Bob is the type of player who always complains of mana flood, mana screw, or just plain bad luck. He'll sit across the table with a hand full of cards, lamenting about the pile of mountains on his side of the table. Each turn he rushes through the untap and upkeep steps, knowing he'll draw another mountain or a card he can't play.

To him, the game of Magic is all about luck. Losing is never his fault; his opponent drew better or he didn't draw well enough. When he finally draws that plains to help him cast his handful of cards, he slams it down in desperation because he's already lost.

This same cry - though the actual words may be different - can be heard from Magic to Texas Hold'em to baseball drafts. Luck. It's that mystical element that allows us to throw our hands up and walk away having taken no credit for our own actions (or given credit to those who best us).

Luck only comes into play when we're buying Lotto tickets. Once we've performed an action enough times with a reasonable chance of success, it becomes a statistic - and statistics can be analyzed, broken down and the results can be used to create an advantage.

Let's look at one simple application of statistics to improve an aspect of Limited that is often fudged up - land-to-spell ratios. Land drops have a very important role in a game of Magic: Cards you can't cast because of a lack of mana or the right color mana is a form of card disadvantage.

The first example should be for a basic, single-color Limited deck. Decks should be forty cards with seventeen land. To calculate all the odds, I've created an Excel spreadsheet so I can just put in the deck total, the number of cards in the deck that I'm looking for, and the break percentage (the probability that I'll see that card) and it spits out a whole host of numbers and even a turn-by-turn chart telling me based on the break percentage when I'll see that card. I'll go over the calculations for this small example and then I'll leave out the statistics, for the most part.

The formula to find the probability of drawing a hand (H) with a particular card or cards (X) out of a certain deck size (D) when there are (Y) of that card in the deck is:

P(X,(H-X)) = (C(Y,X) x C(D-Y,H-X)) / C(D,H)

So if we have a deck of forty cards with seventeen lands in them and we want to know our chance to draw two lands in our opening draw of seven cards (not less or more - only two), then our formula would look like:

P(2,5) = (C(17,2) x C(23,5)) / C(40,7)

P(2,5) = 24.6% chance of drawing only two land in the opening draw.

If you want to know the chances of drawing up to a certain number of cards, then just add all those probabilities before it. If we wanted to know what the chances of drawing up to two land in the opening draw, then the formula would look like:

P(0,7) + P(1,6) + P(2,5) = 1.3% + 9.2% + 24.5% = 35.0%

If you want to look at the chance of having three or more land drawn by turn five, just add four cards to the total:

P(3,8) + P(4,7) + P(5,6).....+ P(11,0) = 79.8%

All of this is fairly easy statistics, though it's best done through a spreadsheet program - but I wanted to clarify the method in case anyone wanted to verify the results.

So what decisions can we make based on our seventeen-land count? The first is in our opening draw on the decision of when to mulligan or not. If we only draw zero or one land, then we'll definitely mulligan. The chances for those are:

Zero or one-land draw with seventeen land - 10.5%

But what if we draw two land? Should we mulligan? In the OLS drafts, the magic number is three. Three is the number of morph and the number shall be three.

Once we've drawn two land, we might have a 2cc creature that has a power of two (to kill morphs) that we might keep. However, if we don't have a 2cc creature or removal spell with a power of two, then we need to look at what our chances are of drawing at least one more land by turn 3, when we'd need to drop that next land. The result is a 71% chance. Compare that to mulliganing down to six cards, which gives us a 76% chance of drawing the third land by turn 3. Considering that we lose a card for only a 5% increased chance, I would not mulligan and hope I drew another land.

Unfortunately, drafting single-color decks is a rarity. The standard draft deck is usually two colors with the occasional splash of a third, though splashing a third color occurs more often in Sealed. Sometimes the toughest question once we've sat down to assemble our draft deck is how many of each land should I put in. As an example, let's look at a Team Sealed deck that Tybuc posted and go through the possibilities:

Boneknitter
Nantuko Husk
Gustcloak Harrier
Whipcorder
Soul Collector
Severed Legion
Daunting Defender
Prowling Pangolin
Aven Soulgazer
Zealous Inquisitor
Nantuko Husk
Smokespew Invoker
Catapult Squad
Grassland Crusader
Battlefield Medic
Whipgrass Entangler
Daru Sanctifier
Clutch of Undeath
Putrid Raptor
Liege of the Axe
Guilty Conscience
Daru Healer
2 Secluded Steppe
Barren Moor
Unholy Grotto
6 Plains
8 Swamps

The most effective way to analyze the deck for mana would be to look at all possible bad combinations - defined as possible ways to draw out an opening seven and not being able to play out a card - and then look at the bad combinations for each successive turn. A total turn-by-turn value could be established, showing only the"castability" of your deck; however, that's a monumental task. Instead, we can take a look at turn 3, which we've established as the magic turn for OLS Limited.

There are 273,438,880 possible draws for the first nine cards. There are 32,346,512 ways to not draw three land by turn 3. There are zero possible ways to draw three or more land and not draw a 3cc or less spell (which is a nice feature of the deck, but it might also mean the deck is underpowered in that it's wasting later mana).

(Zero possible ways? You can't draw ten land in a row? - The Ferrett, confused, but realizing that he may mean a post-mulligan statistics)

Then there are a whole host of ways to draw double-mana spells and not draw the right mana. There are two 2cc drops that are capable of creating a stable board position (the Catapult Squad is a 2/1 and the regenerating Boneknitter can stop the first morph on the board) so those possibilities are added in. The resulting number after about an hour of number crunching (using an Excel spreadsheet) gives us the probability of not having a drop by turn 3 at 22%.

Interestingly enough, of that 22%, 12% of it comes from land drops (namely, not having enough land to make a 3cc drop by turn 3). The another 8% comes from drawing a double-colored spell and not having two of the same land to cast the spell. The final 2% comes from the probability of drawing single-mana spells but only drawing the other color mana. Given that you will need to play between six and nine games to win a draft pod, a 22% chance nearly guarantees us that we'll miss a few turn 3 creature drops - which, when playing a deck that relies on tempo to win, is not a good strategy. Or, compared to a deck that had no double-mana 3cc drops (only a 14% bad draw probability) there is a large difference in reliability (an increase in games played only intensifies the difference between a 14% and 22% probability).

So what is a proper amount of mana to support those 3cc two mana drops? There is a lot of difficulty in doing all the calculations for every variation of bad combinations - especially if we want a turn-by-turn view of it. Instead another method is to only look at the mana base alone rather than including the chance of actually drawing the spell you want to cast. The method here is to calculate the probability of having two lands by turn 3 based on the number of lands of that are in the deck.

Using nine land of one color, we'd see these probabilities:

Turn 1 - 51%
Turn 2 - 59%
Turn 3 - 67%

This is not a stable mana base for double-colored 3cc creatures, since we'll be playing up to nine games with this deck. Using twelve lands of one color, we'd see these probabilities:

Turn 1 - 69%
Turn 2 - 78%
Turn 3 - 83%

So to create a stable condition for our two-mana 3cc creatures, we should use around twelve land. Any deviation from that land count will cause a more unstable mana base.

So what have we learned about some of the basic principles of statistics when applied to mana use for Limited?

• Mulliganing a two-land hand is not a good play.
• Using low-cost, double-mana creatures requires a bigger commitment to that color, manawise.
• Using high-cost, double-mana creatures allows more flexibility in your lands.
• The value of luck in Magic is grossly overrated.

This principles, though simple, can become very complex. If you'd like to learn how to calculate C(X,Y) and P(X,Y), then I would suggest mathforum.org to get some help on it.

Tom Carpenter
carpe@centurytel.net