The mathematics of collector cards

The Sandbox General
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I volunteer at a thrift store and have am responsible for sifting through a number of donated collector cards to see what value can be obtained. Some of these were “Garbage Pail” cards from 10-15 years ago. These were sorted into a variety of subcollections. Some collections were only 20 cards, others were a set of 100 twins, so 200 cards. It got me thinking, how many cards would you expect to buy in order to complete a set of 20?

When you are trying to complete the set of 20, you are guaranteed to need the first card. How many cards are you expected to draw to get the next card? Your probability of success is p=19/20 and the expected number of cards is known to be 1/p = 20/19. Following this string of logic the total number of cards is

20/20 + 20/19 + 20/18 + … + 20/2 + 20/1 = 71.95. It gets progressively more difficult to complete the set. The first 18 cards come from the first 42 draws, and another 30 draws are needed to collect the last 2 (and that’s just the mean, of course any actual trial varies from that). So if you want to complete the set you end up with 20 cards you want and 52 you don’t want, that’s good marketing!

I came up about 7 short on my goal to have a complete set of 200. Luckily for anyone who wants a complete set you can buy what you need online now. I’m in the middle now of sorting through a ton of Magic the Gathering cards for the thrift store. I believe there are 273 to collect and I don’t know how many I have but I’m first sorting them in piles of 25 and then I’ll be trying to put a set together. Excel tells me I should expect 1,689 cards to complete the task! I’m not saying that MTG afficianados are necessarily trying to generate a complete set, but I am.

So I’m trying to put any collections together or clearly mark what is missing and then we’ll see how much random strangers will pay for their kids to have those and the leftovers.

1 Like
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[tan] my grad school advisor was an initial investor, having met the creators while in a post-doc program where the creators also were. Apparently there is an old school card that is “him” whatver that means. [\tan]

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I would expect that many cards are much less common than others, so that’s going to make your estimates more of a best case scenario.

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yeah, assumes each card is equally likely. i can live with that tbh.

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I used to collect Pokemon cards, never Magic the Gathering cards, but I figured they were similar and looked it up to confirm. Magic the Gathering specifically has cards that are Common (Black color set symbol, letter code “C”), Uncommon (Silver, U), Rare (Gold, R), and, introduced in 2008, Mythic Rare (Red-Orange, M). So I imagine collecting all the cards in a Magic the Gathering set would be a herculean undertaking, if the only method is opening packs.

(Back in the day I tried to collect all of the Pokemon cards in the first 2-3 sets, but I probably never got more than about 1/2 or 2/3 of the way there. I didn’t really have spending money as a kid, so I’m not sure how I even got that many to be honest.)

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I put an hour into sorting only to later realize it was THREE different sets of about 270 different cards and I wasn’t going to come anywhere near to a complete set and promptly abandoned the organization.

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Your last step would have been my first step.