The Baseball-Card Collector's Query

Abstract

The following relates my attempt to solve a problem that was asked of a colleague by a baseball-card collector. In attempting to solve this problem, I used a number of techniques from undergraduate mathematics, including series, the exponential function, Newton's method, probability, statistics, simulation, and discrete dynamical systems. I was also reminded about the importance of carefully stating the problem. Suppose that there exist baseball cards for n different baseball players. Assume for simplicity that each card is equally likely to be acquired each time a new card is purchased. One copy of each different card in a collection is put into the original pile. All duplicates, triplicates, etc. are put into the duplicate pile. The cost for obtaining the first original card is just the cost of that card. Once the collector has acquired a large number of original cards, the expected cost for obtaining one more original card will be relatively large, since most cards acquired will be duplicates. There are many questions that could be asked relating to the cost of a collection of baseball cards. In this section, we will investigate one particular question, the baseball-card collector's question (BCQ):