25 per page