Monday, March 24, 2014

Further Examination of Karsten Frank's Analysis

In the previous post, as well as in Karsten Frank's analysis, there is a subtle nuance that we have to touch upon. In both his article and in my initial post we assumed that we can read off the table the number of lands that is required for a spell like Anger of the Gods by looking at double-casting spells on turn 3. But, the table doesn't show that -- it shows the number of colored lands required for casting a CC spell on turn 3, assuming we have enough lands (i.e. 2 lands on turn 3). This is not the same!

It turns out that casting a 1CC spell on turn 3 requires less colored sources than casting a CC spell on turn 3. This sounds absolutely crazy at first, but remember the first assumption, that we have enough lands to cast them in the first place. Consider for a second, that we want 0% color-failure in a 24-land 60-card deck. To have no color failures on turn 2 means that we need to have 24 colored mana. On turn 3, you might think that we can allow non-colored mana, but that is wrong since we are comparing against the chance of having enough lands and that includes the chance of having only 2 lands on turn 3 - in which case there is a non-zero chance that one of them is the off-colored mana source. 

In other words, the pre-condition "have enough lands to be able to cast the spell" over-inflates the colored-land tables that we have presented, and it may be more useful to make a table consisting of the required number of lands in order to avoid color-failures for each of the consecutive spells. So, for instance, a table comprising C, 1C, 2C, 3C, 4C, 5C and 6C for single-colored spells and a table CC, 1CC, 2CC, 3CC, 4CC, 5CC for double-colored spells and so on, given at the first turn you can cast them:

Spell TypeC1C2C3C4C5C6C
90.00%1413119876
As you can see, this is quite different from the previous table. For double-colored spells:

Spell Type-CC1CC2CC3CC4CC5CC
90.00%-201715131110

And finally for triple-colored spells

Spell Type--CCC1CCC2CCC3CCC4CCC
90.00%--2219171513

From this, we can revise our numbers for Anger of the Gods, which requires 17 out of 24 red sources, and Cryptic Command that requires 19 blue sources out of 24.

For completeness, here's a few variations:

------------------------------------------------

16/40    --    1    2    3    4    5    6    7
     C    9    8    7    6    5    4    4    3
    CC   13   12   10    9    7    7    6    5
   CCC   15   13   11   10    9    8    7    6
  CCCC   16   14   12   11   10    9    8    7


17/40    --    1    2    3    4    5    6    7
     C   10    9    7    6    5    5    4    4
    CC   14   12   10    9    8    7    6    5
   CCC   16   14   12   11    9    8    8    7
  CCCC   16   15   13   12   10    9    9    8


18/40    --    1    2    3    4    5    6    7
     C   10    9    8    7    6    5    4    4
    CC   14   12   11    9    8    7    7    6
   CCC   16   14   13   11   10    9    8    7
  CCCC   17   15   14   12   11   10    9    8

------------------------------------------------

22/60    --    1    2    3    4    5    6    7
     C   13   12   10    9    7    6    6    5
    CC   19   16   14   12   10    9    8    7
   CCC   21   18   16   14   12   11   10    9
  CCCC   21   19   17   15   14   12   11   10


23/60    --    1    2    3    4    5    6    7
     C   14   12   10    9    8    7    6    5
    CC   20   17   14   12   11   10    9    8
   CCC   22   19   16   15   13   12   10   10
  CCCC   22   20   18   16   14   13   12   11


24/60    --    1    2    3    4    5    6    7
     C   14   13   11    9    8    7    6    5
    CC   20   17   15   13   11   10    9    8
   CCC   22   19   17   15   13   12   11   10
  CCCC   23   21   18   17   15   14   12   11


25/60    --    1    2    3    4    5    6    7
     C   14   13   11    9    8    7    6    6
    CC   21   18   15   13   12   10    9    8
   CCC   23   20   18   16   14   13   11   10
  CCCC   24   21   19   17   16   14   13   12


26/60    --    1    2    3    4    5    6    7
     C   15   13   11   10    8    7    7    6
    CC   21   18   16   14   12   11   10    9
   CCC   24   21   18   16   15   13   12   11
  CCCC   25   22   20   18   16   15   13   12

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Thursday, March 6, 2014

Further Analysis of Karsten Frank's Mana Consistency

Karsten Frank published a very nice article at Channel Fireball that looked into the question of how many colored mana-sources you need in order to cast a spell or creature.

I thought it would be nice to analyze Karsten Frank's results with my own analysis program. While Karsten programmed a simulation, I have programmed a set of routines that could easily be adapted to his prerequisites. In other words, I'm making use of the hypergeometric distribution function.

First, the numbers I present are all assumed for being on the play, not on the draw. The reason for that is that it is harder to successfully cast spells when you have one less draw step. Second, the "90%" rule refers to the chance of being able to cast your spell, given that you have enough lands in the first place. Third, I will focus on 60-card constructed decks. And finally, he assumed a fixed land count, but in later in this blog I will explore the effects of varying the number of lands in your deck.

But, what if I feel that 90% is too conservative?


A question often asked has been, what happens if you tighten or relax the 90% rule? 90% seems arbitrary, but it is a fairly good number to aim for. It means that 9 out of 10 games, you will not be in the situation where you have the lands to play a spell, but have the wrong colors to cast them.

Spells with a single colored mana:


For the case of hitting a single mana, the table is as follows (the percentages corresponds to $19/20, 9/10, 7/8, 6/7, 5/6$ and $4/5$):

Req\Turn 1 2 3 4 5 6 7
95.00% 16 15 14 13 12 12 11
90.00% 14 13 12 11 10 9 9
87.50% 13 12 11 10 9 9 8
85.71% 12 11 10 9 9 8 8
83.33% 11 10 10 9 8 8 7
80.00% 11 10 9 8 7 7 6

As you can see from this, normally, in order to cast Thoughtseize on turn 1 with the 90% rule, you would have to have 14 sources of black. What this table tells you is that if you want to have that black mana source in 19 games out of 20 (95%), then you better have 16 sources.

Conversely, if you are okay with having no black mana on turn 1 in 1 out of every 5 games, then you may go down to 11 black sources, and even lower if you expect to cast it on later turns.

Spells with double-colored mana:


For double-mana spells we get

Req\Turn1234567
95.00%-222120191817
90.00%-201918161514
87.50%-191817161514
85.71%-191716151413
83.33%-181716141313
80.00%-171615141412

Again, we see that in order to cast Anger of the Gods reliably on turn 3, we need at least 19 sources. If we skimp to 16 red sources then 1 out of every 5 games where we have 3 lands in play, we will not have enough red mana to cast it.

Conversely, if you can fit 21 red sources in your deck, then we will be color screwed only once every 20 games.

Spells with triple-colored mana:


For triple-mana spells such as Boros Reckoner or Cryptic Command, we get

Req\Turn1234567
95.00%--2323222221
90.00%--2222212019
87.50%--2221201918
85.71%--2121201918
83.33%--2120191817
80.00%--2020191817

The same story as before unfolds. With 23 blue sources, you will be color-screwed only once in every 20 games, but by reducing it to 20 you are looking at that happening once every 6 games (83.33%).