Math: The Gathering

I decided I needed to write a post on my, horrible, math. It is easy for me to pick and choose percentages about card draws and such. So lets maybe codify what I'll be looking into when I break a deck down.

First, is the probabilities of drawing at least 1 card/effect. This is accomplished using the HYPGEOMDIST function in Excel. This table is a small piece of what I use. I go down to 60 card/effects.

	Cards drawn
# Cards/Effects	7	8	9	10	11	12	13
1	12%	13%	15%	17%	18%	20%	22%
2	22%	25%	28%	31%	34%	36%	39%
3	32%	35%	39%	43%	46%	49%	53%
4	40%	44%	49%	53%	57%	60%	63%
5	47%	52%	57%	61%	65%	69%	72%
6	54%	59%	64%	68%	72%	75%	79%

So, if you have four copies of a card in your deck, the chance of drawing it in your opening hand is 40%.

But to gauge what a deck is doing, you need to understand what each card is doing. RDW can have 4 Rakdos Cacklers, 4 Stromkirk Nobles, 3 Vexing Devils, and 1 Thunderous wrath in it, but what effect is each card trying to accomplish?

Vexing Devil is not trying to fulfill the role of a Cackler or Noble. It's trying to fulfill the role of the Thunderous Wrath. It's meant to be a 1 mana burn spell to the dome of your opponent.

Thunderous wrath can do that, or it could do 5 damage to another creature. These are two seperate functions/effects. In my short-hand the effect of casting burn at an opponent's life is called "Range" and burning out a creature on the board is just "Removal".

Totaling the cards for each effect in the my example made up list gives us 4 Range spells (3 Vexing Devil and 1 Thunderous Wrath) and 1 Removal spell (Thunderous Wrath). The probabilities of drawing each effect on the list suggest casting Range spells early in the game. Having at least 1 of 4 Range cards in the opening 7 is a 40% chance. Later the deck wants to Miracle Thunderous Wrath when it could take out a creature with high toughness.  The chance of drawing the 1 Removal spell by the 13th card drawn is 22%

Generally I use the number of cards drawn to represent the turns for the player on the play. 7 cards drawn is the opening hand and turn 1. 8 Cards drawn represents the card drawn at the start of turn 2. This math, of course, gets kinda thrown out once card drawing is introduced. I can kind of account for card draw when it is first used. I am not savy enough to account for it for later uses beyond the first.

The above chart is for the chance of drawing at least 1 of a card. Now lets see if an opening hand will more likely have 1,2, or 3 copies of an effect.

	# of Cards/Effect
# in Opening Hand	7	8	9	10	11	12	13
1	42%	42%	42%	41%	40%	38%	36%
2	16%	19%	22%	25%	27%	29%	31%
3	3%	4%	5%	7%	9%	11%	13%
4	0%	0%	1%	1%	2%	2%	3%
5	0%	0%	0%	0%	0%	0%	0%

So, if there are 13 copies of an effect in a deck, the chance of having just one copy of that effect in the opening hand of 7 is 36%. The chance of having 2 of that effect in the opening 7 is 31%. The chance of having 3 or more of that effect in the opening 7 is 16% (13% + 3%)

If, instead, it is turn 4 on the play and there have been no draw spells played (10 cards drawn), the chart looks like this:

 

	# of Cards/Effect
# drawn into hand	7	8	9	10	11	12	13
1	41%	39%	36%	33%	30%	27%	23%
2	25%	28%	30%	32%	33%	33%	33%
3	7%	10%	13%	16%	19%	21%	24%
4	1%	2%	3%	4%	6%	8%	10%
5	0%	0%	0%	1%	1%	2%	3%
6	0%	0%	0%	0%	0%	0%	0%
7	0%	0%	0%	0%	0%	0%	0%

The chance of drawing multiples has gone up as the game progresses, and the chance of drawing just 1 card goes down eventually. If there are 13 cards that perform the same effect, that effect has most likely been drawn 3 or more times (37%) than just 1 (23%) or 2 (33%) times by the time 10 cards have been drawn.

So, when I look at decks, I'll be looking at percentages and at what turn a player could expect to play an effect, and then how often that player will draw the effect. All I have to do is figure what is an acceptable percentage for an effect to be packed into a deck. For that I'll have to continue examining deck lists. .