Using a hypergeometric probability calculator to inform MTG cube design
Foreword: If you are familiar with what a Magic: the Gathering cube is, you can skip this foreword. Magic: the Gathering is a collectible card gaming system with over 25 years worth of cards with which to play. One way to play is to sit at a table with seven others, open packs of 15 cards (one at a time), taking one card from each and passing the remainder to the left or right until there are none left. Retail (using the booster packs sold to distribute cards) Magic is designed to be played this way, but the style of play also lives in the form of cubing. A cube is simply a collection of cards put together for the purpose of drafting, and the customization of which cards to include is a varied, debated & puzzle unto itself. This article explores some of the considerations one might make when doing so.
Ah, the hypergeometric probability distribution. Is there anything you cannot do for Magic players? Frank Karsten has written a great introduction if you are unfamiliar, but its most basic use is for magicians is to determine how likely one is to draw a certain card or any cards that match a criteria in either the opening hand or after seeing some number of cards from the deck (by turn X, for example). We’ll be using this StatTrek calculator.
Let us start by considering an aggro deck, which certainly would love to have a 1-drop in the opener to begin the game. If we had nine 1-drops in our 40 card draft deck, we can see the odds of having at least one in our opener:
85% to have one or more 1-drops if we have nine in the deck is pretty good! If we’re on the draw, we get an 8th card and we can see that jumps up to almost 90%. If we want to have aggro be viable in our cube, and this is a benchmark we want to shoot for, how many (assuming monocolored) 1-drops do we need to run in a given aggro color?
This is another problem the hypergeometric calculator can answer for us!
If our cube has 450 cards and we sit down to draft with the typical configuration of eight players opening three 15 card boosters (a total of 360 cards seen in the draft), the probability of getting nine or more 1-drops from a total of 13 in the entire cube would be 90%. That’s good news for the aggro drafters.
Now, this is an overestimation of some amount because one player does not have access to all 360 cards opened in the draft, firstly because other people will pick cards and might take of the 1-drops we want and secondly because we might want to pick a different card over any 1-drops we do happen to see. However, this does give us an idea for what kind of critical mass we need to properly support a good aggressive deck and have it reliably drafted.
This kind of calculation works best for highly parasitic archetypes, since this probabilistic estimation leans on the assumption that no one else wants the cards we want, which is generally true for many aggressive 1-drops. For other classes of cards, such as removal, that many drafters will want, this calculation becomes less meaningful.
Another parasitic archetype in many cubes is the Upheaval deck, which wants either the titular card or one of the many variants. How many of these variants do we need in our cube to achieve a 90% chance of the deck having access to at least two during a draft?
With three Upheaval-like cards, our drafter can expect to find at least two for their deck if the draft goes perfectly. A rudimentary estimation for the odds of seeing one of the two copies by turn six would be to simply add five or six cards seen on top of our opening hand (of seven) and spin the calculator…
55% to draw our important spell by turn six is not great, which brings me to a final consideration for cube design: cantrips. Magic has no shortage of cantrips and card selection, and any blue deck benefits from having many to help look for key cards and hit land drops. If we cast three cantrips that let us select between two cards each, we’re going to be able to ensure we hit our land drops, play timely mana rocks, and most importantly, find one of those combo pieces! If we run the calculation again, adding six cards from the card selection we had access to, the odds become a healthier 73%.
So, design with confidence and don’t lead your drafters into trap archetypes with sub-optimal support for their decks with the hypergeometric calculator.
