Comparing dealing methods with repeating cards

TL;DR

This paper extends previous asymptotic formulas to analyze the randomness of decks with repeating cards after shuffling and dealing, demonstrating how different dealing methods influence randomness.

Contribution

It combines fixed source and fixed target cases to derive new asymptotic formulas for repeated-card decks, improving understanding of dealing method effects.

Findings

Switching dealing methods enhances randomness similarly to non-repeating decks.

Formulas improve randomness assessment for decks with only two card types.

Dealing method choice significantly impacts deck randomness after shuffling.

Abstract

In a recent work Conger and Howald derived asymptotic formulas for the randomness, after shuffling, of decks with repeating cards or all-distinct decks dealt into hands. In the latter case the deck does not need to be fully randomized: the order of cards received by a player is indifferent. They called these cases the "fixed source" and the "fixed target" case, respectively, and treated them separately. We build on their results and mix these two cases: we obtain asymptotic formulas for the randomness of a deck of repeating cards which is shuffled and then dealt into hands of players. We confirm that switching from ordered to cyclic dealing, or from cyclic to back-and-forth dealing improves randomness in a similar fashion than in the non-repeating "fixed target" case. Our formulas allow to improve even the back-and-forth dealing when the deck only contains two types of cards.