Money and Mathematics: Booster Packs

Written by | April 10, 2011 | 47 comments | 3,884 views | Rating: +18

A friend of mine once told me that at some point in our lives, we were all “n00bs”. Of course, there are no “n00bs” in the Pokémon TCG (my friend was actually talking about an online game). Still, all of us could probably recall the first time we bought our first Pokémon theme deck. I remember mine very well, actually. I think I was eleven or twelve years old, and I was pretty happy getting my very first pre-constructed deck (Overgrowth).

I also remember opening my very first booster pack. During that time, everyone seemed to want a Charizard, and the thought of getting one seemed to be more important that everything else. I remember slowly opening the foil wrapper, careful not to damage the cards. I also remember the feeling of disappointment when I got something else.

Those days, when the fastest internet connection speed only reached a few kilobytes and websites like Amazon and eBay didn’t exist, it was pretty difficult to find the cards you wanted. The easiest way for a grade school student back then had to buy and open booster packs to find the cards he/she wanted (or attend the official leagues and trade cards with other players).

Opening booster packs, however, was a gamble. There was no guarantee you would get the cards you wanted. Back in those days, when I was still in grade school, I only got around $6.00 every week as my allowance (nowadays, I get way much more than that, of course).

Booster packs used to cost around $3.00, and I could only get one once in a blue moon. When I got to high school, I could buy one to two booster packs every week. When I got to college, I started buying booster packs by the dozen every now and then (talk about an increase in purchasing power).

This gave me better chances of finding the cards I wanted, but it also got me spending whenever I passed by the local card store. For a period of time, I found myself buying booster packs whenever I had the money to burn. Opening booster packs became an addiction.

In fact, when you think about it, card stores that sell booster packs are like slot machines. When you buy a booster pack, you’re literally paying for a chance of getting valuable cards. If you find yourself buying booster packs by impulse, you might as well say you have a gambling problem.

For the record, I have nothing against booster packs. I personally think that booster packs are a great way to find the cards you want, though it usually takes a significant amount of money and an extraordinary amount of luck to get very rare cards like Luxray GL LV.X.

Seeing that almost everyone here in 6P talks about deck strategies and the like, I decided to take a different route and talk about something else. You may not like mathematics, but I think you may find the content of the rest of this article a bit helpful (and/or entertaining, hopefully). I hope to give people some insight on booster packs.

A booster pack from the earlier expansions of the Pokémon TCG used to have eleven cards, but after a while, the number of cards were reduced to nine. Booster packs nowadays, however, have ten cards. Each booster pack has five common cards, three uncommon cards, one rare card, and one random card.

The random card is usually a common card or a reverse foil version of a card in the set, but it could also be a Pokémon LV.X, a Pokémon Prime, a Pokémon LEGEND, or some other special rare card like Alph Lithograph. Contrary to what a lot of people believe, these never take the place of the rare card – you would always find one with another rare card inside a booster pack.

It has also been generally accepted that you couldn’t get two copies of the same card in the same booster pack (unless the other one is its reverse foil version). Basic Energy cards are considered common cards, and they have no reverse foil versions.

Knowing this, we could somewhat compute for the probability of getting a certain card by opening a booster pack from a certain set. Please be warned that I’m about to delve a bit into statistics – kindly get some tissue if mathematics makes your nose bleed.

Let’s say you want to know the probability of getting a Ninetales from HeartGold & SoulSilver. You also don’t want its reverse foil version. Since there are thirty-four (34) rare cards in HGSS (excluding Pokémon Prime and Pokémon LEGEND), the odds of you getting a Ninetales is 1/34 or 0.0294.

The odds of getting a reverse foil version is even more grim since a reverse foil version of Ninetales would take the place of the random card in an HGSS booster pack, as there are 157 cards that could appear in the place of the random card – there are 104 cards in HGSS that could have a reverse foil version, 39 common cards (including eight Energy cards), 6 Pokémon Prime, 4 Pokémon LEGEND, and two special rare cards (Gyarados and Alph Lithograph).

Therefore, the odds of getting Ninetales appearing in the random card’s place in an HGSS booster pack is 1/157 or 0.0064. To know the odds of getting either version of Ninetales in a booster pack, simply add the two probabilities – that is, 1/34 + 1/157, or 0.0360.

To know the odds of getting both versions of Ninetales in the same booster pack, simply multiply the two probabilities – that is, 0.0002.

As said earlier, there’s only one rare card in every booster pack (unless you get another one in the random card’s place). What about uncommon cards? There are three uncommon cards in a booster pack. Let’s say you want to know the probability of getting a Pokémon Collector and that you also don’t want its reverse foil version. We know that Pokémon Collector is an uncommon card, and is also from HGSS.

Since there are thirty-eight (38) different uncommon cards in HGSS, the odds of getting a Pokémon Collector in an HGSS booster pack should be 1/38 or 0.0263. There are, however, two other uncommon cards in the booster pack. Pokémon Collector could be any of one the three but cannot appear more than once.

The probability of a Pokémon Collector appearing in a booster pack, therefore, is actually (1/38 + 1/37 + 1/36), or 0.0811. Computing probabilities for common cards generally revolves around the same idea.

At this point, I think some of you are already confused, and that is why I took the liberty of computing all the probabilities for you. As you may notice, the probabilities differ for each set because they all have a different number of cards.

My computations are based on the assumption that you are only looking for one particular card at a time. Kindly note that the probability for the random card in each set may not be accurate, but it should give you a rough idea on how difficult it is to find a card like Luxray GL LV.X in a booster pack.

RISING RIVALS
Common – 16.16% (1/33 + 1/32 + 1/31 + 1/30 + 1/29)
Uncommon – 9.68% (1/32 + 1/31 + 1/30)
Rare – 2.70% (1/37)
Random – 0.65% (1/155)

SUPREME VICTORS
Common – 11.64% (1/45 + 1/44 + 1/43 + 1/42 + 1/41)
Uncommon – 6.52% (1/47 + 1/46 + 1/45)
Rare – 2.08% (1/48)
Random – 0.51% (1/195)

ARCEUS
Common – 16.70% (1/32 + 1/31 + 1/30 + 1/29 + 1/28)
Uncommon – 10.72% (1/29 + 1/28 + 1/27)
Rare – 3.13% (1/32)
Random – 0.75% (1/134)

HEARTGOLD & SOULSILVER
Common – 13.18% (1/40 + 1/39 + 1/38 + 1/37 + 1/36)
Uncommon – 8.11% (1/38 + 1/37 + 1/36)
Rare – 2.94% (1/34)
Random – 0.69% (1/144)

UNLEASHED
Common – 19.29% (1/28 + 1/27 + 1/26 + 1/25 + 1/24)
Uncommon – 11.12% (1/28 + 1/27 + 1/26)
Rare – 3.70% (1/27)
Random – 0.81% (1/124)

UNDAUNTED
Common – 18.57% (1/29 + 1/28 + 1/27 + 1/26 + 1/25)
Uncommon – 3.85% (1/26 + 1/25 + 1/24)
Rare – 4.00% (1/25)
Random - 0.83% (1/120)

TRIUMPHANT
Common – 3.33% (1/30 + 1/29 + 1/28 + 1/27 + 1/26)
Uncommon – 3.85% (1/26 + 1/25 + 1/24)
Rare – 3.23% (1/31)
Random – 0.75% (1/133)

CALL OF LEGENDS
Common – 3.23% (1/31 + 1/30 + 1/29 + 1/28 + 1/27)
Uncommon – 4.00% (1/25 + 1/24 + 1/23)
Rare – 2.56% (1/39)
Random – 0.78% (1/129)

Statistically speaking, booster packs from Rising Rivals and Supreme Victors give you the worst bang for your buck. That being said, it would be much more advisable for you to go and find people selling the card you want from those sets than buying and opening booster packs.

Don’t get me wrong. I’m not discouraging you from buying booster packs; just don’t develop the thinking that buying more booster packs means more chances of getting what you want. This isn’t true, and this kind of thinking is called the gambler’s fallacy.

The probability of getting what you want in one booster pack isn’t affected by the outcome of another. For example, there are twenty-nine different rare cards in Undaunted. Buying twenty-nine booster packs won’t guarantee you getting a Smeargle.

Similarly, buying a box of booster packs won’t guarantee you a Pokémon LV.X. It is more advisable to buy booster packs in bulk, though. Although it won’t exactly give you better chances of finding cards you want, it is a lot more cheaper than buying a booster pack every now and then.

Speaking of boxes, there’s this popular belief going around that there are only one to two Pokémon LV.X in every box of booster packs. That’s not entirely accurate; that’s probably the average number of Pokémon LV.X found in every box of booster packs (there was a time I got ridiculously lucky and found three copies of Machamp LV.X in six straight Stormfront booster packs).

Here’s a little bit of something that might pique your interest. If you ask around, a lot of people would tell you that foil (and reverse foil) cards are a bit thicker than normal cards. Last year, during a moment of insanity, I decided to put this theory to the test.

I brought a micrometre caliper to the card store I frequented and bought around seven booster packs. For those of you who don’t know, a micrometre caliper is a small tool used to accurately measure the degree of thickness of a certain object (a micrometre is 1/1000 of a millimetre) – it is mostly used by those in engineering.

Anyway, for consistency, all booster packs came from the same set. Before I opened each one, I measured the thickness of each booster pack. There was a small discrepancy between some of the booster packs, usually around ten to twenty micrometres.

I initially attributed this to uneven foil packaging, but when I opened the booster packs, the ones that were thicker by ten to twenty micrometres had foil and/or reverse foil cards. I decided to test if there was indeed a difference between them and learned that foil and reverse foil cards were thicker than normal cards by an average of ten micrometres.

If you’re going to buy booster packs, you might want to bring a micrometre caliper on your next trip to your local card store.

In conclusion, opening booster packs to find a card you want isn’t always the a bright idea, especially when you have a small budget. Sometimes, it’s just more practical to get it from someone else.

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  • Derek Coontz

    Interesting.

  • Anonymous

    It’s really cool to see the math behind how much booster packs are a rip off. :( According to your calculations, we have the best chance of getting a rare in the UD expansion. Looks like my luck’s run out though; my best pull was bottom half of Entei and Raikou LEGEND. Everything since has been useless cards.

    • Michael Peter Sison

      The data is subject to one’s interpretation. It depends, really. Let’s take HGSS for example. The set has a lot of good uncommon Trainer cards, but the chance of getting the Trainer card you want is a bit below average. By the way, the editing messed up two of the calculations. The formulas are correct, though.

    • Anonymous

      I tell all the parents at my league that they’re better off buying/trading cards individually than buying packs. It’s such a rip off.

  • Sam Marshall-Smith

    YES!!!!
    I knew I could tell if a pack is good by its weight!

    Also,

    “A booster pack from the earlier expansions of the Pokémon TCG used to have eleven cards, but after a while, the number of cards were reduced to nine. Booster packs nowadays, however, have ten cards. Each booster pack has five common cards, three common cards, one rare card, and one random card.”

    5 common cards and 3 common cards…? Might wanna change that. Otherwise, good article!

    • Michael Peter Sison

      Thank you for noticing. I think there were a few bugs after the pictures were attached. I’ll work on fixing these right away. :-)

  • http://profiles.google.com/nilaypatel2 Nilay Patel

    SOOO, where do you get a micrometre caliper??

  • http://pulse.yahoo.com/_UYSC7MZPCUOUVOG4QULYIJOFGU Scott

    Very interesting article. It’s nice to see things other then a deck analysis or a tournament report every now and again. I did notice a few errors in your calculations though. For example, when you calculated the probability of getting either Ninetales from HGSS, after you add the two probabilities together, you still need to subtract the probability of getting both a regular-holo and a reverse-holo card in the same pack, otherwise you’re effectively counting them twice in your calculation. So instead of a 3.6% chance of getting a Ninetales as you said in your article, it would actually be a 3.58% chance.

    Also when you calculated the probabilities of getting a certain common card in Rising Rivals, or any other set, ignoring the reverse-holo card, the calculation would actually be 5 /number of commons in that set. (the same applies for uncommons and rares, just replace 5 with 3 and 1 respectively). So for Rising Rivals, commons have a 15.15% chance of being the one you want, and uncommons have a 9.38%, etc.

    Other then that, a really good article xD.

    • Michael Peter Sison

      I think you’re right about the first one, but I’m not sure about the second one. I’m no mathematics wizard, though. At any case, thank you for telling me. :-)

  • Anonymous

    I very much agree with the packs being a gambling type thing.

    Maybe not so much now, but when the Level X’s were out and were hard to get but very valuable, I was spending lots of money trying to get lucky.

    Now that the X’s are gone, and most primes are worth between 5-8 bucks, I rarely have the urge to buy packs. Except maybe triumphant because both gengar prime and magnezone prime are fairly valuable.

    But good topic for an article

    • Anonymous

      don’t forget mew prime.

  • http://pulse.yahoo.com/_ASSYSHBBG4XT2GDCDD6VVC2IGU Lukas

    After talking to friends who play other card games, it seems people who play the naruto ccg only buy packs they are sure have their version of a secret rare in them. Most of the players have it down to almost a science without a meter, however it is likely that the naruto ones vary in size more than the pokemon tcg packs do.

  • Matthew Piela

    Pack opening just got REAL.

    • Anonymous

      LOL

  • Anonymous

    This is the kind of article I like to read. Appeals to both my Pokemons and Math interests, and was fairly short, sweet, and to the point. Keep them coming!

    Oh, and I feel like that part about weighing booster packs was a bit like Mythbusters haha. Very intriguing though.

    • Michael Peter Sison

      Thank you. :-)

  • Anonymous

    Booster Boxes can be cracked also, so the location of the ultra rare’s are made more simple to find. The problem is that it cannot always be cracked necessarily across cases.

    Is that cheating? Only of you are skimming the good cards and selling your packs you know have exteremly limited chance of ultra rares.

    • Michael Peter Sison

      I think it’s an ethical dilemma. That’s not a bad topic for another article, actually.

  • Anonymous

    “To know the odds of getting either version of Ninetales in a booster pack, simply add the two probabilities – that is, 1/34 + 1/157, or 0.0360.” <– So if the probability of normal ninetails was 50% and probability of reverse foil ninatails was 50% you would add them to get 100%?

    I don't think that is how it works.
    Lets say the probability of getting normal ninatails is A and the probability of getting foil ninetails is B. The probality of getting none of them is (1-A)(1-B) (they are independent because the reverse foil takes the place of the random card).
    Thus, the probability of getting either is 1 minus the probability of getting none, that is

    1-((1-A)*(1-B)). Thus, if A=50% and B=50% I get the probability of getting either to

    1-((1-0.5)*(1-0.5))=1-(0.5*0.5)=1-0.25)=0.75

    This seems more reasonable, since you can easily realize that the probability of getting one is never 100%. I don't know if this analyses is correct, but it seems more correct than yours.

    • Michael Peter Sison

      You have a point. Thank you for pointing that out to me. :-) I’ll tweak the article when I get some free time. I’m not exactly good with mathematics, and I just based the calculations on what I understood on Wikipedia – I’m not exactly surprised there was an error with one of the calculations. I just wanted to point out that it wouldn’t be a good idea for someone to put his/her faith in chance – opening booster packs, as I’ve said, was a gamble.

      • Anonymous

        No problem, probability theory and combinatorics is always sorta tricky to me, but it is good that we can help each other out.

      • Michael Peter Sison

        I plan to do a second article similar to this one for the next season, and I think I need to brush up a little more on my statistics to make sure my calculations would be more accurate next time. :-)

      • Anonymous

        You need to look up the Hypergeometric distribution. And after you have cracked that then switch to using the multi-variate hypergeometric as that one can be readily applied to opening hand statistics.
        http://en.wikipedia.org/wiki/Hypergeometric_distribution#Multivariate_hypergeometric_distribution

        using the hypergeometric distribution you will be able to see that your figures for commons and uncommons are incorrect.

        ==

        There is also the Coupon Collector Problem to investigate. http://www.ds.unifi.it/VL/VL_EN/urn/urn9.html But beware that the boosters within any given box are not uncorrelated which is why if you are trying to collect a set you are best off buying whole boxes.

      • Michael Peter Sison

        I’m already writing a similar (and more accurate) article. I hope to get it published sometime next month. :-)

  • http://www.bebessearch.com/ BBSearch

    How would one calculate the probability of having an opening hand that had a particular makeup. For example, if a decklist had 4 Sableye and 2 special dark energy, what is the probability that the opening hand would have at least 1 Sableye AND 1 special dark energy?

    • Ahj911211

      X is an interger from 1 to 4
      Y is an interger from 1 to 2

      P(1 or more sableye, 1 or more sp dark)= 1-P(no sableye, no sp dark)-P(X sableye, no sp dark)-P(Y sp dark, no sableye)

      P(no sableye, no sp dark)= (2 nCr 0)(4 nCr 0)(54 nCr 7)/(60 nCr 7) = .45856

      P(X sableye, no sp dark)= [(2 nCr 0)(4 nCr 1)(54 nCr 6)+(2 nCr 0)(4 nCr 2)(54 nCr 5)+(2 nCr 0)(4 nCr 3)(54 nCr 4)+(2 nCr 0)(4 nCr 4)(54 nCr 3)] / [60 nCr 7] = .31997

      P(Y sp dark, no sableye)= [(2 nCr 1)(4 nCr 0)(54 nCr 6)+(2 nCr 2)(4 nCr 0)(54 nCr 5)] / [60 nCr 7] = .14194

      P(1 or more sableye, 1 or more sp dark)= 1-0.45856-0.31997-0.14194 = 0.7953 = 7.95%

      • http://www.bebessearch.com/ BBSearch

        Excellent! Thanks!

      • Anonymous

        That doesn’t look quite right. Mulligans need to be excluded so the denominator of (60 nCr 7) is too high and the opening card draw has not been included as a way of getting a dark energy. So the figure will be higher than 8%. Worse though is that if a trainer card focused deck is made then the expectation for getting to the dark energy is close to 100% and that alone makes all such calculations redundant :(

        ===

        Nice use of the multivariate hypergeometric FWIW I’d have used 1-P(no sableye) – P(no dark) + P(no sableye, no dark) which has fewer calculations.

      • http://www.bebessearch.com/ BBSearch

        If you folks can come up with a formula that passes peer review, I’ll TRY to
        add a tool to BebesSearch.com that allows you to build an “opening hand” and
        then uses the formula to calculate the % chance of getting it.

      • Anonymous

        the formula/math that was used is fine. It seems quite likely that Ahj911211 understands how to use the math and the assumptions that have to be made. But It is the assumptions behind its use that are the problem. And those make including any kind of formula on a website tricky. That said the multivariate hypergeometric distribution is the best way to calculate the odds precisely.

        ====

        To correctly account for mulligans you have to know how many starting cards there are (which wasn’t given in the original question). For a lot of decks that increases the numbers by a factor of around 1.1 (so from 8% to 9%) But if you just run 4 Sableye it is 100%. Big discrepancy there :(

        Including the top card as a candidate for a sp. dark would yield slightly higher probabilities too. Then there is the Uxie drop and other cards drawn T1. So just how many cards will you have access to T1? 10 12 14 or all 46? A deck that used 4 sableye and two special dark is going to use a lot of T1 draw if it is going for the donk.

        —-

        So as with a lot of stats even when the math is sound the assumptions near very careful consideration.

      • http://www.bebessearch.com/ BBSearch

        If I understand you correctly, then I would point out that the formula
        itself should not need to be aware of cards like Uxie or Collector. In terms
        of the website, it would present an interface for the player to select what
        they considered a desirable hand and then use the formula to calculate the
        probability. It would be the players responsibility to choose the hand they
        wanted. For example:

        Player decides they think a good starting hand would be Sableye and special
        dark so they drag those two cards to the interface and set the number of
        cards to 7 to reflect an opening hand (could be 8 to represent Felicity or 4
        for Judge, etc).

        The interface knows their decklist and so knows there are 4 sableyes and 1
        special dark in the deck.

        The player then decides to drag Bebes Search and Collector into the
        interface and the interface knows they have 2 and 4 of
        those, respectively in the deck.

        However, the player doesn’t want to know the probability of starting with
        ALL those cards, what they want to know is:

        (Sableye OR Collector OR Bebes Search) AND (Special Dark)

        Assuming for a moment that the interface could distill the player’s desired
        hand into something like:
        Draw 7 cards out of 60 containing ( (A or B or C) AND (D or E or F) ) where
        number of A=4, B=3, C=1, D=2, E=4, F=4
        ,do you think its possible to calculate the probabilities in
        a meaningful way?

      • Ahj911211

        Yes, it would be similar to the calculation I made up there if you’re just looking for AT LEAST ONE of the cards in each group (or alternatively, the probability of only one from each group). And this calculation would not include the drawn card on the first turn or mulligans?

      • Ahj911211

        Here’s what it would be if you’re looking for the probability of one or more from both group A and group B

        X=A+B+C
        Y=D+E+F

        P(one or more first group, one or more second group)= 1 – [((60-X) nCr 7)+ ((60-Y) nCr 7) - ((60-X-Y) nCr 7)] / [60 nCr 7]

      • Anonymous

        That looks good.

        It isn’t too hard to accommodate mulligans. its only the Numerator that is affected.

      • Ahj911211

        Here’s the start of the game formula, so D=60 and H=7, with the mulligan factored in.

        X= no. of cards in group 1
        Y= no. of cards in group 2

        P(one or more first group, one or more second group)= 1 – [ [((60-X) nCr 7)+ ((60-Y) nCr 7) - ((60-X-Y) nCr 7)] / [60 nCr 7] ] – [ [((60-X) nCr 7)+ ((60-Y) nCr 7) - ((60-X-Y) nCr 7)] / [60-B nCr 7] ]

        where B is the number of basics in the deck.

      • Anonymous

        I knew there was a reason for using simulations :D. had me worried though for a while as I thought I’d missed a simple and precise formula that also represented the real world where mulligans do occur. For the most part the denominator to allow for mulligans changes to [ (60 nCr 7) - ((60-B) nCr 7) ] which isn’t painful.

      • Anonymous

        the ands and or get very awkward quite quickly. It is much better to allow multiple shapes and players fill in the minimum of each card that they want. For most cards this means either a don’t care or >=1 as the criteria for a hand you are interested in.

        multiple shapes are handled with multiple lines. with each shape getting a count but also setting a flag as a good start. Once all shapes are tested you just look at the good start flag to increment a total success count.

        handling mulligans requires valid starters to be identified. As some of these will be interesting cards that are part of the shapes there is a need for a count of the other starters(basics).

        so now you draw your opening hand then check for muligans. Then the T1 draw cards are added. As it is a simulation you can allow for more than a single draw card to represent a typical T1.

        ===

        yes you can calculate the probabilites but as you can see from Ahj911211′s first reply the number of calculations for just two different cards and a very simple this and that opening hand is high.

      • Michael Peter Sison

        My nose is bleeding. :| LOL. Anyway, you might want to write an article about that. That seems a pretty good topic for discussion.

      • Ahj911211

        I calculated it strictly as a 7-card hand, no mulligans or first draw considered. If a tool were to be formed for such a calculation, as you said, both factors would have to be considered. The cards you have access to the first turn and things like that would be too complex to analyze with probability, so either the model has to exclude that or it has to be run strictly through simulation.

  • Anonymous

    “If you’re going to buy booster packs, you might want to bring a micrometre caliper on your next trip to your local card store.”

    LOL

  • http://profiles.yahoo.com/u/ZETUB3TM3ZC4KJULZ4PYPEJR7I Tony

    After reading the article and some of the posts, I estimate the probability of those of us who liked this article to be nerds is 100%. Kinda reminds me of all of those Star Trek episodes when Captain Kirk would ask Spock the probability of survival. Somehow they always beat the odds. Great article, and a fun topic!

  • http://profiles.google.com/atiu.94 atiu.94

    Last week I bought a HGSS booster pack and got one half of Lugia LEGEND. Came back to the same store after about a week and bought another. I got the other half.

  • Brad Snyder

    Does anyone know if reverse holo foils are thicker than reverse holo commons/uncommons/non-foil rares?