What Condition my Condition(al) is in!

The hardest land to speculate impact on in Delver's mana base was Glacial Fortress. It's condition that a plains or island be ont he battlefield presented some troubling math that I hope I deduced correctly. If anything, though, it was fun to work.

Ths first chart is the probability that any land drawn is a plains or island:

	# Island or plains drawn
Drawn:	>=1	0	1	2	3	4	5	3+
1	47%	53%	47%	0%	0%	0%	0%	0%
2	74%	26%	53%	21%	0%	0%	0%	0%
3	88%	12%	42%	37%	9%	0%	0%	9%
4	95%	5%	28%	42%	22%	3%	0%	25%
5	98%	2%	16%	37%	33%	11%	1%	44%
6	99%	1%	8%	28%	37%	21%	5%	63%

So, if you draw 1 land in your opener, there's a 47% chance it is a basis land. When you draw 2 lands, there's a 53% chance just 1 of them is basic, and a 21% chance both are basic.

Next is the chances of drawing a Glacial Fortress as your lands:

	# Glacial Fortress Drawn
Drawn:	>=1	0	1	2	3	4	3+
1	21%	79%	21%	0%	0%	0%	0%
2	39%	61%	35%	4%	0%	0%	0%
3	53%	47%	43%	9%	0%	0%	0%
4	65%	35%	47%	16%	2%	0%	2%
5	74%	26%	47%	23%	4%	0%	4%
6	82%	18%	44%	30%	7%	0%	7%

There's a 21% chance that the first land drawn by the deck is a glacial fortress. 35% chance that one land drawn is a glacial fortress by the time two lands are drawn.

Conditionally, if you have 2 lands drawn, the probability of it being a glacial fortress AND a basic land is 21%. If you have 3 lands drawn, the chance of having 1 glacial fortress and at least 1 basic is 37%. Whether it'll be first or second is a mystery to me. I am not that good at this math, yet.

Next, I want to determine whether 0,1,2,3, or 4 glacial fortresses would be best based for the deck. I'm just going to speculate on this and try to use some sort of deductive reasoning. I made a chart that will compare 8 criteria for combinations of Plains, Island, and Glacial Fortresses in a Delver deck.

1.% First land tapping for {U}
2.% First land being Glacial Fortress
3.% First land being a basic land
4.% Having a land that taps for {W} by 3rd land drawn
5.% Having a fortress and basic by land drop 2
6.% Having a fortress and at least 1 basic by land drop 3
7.% Having the Geist in hand to play by 9th drawn card
8.% Having the Delver or Ponder in hand to play in opening hand

There is a lot of combinations here, too, because we have to determine what basics we'd replace the fortresses with. There's:
UUUU, UUUW, UUWW, UWWW, WWWW
GUUU, GUUW, GUWW, GWWW
GGUU, GGUW, GGWW
GGGU, GGGW
GGGG

15 column chart! :D

No, just kidding. I want to maximize my chances of castign 1st turn Delver. It's the name of the deck! So I only want combinations that have maximum {U} tapping power. That's:

UUUU, GUUU, GGUU, GGGU, and GGGG

5 columns. I have a :(

	Land Combo
Crriteria	UUUU	GUUU	GGUU	GGGU	GGGG
1	84%	84%	84%	84%	84%
2	0%	5%	11%	16%	21%
3	68%	63%	58%	53%	47%
4	62%	70%	77%	83%	88%
5	0%	7%	13%	18%	21%
6	0%	14%	25%	33%	37%
7	51%	51%	51%	51%	51%
8	68%	68%	68%	68%	68%

You have 16 more percentage points of having your first land tap for {U} than having a delver or ponder in your opening hand. Applying the same metric, 16 pecentage points more than the 51% chance of having Geist in your hand by the 9th card drawn is 76%. The first combination that has a 76% or greater chance of having a {W} as the third land drop is GGUU. If you treat this proportionally, though, you'd be looking at a percentage drawing {W} starting at 68%, which is GUUU.

I'm just going to state that while its good to have the highest percentage possible of having {U} and {W} sources in the deck, the loss of tempo that could be generated by running 4 Glacial Fortresses is not worth that maximization. Delver should have ran either 1 Fortress, and 3 extra islands, instead, or 2 fortresses and 2 islands.

What's just so bad is I can't test this theory since Delver does not exist as a deck archetype yet. However! Having this chart and knowing the conclusions I drew from it, and why, contribute to my budding manifest on deck design.

What a Mana

So, here's is a chart:

 

	Lands in hand
Drawn:	>=1	0	1	2	3	3+
7	96%	4%	19%	32%	28%	44%
8	97%	3%	14%	28%	30%	56%
9	98%	2%	10%	23%	30%	65%
10	99%	1%	7%	19%	28%	74%
11	99%	1%	4%	14%	25%	81%
12	100%	0%	3%	11%	22%	86%
13	100%	0%	2%	8%	18%	90%
14	100%	0%	1%	5%	14%	93%
15	100%	0%	1%	4%	11%	95%
16	100%	0%	0%	3%	8%	97%
17	100%	0%	0%	2%	6%	98%
18	100%	0%	0%	1%	4%	99%
19	100%	0%	0%	1%	3%	99%
20	100%	0%	0%	0%	2%	100%

This is just the probabilities of having mana drawn in hand playing with the sample Delver deck. This is the math for a 56 card deck b/c Gitaxian Probe is just air to make the deck seem meatier. This is similar to how Lays fills bags of Baked Lays with air and, like, 7 chips. I hate Lays so much. Their baked chips are just so tasty. Nom. My sample Delver deck ran 19 lands. You have a 96% chance of having at least 1 land in hand in your opener, but a 19% chance of having ONLY 1 land. You're more likely to have 2 land in hand than any specific number, and most like to have 3 or more. You have a 46% chance of having 3 or more mana in hand in your opener running Delver. You have a 56% chance of having 3 or more mana in hand
by the draw of the 8th card.

16 of the 19 lands in my sample deck tap for {U}, 9 tap for {W}, and 2 for {1}. You really have a very small chance of all lands in your hand not being a source of {U}. So I'm primarily interested in when this deck will draw a {W} source and its Moorland Haunts, the 2 {1} sources. The deck needs the white to cast Geist of St. Traft and activate Moorland Haunt's ability. So I want to know the probability of there being a single white source on the board by the third land drawn.

	White Sources Drawn
Drawn:	>=1	0	1	2	3
1	47%	53%	47%	0%	0%
2	74%	26%	53%	21%	0%
3	88%	12%	42%	37%	9%

By the third land in hand, there is an 88% chance that one land wil tap for {W}

My sample Delver deck ran 2 copies of Moorland Haunt. Here's the chart for drawing that card:

	Moorland Haunts Drawn
Drawn:	>=1	0	1	2
1	11%	89%	11%	0%
2	20%	80%	20%	1%
3	30%	70%	28%	2%
4	39%	61%	35%	4%
5	47%	53%	41%	6%
6	54%	46%	46%	9%
7	61%	39%	49%	12%
8	68%	32%	51%	16%
9	74%	26%	53%	21%
10	79%	21%	53%	26%
11	84%	16%	51%	32%
12	88%	12%	49%	39%

The deck has a better than half chance of having a Moorland Haunt drawn by the 6th land drawn. This most likely happens around the 17th-19th card a player would draw all game (see chart 1). 

The deck does run 2 conditonal lands so it can have more access to white sources of mana. Seachrome coast you need out before you hit your 3rd land drop, and Glacial Fortress needs you to have an island or plains out for it to come out untapped. Figuring out the probabilities of drawing and playing Seachrome Coast is the easiest to do.

	Seachrome Coast in hand
Drawn:	>=1	0	1	2	3
3	53%	47%	43%	9%	0%

Since it needs to be at most the third land, we can stop there. By the time you've drawn three lands, you have a 53% chance that at least 1 card is a Seachrome Coast. If it doesn't hit the table by the 3rd land drop, it'd only be a hindrance to casting Restoration Angel, anyway, if it was your only option or your 4th land.

The hardest to speculate on, for me, is Glacial fortress. It needs to have at least one plains or island in play already. There are 9 of them in the sample. That math I'll present tomorrow.

Delver

I've been writing this blog for about a week, but I've been developing my metrics for Magic over the prior 3-4 months. I'm finding that each day I return to this blog I grow more excited to create and test my theories. New ideas continue to burst into my brain as I delve into my work.

Starting today I'm going to look into Delver. I'm interested in dissecting this deck since I am fairly certain I'll be playing it once Gatecrash comes out and get my hands on Dimir Charm. I want to work on studying Delver over several days. I am very interested to poke at the mana base and study its spell curve. I hope to discover principals of deck design within its construction.

4 Glacial Fortress

8 Island

2 Moorland Haunt

1 Plains

4 Seachrome Coast

4 Delver of Secrets

4 Geist of St Traft

3 Restoration Angel

4 Snapcaster Mage

4 Gitaxian Probe

3 Gut Shot

3 Mana Leak

1 Mutagenic Growth

4 Ponder

3 Runechanter's Pike

4 Thought Scour

4 Vapor Snag

The top of Delver's spell curve is Geist of St. Traft and Restoration Angel. The sample deck I am using has 4 Geists and 3 Angels. The difference kind of drew my attention. If a deck runs 4 copies of a card, of course it WANTS to play that card. So the deck obviously would love to cast Geist of St. Traft. I wonder whether the deck runs 3 Angels because it wants to limit the likelihood of drawing the very top of its curve, or it runs 3 angels because of their speed and use? So, I'v developed a few hypotheses considering those questions. My first hypothesis is, obviously, if a deck runs 4 copies of a card, then the deck really wants to play that spell. My second hypothesis is that a deck is looking to curve its mana to cast its sorceries. I'm guessing that is because spells that can be cast as instants give players options. Generally a player will keep his options open until he/she HAS to make a play. Sorceries ask players to take options away from themselves and commit to a plan of action on their turn. So, a deck that has 4 copies of a 3 mana sorcery speed spell obviously wants to cast the spell as soon as the have 3 mana and further their game plan. 
I do need to consider, too, that the "4 of" rule I'm developing breaks at higher mana costed spells. A deck may really want to play an 8 mana spell, but does not want that spell languishing in his/her hand as they acquire their mana. Higher costed spells need fewer copies of themselves in a deck. They can be casted at reasonable times along a curve without having 4 copies of themselves. I doubt this is the case with Restoration Angel, though.

Returning to the budding manifest. I said that you can discern a deck's plan through its sorcery speed spells.
The deck runs:
4 Delver of Secrets {U}--1CMC
4 Geist of St. Traft {1WU}--3CMC
4 Gitaxian Probe {U/P}--1CMC, P is for "Phyrexian mana"
4 Ponder {U}--1CMC
3 Runechanter's Pike {2}--2CMC, 2 Equip

Seeing Gitaxian probe, we can now state that Delver is actually a 56 card deck. Gitaxian probe can cost 0 mana to cast and it replaces itself thanks to being a cantrip.

It runs 2 different 1CMC spells in Delver and Ponder. 3 copies of a 2 CMC spell in Runechanter Pike, and 4 copies of a 3 CMC spell in Geist. I'm guessing Delver wants to play either Ponder or Delver turn 1, the other card turn 2, and Geist on turn 3.

Lets see what the probabilities are of Delver having the 1 CMC spells in hand:

	# of Delvers and/or Ponders in Hand
Drawn:	>=1	0	1	2	3	4	>1
7	68%	32%	42%	21%	5%	1%	27%
8	73%	27%	41%	24%	7%	1%	32%
9	78%	22%	40%	27%	9%	2%	38%
10	82%	18%	38%	30%	12%	2%	44%
11	85%	15%	35%	32%	14%	3%	49%
12	88%	12%	32%	33%	17%	5%	56%
13	90%	10%	29%	33%	19%	6%	59%
14	92%	8%	27%	34%	22%	8%	66%

So, we're looking for Delver to have 2 cards in hand by turn 2, 1 Delver and 1 ponder. Combined, the chance of having any combination of at least two of any of these cards, 2 Delvers, 2 Ponders, or 1 of each, is 27% in the opneing hand. There's a 32% chance of having 2 or more of these cards by the 8th card drawn. There's a better than 50% chance of ay least 2 cards drawn by the 12th card drawn.

Next is Runechanter's pike. There's 3 copies of that in the deck. The chances of drawing it are:

	# of Pikes in Hand
Drawn:	>=1	0	1	2	3
7	34%	66%	30%	4%	0%
8	38%	62%	33%	5%	0%
9	42%	58%	35%	6%	0%
10	45%	55%	37%	7%	0%
11	49%	51%	39%	9%	1%
12	52%	48%	41%	10%	1%
13	55%	45%	42%	12%	1%
14	59%	41%	43%	14%	1%
15	62%	38%	44%	16%	2%
16	64%	36%	45%	17%	2%
17	67%	33%	45%	19%	2%
18	70%	30%	46%	21%	3%
19	72%	28%	46%	23%	3%
20	74%	26%	45%	25%	4%

So, the deck has a better than 50% chance of drawing the pike by the 12th card drawn. It is as likely to be drawn by the 17th card as just 1 copy of Delver or Ponder in the opening hand, ~68%.

Lets grab some data about the top end of Delver's sorcery speed spell curve. Delver wants to play Geist of St. Traft, and wants its mana available to do that. So, here's the chance of drawing at least 1 Geist:

	# of Geist in Hand
Drawn:	>=1	0	1	2	3	4
7	42%	58%	35%	7%	0%	0%
8	47%	53%	38%	9%	1%	0%
9	51%	49%	40%	11%	1%	0%
10	56%	44%	41%	13%	2%	0%
11	59%	41%	42%	15%	2%	0%
12	63%	37%	43%	17%	3%	0%
13	66%	34%	44%	19%	3%	0%
14	70%	30%	44%	21%	4%	0%
15	72%	28%	44%	23%	5%	0%
16	75%	25%	43%	25%	6%	0%
17	78%	22%	42%	27%	7%	1%
18	80%	20%	41%	29%	8%	1%
19	82%	18%	40%	31%	10%	1%
20	84%	16%	39%	33%	11%	1%

In an opening hand of 7, Delvers has a 42% chance of having Geist in its hand. As Delver digs deeper into its deck, that probability goes up. 8 cards deep into its deck there's a 47% chance that Delver has drawn a Geist. Delver has a better than 50% chance of having a Geist in hand by the 9th card drawn. The chances of Delver having EXACTLY 1 Geist in hand top out at 44% by the 15th card drawn. That starts to go down as the probability of having multiple Geists goes up. 

If we intuit the deck's plan on the cards that reach 50% probability of having at least 1 card in hand first, we come up with:at least 1 of Ponder/Delver in the opening hand, then 1 Geist by the 9th card, and finally playing a Pike by the 12th card drawn. I want to dig more into this progression later. The combination of Ponder and Delver is very synergistic and once Ponder left the format Delver dropped off as a deck choice. I want to determine, for myself, if that is really because of Ponder leaving.

Just for fun, I looked at the highest costed spell in Delver, Restoration Angel. It is not a sorcery so the deck does not have to plan to cast it at any specific point. It just needs 4 mana to cast it. The chances of Delver having Restoration Angel in hand are:

Drawn:	>=1	0	1	2	3
7	34%	66%	30%	4%	0%
8	38%	62%	33%	5%	0%
9	42%	58%	35%	6%	0%
10	45%	55%	37%	7%	0%
11	49%	51%	39%	9%	1%
12	52%	48%	41%	10%	1%
13	55%	45%	42%	12%	1%
14	59%	41%	43%	14%	1%
15	62%	38%	44%	16%	2%
16	64%	36%	45%	17%	2%
17	67%	33%	45%	19%	2%
18	70%	30%	46%	21%	3%
19	72%	28%	46%	23%	3%
20	74%	26%	45%	25%	4%

In an opening hand of 7, Delver has a 34% chance of having at least 1 Angel in hand. There is a better than 50% chance of Angel being in hand when the 12th card is drawn. I looked further down my chart and discovered Delver has a better odds of having drawn 2 or 3 Angels than 1 by its 26th card and a better than 50% chance of having 2 or 3 than 1 or less by the 29th card drawn. 

Next post I'm going to spend my time dissecting the mana base for delver.