Magic the Gathering: Pack Cracking Statistics (Oath of the Gatewatch)

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In addition to writing actuarial exams, I am a massive nerd when it comes to the game Magic: The Gathering. One thing that has always irked my statistician side about M:TG journalism is that when it comes to statistics, the only thing ever talked about is the expected value of opening packs of cards. I’m here to set that straight by demonstrating an even more important factor, variance.

Hold up, what is M:TG?

Magic: the Gathering is a trading card game that began in 1993 and has since exploded in popularity to become the most popular physical card game on the planet. If you have never played before I encourage you to find a friend that does (might be easier than you think!) and get them to teach you the basics. The game has something to offer for all skill levels and is a fantastic social activity.

The Basics of Pack Cracking

Each regular pack of Magic cards contain 15 cards distributed over the four main rarities: Common, Uncommon, Rare and Mythic. A normal pack will contain 10 Commons, 3 Uncommons, 1 Rare OR 1 Mythic and 1 “Land” card. There is also a chance that there will be a “premium” foil card taking the place of one common. These probabilities are as follows:

  • Chance of a Mythic Rare: 1 in 8 packs, or 12.5%
  • Chance of a Foil card (any rarity): 1 in 6 packs, or 16.7%

If you buy a “box” of cards, you get 36 packs. From this we can calculate the expected number (the Expected Value) of Mythics and Foils we get in our box:

  • Expected Mythics: 0.125*36 = 4.5
  • Expected Foils: 0.1666*36 = 6

Most people will stop here with the calculations. I’m going to go a step further and show you some other ways to look at what we expect from our boxes of cards.

The Simple Model

I will be using “RStudio” to complete the math for the rest of this article, feel free to skip over the coding. I’m adding it in for anyone who would like to reproduce my results.

When we open a pack of cards, we are actually performing a Bernoulli Trial. Put simply this says we have two possible outcomes: a “Rare” or a “Mythic”. Using the probability from above, let’s simulate cracking an entire box of cards:

p.mythic <- c(1/8)

# Simulate a box (36 packs) for Rares
packs <- rbinom(36,1,1-p.mythic) > results
[1] “Rare”   “Rare”   “Mythic” “Rare”   “Rare”   “Rare”   “Rare”   “Rare”   “Rare”   “Mythic” “Rare”
[12] “Rare”   “Rare”   “Rare”   “Rare”   “Rare”   “Rare”   “Mythic” “Rare”   “Rare”   “Rare”   “Rare”
[23] “Rare”   “Rare”   “Rare”   “Rare”   “Rare”   “Rare”   “Rare”   “Rare”   “Rare”   “Rare”   “Rare”
[34] “Rare”   “Mythic” “Rare”

In a nicer form:

results
Mythic   Rare
     4     32

Congrats! You got four mythics in your box. We can do the same thing to simulate our foils:

results2
   Foil No Foil
      4      32

Hmm, looks like we came up with a disappointing 4 foils (we expected 6). Ask yourself, why did this happen? You’re right, it’s variance.

Simulating 100,000 Boxes

To visualize how the distribution of how many mythics you get will change, we have to repeat this experiment many times. Many, many times. For those interested, check out The Law of Large Numbers. Let’s take a look at the results of this simulation:

 boxes.mythics <- rbinom(10000,36,p.mythic) hist(boxes.mythics, col = “lightblue”, xlab =“Number of Mythics”, ylab = “Number of Boxes”, main = “Distribution of Mythics in Boxes”) 

mythics
We can do a similar analysis for foils:

Foils

I think this is a pretty great visualization as we can see that even though we “Expect” to get 6 foils, we can clearly see that it’s actually more likely that we get a number other than 6. Some poor soul may only get 0-2 foils, where some lucky duck will walk away with 8 or more! Sure, we “knew” this already, but this proves it.

In fact, we can calculate the exact probability of getting a specified number of Mythics or Foils using a Binomial DistributionThe binomial distribution results from performing consecutive, independent Bernoulli trials. The resulting chart is depicted here:

chances

The Complicated Model

In the most recent Magic expansion, Oath of the Gatewatch, there are 70 commons, 60 uncommons, 42 rares and 12 mythics for a total of 184 cards. There is also a very small chance of opening an Expedition Land, which command a high price. I will be using the same assumptions as above, with the following additional foil assumptions:

  • Commons and uncommons are equally distributed over the foils
  • The chance of your foil being rare or mythic is equal to their proportion of over all cards (52/184), and then the card has ~4 times chance of being a Foil Rare over a Foil Mythic
  • Expeditions will be treated the same as the calculated value for Foil Mythic Rares (the article states it is slightly less rare, but I have no way to make an educated guess as to how much and it is a very small amount anyway)

I know what some of you are saying. There is an often repeated assumption that a Foil Mythic is a 1:216 pack occurrence and that’s that. I scoured the internet to find a source and everything seems to back to a single article from StarCity who themselves source a website that is now offline. Additionally, those numbers are for a Large set (200+ cards) whereas Oath is a small sized set. From my numbers below I hope you can agree that these seem like reasonable jumps.

When we open a pack, we have multiple different possibilities:

  • Rare or Mythic
  • No Foil, Common Foil, Uncommon Foil, Rare Foil, Mythic Foil
  • No Expedition, Expedition

And any combination of the three. If you trust my math, here are the probabilities for each of the combinations:

tablecards2

What we have here is a Multinomial DistributionJust like before, we can simulate boxes. Let’s open 10 boxes and see what we get:

                                   [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
Rare|No Foil|No Expedition           25   29   24   26   29   30   24   22   26    29
Mythic|No Foil|No Expedition          4    3    2    4    4    2    6    7    2     4
Rare|Common Foil|No Expedition        2    2    4    3    1    2    2    1    4     1
Rare|Uncommon Foil|No Expedition      4    0    1    0    0    1    2    2    1     0
Rare|Rare Foil|No Expedition          1    0    3    1    1    0    1    1    0     2
Mythic|Common Foil|No Expedition      0    0    0    1    1    0    1    3    1     0
Mythic|Uncommon Foil|No Expedition    0    1    1    0    0    0    0    0    1     0
Rare|Mythic Foil|No Expedition        0    0    1    1    0    1    0    0    0     0
Rare|No Foil|Expedition               0    1    0    0    0    0    0    0    1     0
Mythic|Rare Foil|No Expedition        0    0    0    0    0    0    0    0    0     0
Mythic|Mythic Foil|No Expedition      0    0    0    0    0    0    0    0    0     0
Mythic|No Foil|Expedition             0    0    0    0    0    0    0    0    0     0
Rare|Common Foil|Expedition           0    0    0    0    0    0    0    0    0     0
Rare|Uncommon Foil|Expedition         0    0    0    0    0    0    0    0    0     0
Rare|Rare Foil|Expedition             0    0    0    0    0    0    0    0    0     0
Mythic|Common Foil|Expedition         0    0    0    0    0    0    0    0    0     0
Mythic|Uncommon Foil|Expedition       0    0    0    0    0    0    0    0    0     0
Rare|Mythic Foil|Expedition           0    0    0    0    0    0    0    0    0     0
Mythic|Rare Foil|Expedition           0    0    0    0    0    0    0    0    0     0
Mythic|Mythic Foil|Expedition         0    0    0    0    0    0    0    0    0     0

Two Expeditions in 10 boxes. I hope they were good ones.

I plan on doing more Magic related posts in the future! If you liked this post and want to see more, message me your ideas on Reddit /u/NaturalBlogarithm and be sure to follow me @NatBlogarithm on Twitter so you don’t miss out.

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