# Baseball Card Collecting

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This Demonstration compares collecting baseball cards one by one (red) to collecting packs of distinct cards (blue) in an attempt to assemble a complete set.

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In the beginning, baseball cards could only be purchased in packs of one, with a typical set consisting of 350 cards. The number of purchases required to assemble a complete set has already been demonstrated by Ed Pegg Jr [1]. Baseball cards eventually were packaged five to a pack and then later 14 and higher. It might seem as though there were a great advantage to being able to buy packs of cards guaranteed to be different from each other in order to fill out a set.

Use the slider "total cards" to adjust the size of the complete set, "pack size" to set the number of cards in each pack and "seed" to set a random number. When the pack size is equal to one, it is the original coupon collector problem.

There are two plots on the graph. The red plot shows the theoretical value of the number of singletons required and the jagged blue path shows counting unique cards against the number purchased. When packs are superior to the theoretical value, the region is shown in green. If the packs lead to an inferior value, then the region is red. It is interesting to see how often the packs fail to break even against the theoretical single-card method.

By the way, cards can now be purchased in packs but the season is broken into two disjoint 350-card sets, making them more difficult to complete than one 700-card set.

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Contributed by: Carl McCarty (January 2021)
Open content licensed under CC BY-NC-SA

## Details

Reference

[1] E. Pegg Jr. "Coupon Collector Problem" from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/CouponCollectorProblem.

## Permanent Citation

Carl McCarty

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