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


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


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%).

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