A Useful Solution of the Coupon Collector's Problem

TL;DR

This paper introduces new assumptions for the Coupon Collector's Problem that better reflect real-world sticker collecting, providing formulas to estimate the average cost and variability of completing an album considering practical effects.

Contribution

It develops practical formulas for the Coupon Collector's Problem that incorporate effects like replacement and swapping, improving real-world applicability.

Findings

Derived formulas for mean number of stickers needed

Calculated standard deviation considering practical effects

Provided a method for cost estimation in collecting

Abstract

The Coupon Collector's Problem is one of the few mathematical problems that make news headlines regularly. The reasons for this are on one hand the immense popularity of soccer albums (called Paninimania) and on the other hand that no solution is known that is able to take into account all effects such as replacement (limited purchasing of missing stickers) or swapping. In previous papers we have proven that the classical assumptions are not fulfilled in practice. Therefore we define new assumptions that match reality. Based on these assumptions we are able to derive formulae for the mean number of stickers needed (and the associated standard deviation) that are able to take into account all effects that occur in practical collecting. Thus collectors can estimate the average cost of completion of an album and its standard deviation just based on elementary calculations. From a practical point of view we consider the Coupon Collector's problem as solved. ----- Das Sammelbilderproblem ist eines der wenigen mathematischen Probleme, die regelm\"a{\ss}ig in den Schlagzeilen der Nachrichten vorkommen. Dies liegt einerseits an der gro{\ss}en Popularit\"at von Fu{\ss}ball-Sammelbildern (Paninimania genannt) und andererseits daran, dass es bisher keine L\"osung gibt, die alle relevanten Effekte wie Nachkaufen oder Tauschen ber\"ucksichtigt. Wir haben bereits nachgewiesen, dass die klassischen Annahmen nicht der Realit\"at entsprechen. Deshalb stellen wir neue Annahmen auf, die die Praxis besser abbilden. Darauf aufbauend k\"onnen wir Formeln f\"ur die mittlere Anzahl ben\"otigter Bilder (sowie deren Standardabweichung) ableiten, die alle in der Praxis relevanten Effekte ber\"ucksichtigen. Damit k\"onnen Sammler die mittleren Kosten eines Albums sowie deren Standardabweichung nur mit Hilfe von elementaren Rechnungen bestimmen. F\"ur praktische Zwecke ist das Sammelbilderproblem damit gel\"ost.