If Timmy has a jar with 30 marbles in it – 4 white, 6 blue, 8 black, 7 red, and 5 green – what are the odds of him pulling a mythic red marble?
As Mirrodin Besieged was spoiled and some cards seemed more interesting than others, I had a choice – do I buy a box or acquire singles? I chose the latter, going to a prerelease ($25), drafting after I dropped ($15), and buying about $110 worth of singles (surprise…about $110), and I’m glad I did. Doing the math, if you’re a casual player who can afford around a box at every set release, it’s better to spend that much in singles, at least of a small set. And binomial distributions helped me get there.
Put simply, binomial distributions the big league version of what you were doing in school when they told you how many marbles were in a jar, wanting to know the odds of drawing a specific color marble or something or other like that. (They taught you this because drawing marbles from jars competently is a skill we all use in our everyday lives.) If you use Excel, you can ask for the percentage odds of getting exactly X amount of a certain card given how many chances you get at it and the odds of getting a specific one. I think you can also find the odds of getting X or less with some tweaks, but I find it easier to just run it a couple of times at different numbers and sum up as I go along.
For example, there are 40 uncommons in Mirrodin Besieged. A booster box has 108 uncommons (36 packs x 3 uncommons per pack). So how many Go for the Throats should you expect from a box? If you want to run these yourself, you could go into Excel and put =BINOMDIST(1,108,1/40,FALSE), the first 1 asking for the odds of getting exactly 1 Go for the Throat in your box (or 1 of any specific uncommon). For Besieged, here are the uncommon odds for each number:
Everything else is under 1%.
So expect 1-4 of most of the uncommons. This is also the same for $1 uncommons like Signal Pest and Sphere of the Suns; these and Go for the Throat I’ll call power uncommons, since uncommons $1 and up tend to be the high demand ones, particularly as the supply dwindles over time. Generally, if you’re the type who gets a box and wants to trade for other parts, note here that Besieged will get you about $8 worth of these three cards.
Moving onto the rares and mythics…this gets a bit trickier. Let’s start with the simplest bit: how many mythics ought your box to contain? Again, this part is very important for those who buy boxes and trade the chase mythics later, because that’s where you tend to fill out playsets of things the box didn’t help you with.
If a mythic replaces 1/8 of the rares, what are the odds of mythic amounts in your box? With 36 chances and 1/8 odds…
This is a greater spread than I anticipated. I’ve seen boxes with 3, 4, and 5 mythics, but I have yet to hear of 2 or 6.
With 1/8 odds of a booster having a mythic and 10 mythics in Besieged, odds of getting a specific mythic would be 1/80. As with the rares, there are 36 chances to get the cards, but mythic odds are 1/80 instead of 1/35 with the rares. So here are your Tezzeret (or Thrun, or Jace if you’re opening Worldwake) odds:
Those are reasonable odds of getting a chase mythic, but they’re not great. Just to complete the value train, here are the rare odds from Besieged (with 35 rares):
So your odds are better here, but still nothing’s guaranteed. You’re in the 0-2 range per rare instead of the 0-1 range per mythic.
To sum up this math, in a Mirrodin Besieged booster box you have 108 chances at a 1/40 uncommon, 36 chances at a 1/35 rare, and 36 chances at a 1/80 mythic.
With these odds and the price list of your choice, you can create an expected value of a potential box, i.e. what a typical box would be worth. Why bother? Because it’s easy to dream. It’s easy to say that the box will pay for itself if it gets a Jace or a Tezzeret, but that crack-a-pack urge fools us into unjustified optimism. We tell triumphant stories about the good pulls and forget the bad pulls, never putting them into a bigger statistical picture. It’s part of the thrill and the adventure to do things that way, but it’s also the thing that can get us most in trouble with our parents, wives, bill collectors, etc.
The expected value of anything is the combined probabilities of the individual events in it – in other words, a weighted average. (You can look it up on Wikipedia; their initial explanation and formula are generally in layman’s terms.) We’ve already seen that, with 2-3 of each uncommon being the norm in a small set and 3 “power uncommons” (i.e. the high-priced, highly tradable cards) in the set, that you’ll get about $8 of value from the uncommons. Taking any card starting at $2 or higher as a “power rare” for now (this is Slagstorm and up), there are 8 of these in Besieged as ChannelFireball defines them. How many “power rares” can you expect in a Besieged box?
So if you’re a casual player wondering if a Besieged box is for you, you should expect
6-9 power uncommons;
6-10 power rares; and
to stand out from the normal stuff.
The power rares from Besieged are averaging about $5 and the mythics about $10. Straight down the middle, then, your box will give you floods of commons, random uncommons, junk rares, and $78 worth of stuff different people might want from you. That’s a reasonable value, and I wouldn’t fault you for buying a Besieged box.
But what else is available? Well, ChannelFireball sells common/uncommon sets for $10 apiece (and sometimes common/uncommon playsets for $35). That would give you 12 power uncommons, which is quite reasonable. Compared to a box, that leaves you $64 (accounting for shipping) to get exactly the rares/mythics you want which, if you’re a casual player, probably suits you more anyway. You can buy playsets of junk rares for $2, or get a playset of Green Sun’s Zenith for $40 (and, let’s face it, it could take awhile to find enough people to trade you an eventual 4), or anything in between.
I did a hybrid of these things for Besieged, as noted at the beginning. I was interested only in a few rares, so here’s what I bought playsets of:
Green Sun’s Zenith $32 (I had one from a trade with a friend)
Glissa, the Traitor $20
Phyrexian Revoker $15 (Had one from the same trade)
Spine of Ish Sah $3.20
Mitotic Manipulation $2
Phyrexian Hydra $2
Creeping Corrosion $2
Knowledge Pool $0.75 (Had one from prerelease)
This covers almost everything I wanted, apart from Black Sun’s Zenith, which I’m hoping will go down in price at some point. But take a look at that rares list. If you subtract the Zeniths at the top of the list, those rares plus the playsets of commons/uncommons comes out to less than a box by $7. It’s basically the same amount of rares as a box, but it’s all playsets, so it’s stuff that I know I’ll use, rather than singleton rares I might use or hope to trade to someone to obtain playsets. I make Commander decks based on “what are good cards I only own 1 of?” rather than fine-tuning; I’m much more focused on obtaining playsets than single rares so I can build 60-card oddities.
Now why did I give all this math instead of just showing the deal I got? Two reasons:
1) It needed to be shown that you’re not likely to get playsets of uncommons out of a box, which can be very frustrating;
2) You can use this math to calculate the expected value of any box.
Case in point: Mtgmintcard.com sells on Ebay and their website several Chinese-language boxes of older sets at discounted prices. Torment is at $60, Legions is at $40, and Eventide is at $49. Are those good deals or not? I can use the math to figure it out:
- Check the set’s uncommon/rare/mythic breakdown;
- If you’re just looking for sheer acquisition and don’t expect to trade, calculate the odds of getting specific cards in various amounts and see if that suits you;
- If you’re looking for some of the expensive cards to add to your trade binder value or because you think it might be more economical to get them in a box rather than as singles, add those odds up separately from the chaff.
If you’re math-inclined, you know what all this means. If you’re not, I’m guessing you’ll just want the Excel formulas, so here’s the rundown:
In a booster box with 36 packs and therefore 108 uncommons and 36 rare/mythic slots…
The odds of getting exactly x of a specific uncommon from a set with y uncommons is
=BINOMDIST(x, 108, 1/y, FALSE)
The odds of getting exactly x of a specific rare from a set with y rares is
=BINOMDIST(x, 36, 1/y, FALSE)
The odds of getting exactly x of a specific mythic from a set with y mythics is
=BINOMDIST(x, 36, 1/[8y], FALSE)
If a set has a fixed amount of power uncommons, power rares, or power mythics, and you want to see how many you’re likely to get, substitute that amount for the 1 in the numerators above. I’d calculate power uncommons as $1 and power rares/mythics anywhere from $2 to $5 depending on how many trading partners you have, the age of your group’s card pool (my collection predates a lot of my playgroup by several years, so older sets would be more in demand to my pool than others), how much they play Commander (to want your 1-of rares from the box), and so forth. The more these factors swing in your favor, the lower the threshold for a power rare/mythic, because it’s less likely your trading partners have what you have.
For example, take the Chinese box of Legions mentioned earlier for $40 (plus shipping; the Ebay listings ship it for $7, while Mtgmintcard ships them for $22, which is quite a difference). Using Starcitygames.com for this (because they tend to have good stock of older cards and they have a search function that lets me filter prices), 10 of its 45 rares are at least $2. I can use $2 here because the minority of my playgroup played in Legions and there’s a fairly high demand for Commander cards among them. Using (x, 36, 10/45, FALSE), I should get 5 to 13 such rares, with 7, 8, and 9 being almost equally likely. For sake of argument, let’s say you get 9 power rares and one of them is worth $4; that’s $20 of tradable value.
There are 6 power uncommons in the set. Using (x, 108, 6/45, FALSE), you should get 10 to 18 from this group, which is fairly substantial. We’ll say you get $15 out of 15 uncommons. Thus, for your $40+shipping, you should get about $35 of tradable material, 18 junk rares, and uncommons and commons you can get from anyone for a pittance. If you’re a Commander player looking for Slivers, this might be a good deal for you. If you don’t have any Legions and therefore can use the filler, then that works too. Otherwise, you might break even, but there’s no great deal hidden in there.
What about Torment, a great set for power uncommons? The set has 44 rares and 44 uncommons. There are 8 power rares, though half of them are at least $4 and one of those is Grim Lavamancer, so that ticks their average value up. Similarly, there are 7 power uncommons, but only one’s in the $1 range; 4 are at $1.50, and Cabal Coffers is at $6. That makes for 3 to 9 rares and 13 to 21 uncommons. So on balance you’ll have $18 of easily tradable rares and $22 of tradable uncommons, though with 72% odds of at least 2 Cabal Coffers that can increase easily. Is that better than just buying a bunch of the power uncommons? It depends on how much of the set you already have.
Eventide is slightly larger than a normal small set (180 cards) but since it’s still in Extended there’s a higher demand for it generally, though not necessarily in my playgroup. There are 60 rares and 60 uncommons in the set, so the odds aren’t great per card, but there are a whopping 26 power rares, most of which are above $3. (Having dual lands always helps.) This would give you 12 to 19 power rares and around $45 of value. On the flipside, there are only 2 power uncommons, so it’s not even worth calculating that. (For the record, they’re Gilder Bairn and Quillspike.)
Each set is costed differently and has its value in different places, but you can use the same method to see if you’re getting a reasonable deal. Casual doesn’t equal budget, but there’s no point in acquiring cards en masse if le masse is worthless. If you figure out these things before you buy, your collection will fit your needs better, you won’t clutter your binders with useless items, and your Magic expenditures will be almost entirely free of waste. And who doesn’t want that outcome?