opening hand analysis: WWYD?

[Liam-K]

This thread originally started as a footnote to this one, but became a veritable appendix.  I decided to give it its own thread, as really it is a topic in its own right and I didn't want to clutter the other thread with math and unrelated conversation.

You are playing GAT.  Your opponent is playing GAT.  You are on the play.  You fan open your 7 and see:
flooded strand, duress, gush, gush, misdirection, mystical tutor, mana drain

What do you do?

Personally I feel this hand has the potential to be quite strong assuming a land is found quickly.  Though it does not contain brainstorm, it does contain turn 2 eot mystical -> turn 3 ancestral, and most enough disruption to back it up.  First turn duress buys time and defends the hand's main vulnerabilities: opposing duress or too much early pressure... playing first makes making your second land drop turn 3 much more acceptable.  Frankly, I think the hand is stronger than your average 6 card hand assuming you can play a second land turn 3 at the latest.

Here's some math, analysing the probability of topdecking a land or a cantrip that digs into land on your second turn, and the probability the mystical -> ancestral play succeeds if the first scenario does not occur.  It is probably not perfect, I'm not up on my probability algebra, but it should be a reasonable guideline.


Situation A: drawing a land in your second turn draw step
10/53 = 0.188679245283018867924528301 88679% chance land drawn turn 2

0.188679245283018867924528301 88679% chance of Situation A

if this does not occur, my next favorable possibility is digging into a land.  Since I will have U/B available if I draw a spell that could find a land, there are 3 spells which could possibly find me my second land drop on turn 2

Situation B: drawing a cantrip in your second turn draw step and finding a land with it
4/53 = 0.075471698113207547169811320 754717% chance brainstorm drawn turn 2
- 10/52 = 0.192307692307692307692307692 30769% chance land drawn card 1
-- 10/51 = 0.192307692307692307692307692 30769% chance land drawn card 2
--- 10/50 = 0.2% chance land drawn card 3
these are all favorable outcomes... the chances of drawing a second land don't matter because that only happens if you've drawn the first and the brainstorm is already favorable.  So I believe the math here is additive...
= 0.584615384615384615384615384 614% chance brainstorm nets land
chance of drawing brainstorm * chance brainstorm nets land =
0.044121915820029027576197387 517619% chance of brainstorm -> land

3/53 = 0.056603773584905660377358490 566038% chance opt drawn turn 2
- 10/52 = 0.192307692307692307692307692 30769% chance land drawn card 1
-- 10/51 = 0.196078431372549019607843137 2549% chance land land card 2
= 0.384615384615384615384615384 614% chance opt nets land
chance of drawing opt * chance opt nets land =
0.021770682148040638606676342 525306% chance of opt -> land

1/53 0.018867924528301886792452830 188679 chance ancestral drawn turn 2
- 10/52 = 0.192307692307692307692307692 30769% chance land drawn card 1
-- 10/51 = 0.192307692307692307692307692 30769% chance land drawn card 2
--- 10/50 = 0.2% chance land drawn card 3= 0.584615384615384615384615384 614 land from ancestral
= 0.584615384615384615384615384 614% chance ancestral nets land
chance of drawing ancestral * chance ancestral nets land =
0.011030478955007256894049346 87911% chance of ancestral -> land

again, the chances of drawing each of the spells is mutually exclusive, so again unless I am mistaken the math is additive again:
chance of brainstorm -> land + chance of opt -> land + chance of ancestral -> land =
0.076923076923076923076923076 921112% chance of spell -> land

0.076923076923076923076923076 921112% chance of Situation B

now, if none of this occurs, my third turn draw phase will net me ancestral recall via mystical tutor and I will see 3 cards after having drawn 2.

Situation C: tutoring for ancestral, casting it turn 3 and finding a land with it
10/51 = 0.2% chance of drawing land card 1
10/50 = 0.204081632653061224489795918 36735% chance of drawing land card 2
10/49 = 0.208333333333333333333333333 33333% chance of drawing land card 3
= 0.612414965986394557823129251 70068% chance ancestral nets land

So now, situation A and B are mutually exclusive so I believe they are additive. That means about a 26.5% chance of making my turn 2 land drop.  If that fails (73.5%), I have a 61% chance of that fraction leading to a land drop on turn 3, which settles out at about 45% of all possible scenarios.  These two probabilities are mutually exclusive so an overall 28.5% chance the hand doesn't find land before the fourth turn and consequently be awful (71.5% chance of success). 

I'm going to go ahead and assume that not finding land turn 3 probably leads to a game loss, even though the game I played with this hand didn't pan out that way.  I'm also going to assume the duress + misdirection is enough to defend your land-finding attempts.  These two ignored possibilities probably cancel each other out, or pretty close.

Feel free to correct my math but I'd still put 71.5% chance of playability as comparable to the average 6 card mulligan hand... I elected not to mulligan as I felt my chances of the new hand being playable were comparable. 

What's your call?

[PETER FLUGZEUG]

So now, situation A and B are mutually exclusive so I believe they are additive. That means about a 26.5% chance of making my turn 2 land drop.  If that fails (73.5%), I have a 61% chance of that fraction leading to a land drop on turn 3, which settles out at about 45% of all possible scenarios.  These two probabilities are mutually exclusive so an overall 71.5% chance the hand doesn't find land before the fourth turn and consequently be awful. 

What's your call?

I have a question regarding this: Do I misread this part when I see that you write there is a 71.5% chance you don't find land until turn 4? that's the way this sentence looks to me. Or did you intend to write that there is only a 28.5% chance you don't find land? Can you clarify this up?

Thank you very much.

[Liam-K]

yeah, I wrote in the wrong number.  71.5% chance that you play a land turn 2 or 3.

[desolutionist]

That's a really bad hand to have in the mirror match.  The only way I can see you winning that game is if you Duress first turn, draw into a land, and then Gush into Sapphire + Dryad because Mystical->Ancestral just won't get there.

My call?  Mulligan.

[Outlaw]

This is a ballsy opening hand, it contains nothing that actually digs for the 2nd land.  In reality I guess if you decided to keep the hand you could go over a few routes.

Hand contains: flooded strand, duress, gush, gush, misdirection, mystical tutor, mana drain

Turn 1 -> Fetch, Duress (pray they have have 1 force and they can somehow get ancestral and cast into your misdirection, unlikely)
Turn 2 -> Pray to draw 2nd land, initiate lose phase if unsucessful.


Turn 1 -> Fetch
EOT Opp Turn 1 -> Mystical into Recall praying one pitch counter backup is enough.

[Smmenen]

i would definitely keep that hand and begin with turn one duress.

i would not play Mystical tutor, but would draw a card on turn two.   if the card is not a land, I would then play mystical eot or something.   the idea is that Durses + misd you should have little trouble forcing through your acall. 

[TheKing23]

I agree.  Any opening hand with Duress in it and being able to play it on turn one in vintage is great.  If nothing else, you go: fetch a Underground Sea, play Duress, pass the turn, next turn: draw, and, depending on what you draw and what they play,  Mystical Tutor on their EOT for an Ancestral Recall, then get it through with MisD/whatever you draw.  Remember, you don't need your draw to be a land, it can be any mana-producing card.