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arXiv · 1702.08874

A Useful Solution of the Coupon Collector's Problem

Niklas Braband·Sonja Braband·Malte Braband
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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 regelmäßig in den Schlagzeilen der Nachrichten vorkommen. Dies liegt einerseits an der großen Popularität von Fußball-Sammelbildern (Paninimania genannt) und andererseits daran, dass es bisher keine Lösung gibt, die alle relevanten Effekte wie Nachkaufen oder Tauschen berücksichtigt. Wir haben bereits nachgewiesen, dass die klassischen Annahmen nicht der Realität entsprechen. Deshalb stellen wir neue Annahmen auf, die die Praxis besser abbilden. Darauf aufbauend können wir Formeln für die mittlere Anzahl benötigter Bilder (sowie deren Standardabweichung) ableiten, die alle in der Praxis relevanten Effekte berücksichtigen. Damit können Sammler die mittleren Kosten eines Albums sowie deren Standardabweichung nur mit Hilfe von elementaren Rechnungen bestimmen. Für praktische Zwecke ist das Sammelbilderproblem damit gelöst.

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Niklas Braband, Sonja Braband, Malte Braband. 2017-02-26. A Useful Solution of the Coupon Collector's Problem. https://doi.org/10.7795/320.201908

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