MDFC Hyperdrive

(Read when you have time... like... a lot of it)

The long-awaited (ok long, not necessarily awaited) third in a series of pseudo-articles has arrived. And, spoilers, it’s basically just an excuse to rant about a pet peeve of mine and pretend to justify it with some shoddy math (hey, my doctorate is in witchcraft, not mathematics).

And if you want to skip this article, just memorize the following chant and recite it on the next new moon: “MDFCs do not use land slots.”

The Tools

For this ritual, the main tool we will use is called a Hypergeometric Calculator. This is a tool used to find the Hypergeometric Distribution. Wikipedia describes this as:

In probability theory and statistics, the hypergeometric distribution is a discrete probability distribution that describes the probability of k successes (random draws for which the object drawn has a specified feature) in n draws, without replacement, from a finite population of size N that contains exactly K objects with that feature, wherein each draw is either a success or a failure. In contrast, the binomial distribution describes the probability of k successes in n draws with replacement.

Alright, allow me to make that a little simpler. Basically, if you have a stack of cards and every time you draw from them you don’t replace them, it lets you find out how likely you are to have drawn a specific amount of something at a given point. For example, if you want 4 lands by turn 4, you could figure out what the probability of that is.

Let’s get Familiar

So for those math majors who have never used a hyperdrive collider before, let’s put in a few examples. First, let's note what our input variables will be:

• Population size: This is the number of cards in the deck. 99 (or 98 if you have a partner)

• Number of successes in population: How many that you have in the deck.

• Sample size: How many cards you get to see. For example, 7 cards in your opening hand, 8 cards on your first turn, etc.

• Number of successes in sample: How many that you're hoping to draw.

So, for example, if you ran 40 lands in your EDH deck and wanted to see 3 of them by turn 3, you'd enter:

• Population size: 99

• Number of successes in population: 40

• Sample size: 10 (7 card opener, +3 draws)

• Number of successes in sample: 3

Then, our hypermagnetic cultivator gives us the following output:

• Hypergeometric Probability P(X = 3): 0.216348692

The way that we read this is by shifting the decimal over two places, so our chance of having EXACTLY THREE lands in our opening hand + 3 draws is 21.6348692%, or simply truncated to 21.6%. But wait, we don't want exactly three, per se, we want AT LEAST three, so to figure that out, we will find another output from the list: Cumulative Probability: P(X ≥ 3).

• Cumulative Probability: P(X ≥ 3): 0.852683621

Again, we would read this as having an 85.2683621% chance, or simply an 85.2% chance (again, truncated). That's a pretty good chance, which is why everyone plays landfall (an archetype where you aren’t punished for running nearly 40 lands). To increase the turn count, simply re-enter the same numbers, but edit the sample size (to represent the total number of cards drawn.)

The number goes down as we expect more land cards. For example, on turn 10 we've seen 17 cards, so if we want to play a land drop every turn for 10 turns, we have to set the "Number of successes in sample (x)" to 10 as well. We'll find our chance of hitting land drops every turn for ten turns even in this 40 land deck is only 0.077472201 or 7.7%.

Of course, that's not the other way to mess with the numbers: we can also increase the sample size. In this case, that means increasing the number of cards that we've drawn per turn. Let's say I want to reach that 10 lands by turn 10 metric (I don't know - I'm playing Progenitus and my game plan is to get there with one land drop per turn), and I want to do so more than 50% of the time. Fudging around with the sample size can tell me how to do it. After entering some numbers and seeing what I can get, I found that the jump over 50% happens in-between sample size 23 (45.6%) and sample size 24 (53.4%), which means I'll have to aim to draw 24 cards over the course of those ten turns. Since I'm already drawing 17 (starting hand of 7 + 1 per turn), then I only have to draw 7 additional cards. Over 10 turns, that's less than one extra card a turn. That's right!! 7 additional cards bump it from a 7.7% chance to a 53.4% chance - that should be a testament to the power of card draw!

Caveat: Some might argue that you actually have 14 cards in your opening hand in EDH (7 cards + 1 free mulligan), but the math gets weirder since, in order to do that math, we'd have to do a binomial distribution for the first round of cards (or any number of other cards we add via "mulligan"), followed by a geometric distribution for the second 7. If you want to do that, go ahead spend your time with that. I find it to not be as important... just remember that if you keep a hand with 2 lands, your chance of getting to 3 on turn 3 is a lot greater than if you keep a hand with 1 land - and don't use this math as an excuse to keep bad hands.

What does this have to do with MFDCs?

Well, it's simple. The power of an MDFC is that it's a relevant spell that can let you pretend to have an extra land card in your deck. For example, if your deck is running 35 lands and an MDFC, and you want at least 4 lands by turn 4... then you can use the above math to see that the cumulative probability (using 35 as the successes in population) is about 59.2%. If you do the same math counting the MDFC as a land, (using 36 as the successes in population), the odds are 62.0% - a near 3% better.

Similarly, if we have 36 lands and one MDFC... our odds just counting lands (36) to counting the MDFC (37), jump from 62% to 64.8% - another near 3% better. Going up to 38 bumps it to 67.5%. So what am I ranting about? In my opinion, people are using MDFCs wrong when they cut land slots for MDFCs... they are not taking a decent mana base and fudging the math in their favor, instead they're replacing untapped lands (like the most powerful land in the game) with slower worse lands that they rarely have the intention of casting on the spell side. If you’re using it in a land slot then, statistically, you’re intending on using the card as a slower, worse land that will, occasionally, be a spell. Even the shocks (though there is more of an argument to be made for them) are worse than basics - lacking land types for stuff like coffers and requiring you to pay life for 0 upside in that game (which, again, is how you’re statistically intending to play them if you’re using them in a land slot). The power of an MDFC is that it's a relevant spell that can let you pretend to have an extra land card.

(In hindsight maybe I should've stuck with "3 lands on turn 3" (the numbers I used in the previous stanza) for clarity - but I'll just trust you to keep in mind that the more lands you're asking for causes the likelihood to go down. Something else you might notice is that the further you decrease/increase your land count, the bigger jumps you're likely to make for P(X ≥ x), it's not a linear 3%.)

What about 7x9?

For those of you who read my article about how to build a deck, you’ll know I’m a pretty big fan of using the 7x9 method… but flipping it a little bit. Instead of using 7 cards in 9 categories, I use 9 cards in 7 categories, because usually a 60 card deck would run more than a 4 of if it has the chance and also because an EDH deck trying to do 9 different things is too split compared to an EDH deck trying to do just 7 or fewer things. Of course, this is just a starting point, but let’s plug it into the Hypergalactic Conductor to see what (s)he has to say!

Let’s start with the slim model, if you run 7 interaction pieces, then the chances you have of having at least one by turn X are:

• Turn 0: 41.1%

• Turn 1: 45.6%

• Turn 2: 48.9%

• Turn 3: 53.7% (Turn 3 your odds are slightly better than a coin flip)

• Turn 4: 57.3%

Next, my proposed method, if you run 9 interaction pieces, then the chances you have of having at least one by turn X are:

• Turn 0: 48.5% (Almost better than a coin flip for it being in your opening hand)

• Turn 1: 54.7% (Almost the same odds as turn 3 with just 7)

• Turn 2: 59.2% (Better than turn 4 odds with just 7!)

• Turn 3: 63.2%

• Turn 4: 66.9%

Many of my decks have a "double feature" category, i.e. a category that takes up two category slots. If I am double featuring interaction (18 pieces of interaction in the deck), for example:

• Turn 0: 76.6%

• Turn 1: 81.2%

• Turn 2: 84.9%

• Turn 3: 87.9%

• Turn 4: 90.3% (That's pretty good!)

Of course, the odds go up or down depending on exactly what you're asking for and remember that tidbit from earlier - drawing more cards increases your sample size without increasing the turn counter. (For example, running 9 interaction pieces, your turn 2 odds are 59.2%. But if you draw two extra cards - e.g. Night's Whisper - your odds go up to what it says for turn 4: 66.9% even though it's only turn 2).

Naturally, you can apply the same technique to any type of card - a testament to the power of sorting your cards into categories by function, allowing you to easily plug in these numbers (or estimate once you get used to it).

Further Tinkering

Thanks to /u/632146P, who posted a comment featuring another Hypertonic Converter that's fun to play around with because it has a nice graph you can look at to see what possible hands really look like (and also makes it easier to see if you might flood, without having to do much math). It does make you re-enter x all the time which is frustrating.

Another friend of mine gave me this code for a multivariable version of the HGDC which you can save as a HTML file and play around with.

There’s a lot of stuff you can tinker around with on any of these great calculators. One example is conditional probability. For example, if you run 36 lands and 12 of them are black sources, you can plug 36 as the population size, 12 as the successes in population, 3 as the sample size, and 2 as successes in sample… and find you have X chance of being able to cast a double-black-pipped spell IF you’ve drawn 4 lands.

Edit: The best Hyphen-Comma for MTG I’ve found to date is here.

Credits

Thank you for wasting your time with my article about the Hyper Cruncher. All proceeds go. I’d like to thank the universe for spawning me, kicking and screaming, into existence. My daily tarot today was The Star, in the reverse position.

Happy Halloween.