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That First Hand

This article is about your first hand, and gives hard data on how many lands you want to run in your 60 card deck.

Your first hand in Magic: the Gathering is important. It's 7 cards all at once, an effect not seen on too many cards. Sure, a great first hand can be crippled by bad draws, but those happen one at a time and the game can change a lot from draw to draw.

One of the most important things about you opening hand is the number of land (or, more generically, low/no cost mana sources). To cast virtually any spell, you'll need mana, which is generally gained by playing and tapping lands.

How many lands do you need in your opening hand? It depends on your "curve". Detailed curve analysis is beyond me, right now, although just the act of arranging your deck my casting cost can give you some intuition. Play testing is best.

Once you know how many lands you want in your opening hand, how many lands do you want to play in your deck? Sure, play testing can give you some data here too. But, play testing has the disadvantage of testing only one opening hand at a time. Even if you are just dealing out 7-10 cards, shuffling, and repeating to get an idea you are constantly struggling against the fact that the number of possible arrangements for your deck is HUGE.

How do I like to deal with large numbers? Math. In this case, some probability / combinatorial math. Let's consider all possible draws from your deck at once, and work out the chance you get a certain number of lands.

Assume a 60 card deck. The math is the same for any size deck, but for examples, it helps to have specific numbers. 60 is the minimum number of cards you can have in your deck in a constructed event, and most players shy away from running more since it makes the deck less consistent. If "less consistent" means "there are more ways your deck can be randomized" then that statement is well-supported by the math, although it also depends on the card counts in your deck. For example, running 4 copies of 9 different cards has MANY fewer arrangements than running 3 copies of 12 different cards (about 1200 TIMES fewer). That's a different article though.

Assume you are running 24 lands. The chance of each card being a land is 24/60 or 2/5. However, each card you draw changes this number. If the first card is a land, the next card only has a 23/59 chance of being a land. If the first card is not a land, the next card has a 24/59 chance. So, the chance of land, land, land, non-land, non-land, non-land, non-land (LLLNNNN) might be different than NLNLNLN, right?

Turns out it isn't. LLLNNNN has a chance of 24/60 * 23/59 * 22/58 *36/57 * 35/56 * 34/55 * 33/54 = 24 * 23 * 22 * 36 * 35 * 34 * 33 / 60 * 59 * 58 * 57 * 56 * 55 * 54. NLNLNLN has a chance of 36/60 * 24/59 * 35/58 * 23/57 * 34/56 * 22/55 * 33/54 = 35 * 24 * 35 * 23 * 34 * 22 * 33 / 60 * 59 * 58 * 57 * 56 * 55 = 24 * 23 * 22 * 36 * 35 * 34 *33 / 60 * 59 * 58 * 57 * 56 * 55.

Ug. Those numbers are big. Too long to type. Time for some notation. Let's define the "pick" function, by example P(3,24) = 24 * 23 * 22. That's the number of ways to pick 3 things, one at a time, from a pool of 24 things with no replacement. P(4, 36) = 36 * 35 * 34 * 33. Now, we can write the chance of drawing LLLNNNN or NLNLNLN as our first hand as P(3,24) * P(4,24) / P(7,60). Pick 3 of 24 lands and 4 of 36 non-lands out of pick 7 of 60 cards. Make sense?

So, we now know the chance of drawing some specific order involving lands and non-lands. We can apply this of any arrangement. In that 24-land, 60-card deck, we draw seven lands (LLLLLLL) in our opening hand exactly P(7,24) / P(7,60) or less than 1% of the time.

Also, we know that LLLNNNN and NLNLNLN have the same probability, but how many ways can we arrange those seven letters so that we get 3 Ls and 4 Ns? Well, if they were all different, we'd have P(7,7) ways. That would count L2 L1 L3 N1 N2 N3 N4 as different from L1 L2 L3 N3 N2 N1 N4. Well, if we divide by the ways to pick the lands P(3,3), and the ways to pick the non-lands P(4,4) we eliminate all the duplicates. P(7,7) / (P(3,3) * P(4,4)) is a little bit long. Let's call this the "choose" function. C(k, n) = P(n,n) / (P(k,k) * P(n-k, n-k)).

So, if you have a deck with "C" cards in it and "L" of those are lands and your open hand is "D" cards, the chance of you getting "N" lands is simply: P(N, L) * P(D-N, C-L) * C(N, D) / P(D, C).

Here's the Haskell functions:

permutations n k | k  n = 0
                 | otherwise = product [n-k+1..n]
combinations n k = (permutations n k) `quot` (product [2..k])
chance_of_drawing c l d n = (permutations l n)
                          * (permutations (c-l) (d-n)) 
                          * (combinations d n) 
                          Ratio.% (permutations c d)

Haskell is a nice functional programming language with a freely available standard. That doesn't matter so much for this purpose. What we need is the ability to deal with truely large integers and ratios of integers easily. A lot of scripting and interactive languages provide the means to do that. These features avoid having to deal with overflow or accumulated rounding errors. Haskel also lets us check our math fairly quickly. First the chance we draw 8 lands from 7 cards (should be zero):

Prelude> chance_of_drawing 60 24 7 8
0 % 1

"0 % 1" is how Haskel represents a Ratio of integers that is 0 / 1 or 0. So, that worked.
Lets check out chanced to draw between 0 and 7 lands, inclusive (should be 1).

Prelude> sum [chance_of_drawing 60 24 7 n | n 

"1 % 1" is of course 1 / 1 or 1. So another success. Some other tests might be good, too:

Prelude> chance_of_drawing 7 7 7 7
1 % 1
Prelude> chance_of_drawing 400 0 7 1
0 % 1
Prelude> chance_of_drawing 2 1 1 1
1 % 2
Prelude> chance_of_drawing 2 1 1 0
1 % 2

Those make good sense.

What's the chance of a bad hand (either less than 2 or more than 4 lands) with 25 land in a 60 card deck?

Prelude> sum [chance_of_drawing 60 25 7 k | k 

(That's 22.2%.) Can we get any lower?

Prelude> (foldl1 min [
sum [chance_of_drawing 60 l 7 k | k 

(That would be a "no".)

Here's a nice table of probabilities, if you don't want to fiddle with Haskell or another similar language. Chances are rounded to the nearest 0.1%.

Chance Lands in 7 card hard
0 1 2 3 4 5 6 7
Lands in Deck 16 9.9% 29.2% 33.7% 19.7% 6.2% 1.1% 0.1% < 0.1%
17 8.3% 26.8% 33.9% 21.7% 7.6% 1.4% 0.1% < 0.1%
18 7.0% 24.4% 33.7% 23.6% 9.1% 1.9% 0.2% < 0.1%
19 5.8% 22.1% 33.2% 25.4% 10.7% 2.5% 0.3% < 0.1%
20 4.8% 19.9% 32.4% 27.0% 12.4% 3.1% 0.4% < 0.1%
21 4.0% 17.7% 31.3% 28.3% 14.2% 3.9% 0.5% < 0.1%
22 3.3% 15.7% 30.0% 29.4% 16.0% 4.8% 0.7% < 0.1%
23 2.7% 13.8% 28.6% 30.3% 17.8% 5.8% 1.0% 0.1%
24 2.2% 12.1% 26.9% 30.9% 19.6% 6.9% 1.3% 0.1%
25 1.7% 10.5% 25.2% 31.2% 21.4% 8.2% 1.6% 0.1%
26 1.4% 9.1% 23.4% 31.2% 23.2% 9.6% 2.0% 0.2%