You Make the Play: Statistics for Dummies

So a couple of weeks ago, I presented the following You Make the Play:

This is a seven card hand. You lost the flip so there are 53 cards in your deck and you are playing second. The deck list is the one we have been bandying about the past week or two — Jund Mana Ramp.

So… Keep or no?

The responses were interesting and varied. I was using this You Make the Play as a lead-in for some basic statistics (which we will get to) but my intrepid readership took typically galvanized positions as well as the opportunity to stand on soap boxes.

When the dust cleared (out of nineteen responses), we had a little under 2:1 ratio in favor of keeping the hand… and a handful of people who just don’t like Gift of the Gargantuan :)

* Zvi, by the way, said that he didn’t have to look at the hand, and if I were asking, I should mulligan it.

However, like I said, I wanted to use this as a lead-in for some basic statistics rather than a critique of the deck.

When we make plays we often do things on “gut” or fear that end up being terrible decisions. One of the worst mistakes in my entire career was in a Feature Match at US Nationals 2000.

I was playing trusty Napster in its best tournament, and riding a 2-0 open on the day, I found myself in the Feature Match area in a semi-mirror against Donnie Gallitz.

I opened on Swamp, Dark Ritual, Phyrexian Negator and Donnie started with a Duress for my Vicious Hunger. I Duressed Donnie back; his hand was all garbage – Unmask, Stupor, Masticore, and Skittering Skirge. Skittering Skirge was the best card in the hand but I couldn’t take it; Masticore was borderline unplayble in the matchup but happened to be good against my Phyrexian Negator (again I couldn’t take it), so I decided to take the Unmask.

Donnie dropped the Skirge to defend, then mized into Ritual + Persecute. I responded with Vampiric Tutor for Vicious Hunger and got in more and more.

Donnie played his unplayable Masticore (which nevertheless prompted me to sacrifice my Phyrexian Negator). Then I Eradicated the Masticore via Vampiric Tutor, then Tutored again to set up Yawgmoth’s Will.

Ritual + Vicious Hunger to start, smashing Donnie’s fresh Skitting Skirge, then I set up a Phyrexian Negator. Donnie mized into a Yawgmoth’s Will of his own, but if you go back and read the cards they were pretty pointless so all he did was make a dude.

Okay here’s my mistake.

I topdecked a Skittering Horror.

Now of course I played it pre-combat.

My plan was to smash Donnie with my Negator and sacrifice down to just the Negator, and then just kill him the next turn. But the Horror gave me another permanent and another option. So because I drew the Horror and correctly played it, I smashed in and sacrificed down to the Horror and one land instead. I figured if Donnie drew a Vicious Hunger he could make me sacrifice down to nothing, but the same wouldn’t be true of a Horror.

Instead Donnie — no cards in hand — picked up a Skittering Skirge that held me (down to essentially no permanents) off until he came back to win the game… From about three life.

In the same spot I would have just trampled him to death.

I had what Dan Paskins calls The Fear, and made a terrible decision.

I should probably have sacrificed down to double guys; next best would have been Negator and land (which was my original plan).

Dave Price described this as bad because of probability. Donnie had what? Forty cards or more in his deck? What were the chances of his drawing a Vicious Hunger (bad for two permanents if one is a Negator) versus any creature that could stop a Skittering Horror?

It gets worse.

Donnie played his unplayable Masticore (which nevertheless prompted me to sacrifice my Phyrexian Negator). Then I Eradicated the Masticore via Vampiric Tutor, then Tutored again to set up Yawgmoth’s Will.

Eradicate allows you to look through the opponent’s entire deck.

… Where I could have seen that he played no Vicious Hungers at all main deck!

I won Game Two in dramatic fashion, but just had no resources in Game Three whereas Donnie got the fast Persecute. But the fact is: It shouldn’t have gone three games.

So how does this come back to our discussion of whether or not to mulligan this hand on the draw?

The deck plays 23 lands and four Rampant Growths. I was operating under the idea “If I have three lands untapped on my third turn, I can basically make my land drops all the way to six without interruption” (six land being Broodmate range). I understand some of you think this hand is not strong v. Tokens, but you probably haven’t played the matchup as much as I have. Tokens is often a Batman / Vs. System battle where you just play something better than what they play, top up on your six, and then play sixes every turn while they are still piddling around with three 1/1 creatures (which actually get soundly stomped by some of your sixes).

So…

How do we get to three untapped lands on turn three?

1) We can draw any land on turn one or turn two.
2) We can draw Rampant Growth on turn one or turn two.
3) We can topdeck one of 13 comes into play untapped lands on turn three.

So here are our probabilities.

Turn One – 25/53

Turn Two – 25/52

Turn Three – 13/51

You have twenty-five options on turn one – any of the twenty-one remaining lands, plus any of the Rampant Growths.

You have twenty-five options on turn two (assuming you did not already fulfill your minimal requirement on the first turn) – the same twenty-one lands and the same Rampant Growths. Note that this only works because you have two lands that come into play untapped in your opening hand; the math changes dramatically if you have a different land configuration… For example if both lands came into play tapped, you could not count Rampant Growth without an intervening untapped land pull (which itself would have fulfilled what we need fulfilled).

Turn three you still have options but they decline sharply. You lose eight of your lands (Treetop Village and Savage Lands come into play tapped, so drawing them on turn three is useless in the short term; ditto on Rampant Growth).

Most players can evaluate a situation like this one and look at the first turn. You are under 50% likely to pull a relevant piece of mana on turn one.

You are similarly less than 50% likely to pull a relevant land on turn two. But what about the fact that you get turn one and turn two both?

Turn three is much less likely than turn one or two, but you still have a nice lift… That is an “advantage” of going second in this hypothetical.

So how likely are you to pull the right land?

  • You start with 25/53, or about 47%… That’s yours, that slight dog / coin flip.
  • Of the remaining 53%, you get 25/52 (or about 48%), an addition 25%.
  • So here is the super tricky part. Of the 75% of that lost 53%, you get there another 13/51 (~25%) of the time… about 7%.
  • Ultimately you’re in at about 79-80% likely to have three untapped lands on turn three.

A faster and arguably easier way to come to the same conclusion is to figure it out in the negative. How unlikely are you to have the land you need on turn three? Your likelyhood of actually having the goods is whatever is left.

Relative likelihoods of drawing non-relevant cards:

  1. 28/53
  2. 27/52
  3. 38/51

Multiply all those together and you’re a hair over 20% not likely to get there… Or 79-80% to have the mana you need (just like we said).

So what happened?

80% is a pretty good bet, so I kept.

It turned out that my opponent was Reflecting Pool Control, one of the deck’s best matchups, and of all the matchups in Standard, the most vulnerable to this type of hand (incremental card advantage via small threats).

So of course I missed my third three times, discarded, and lost one of my best matchups :)

But at least I kept when I should have.

LOVE
MIKE

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13 comments ↓

#1 starwarer on 05.28.09 at 6:54 pm

Now, will you remember to fight the fear next time you draw a hand like this? Seeing as it is of the story form: Man, remember how I got screwed last time I kept a hand like this!

Or is this your way of helping yourself remember it as a statistical anomaly? A little positive re-inforcement, perhaps? To help counteract the impulse to act negatively.

#2 StaplerGuy on 05.28.09 at 7:56 pm

Last time I heard that story, you made fun of him for not playing Vicious Hunger maindeck and still forgot. That’s the story I’m thinking of, right?

#3 admin on 05.28.09 at 8:13 pm

@starwarer
Jonny Magic says that the tech is to make the right decision five times in a row, get burned every time, and still make the right decision the sixth time (or however many X).

@StaplerGuy
Sounds like something I would do.

#4 BitterSting on 05.29.09 at 6:12 am

Wow. This really looks like bad misleading post.

The stats of drawing a land/land equivlant are pretty meaningless by themselves. You don’t even mention what happens if you fail to draw your third land until the fourth turn. Going purely by the writeup, one would think that the thinking about keeping/tossing should be based purely on whether or not you get that third land. This post says “drawing a third land is the important thing, winning or losing is a footnote to making sure I got the math right.”

That probably isn’t your message. The decision to keep should be based on the likelyhood of WINNING not on the likelihood of getting a third land in time. You don’t even mention winning or losing. Just wow I was probably going to get that third land; here is the exact details about how likely that is. Would you have kept if you’d had a 76% chance of drawing the land but thrown it back if the odds were only 74%?

Were you definitely going to lose without the third land on turn 3? What happens if you get it on turn 4? Were you definitely going to win if you got it on turn 3?

I’m pretty sure you were thinking clearly at the time, but this blog post presents the math about some relative trivia and not the decision you were actually making. It. You are yak shaving (worth looking up if you haven’t seen that expression, it happens a lot in magic and magic articles).

The article should be “I figured I’d win if I drew that third land in time. Roughly half my deck was land/equivlants, so my odds were pretty good.” The exact numbers are pretty trivial.

You had a pretty good shot of getting the third land, and you thought you’d win if you drew it.

Isn’t that all you need to say?

#5 vpreacher on 05.29.09 at 8:00 am

With the emergence of Swans do you think this deck could be better suited in the meta? Turn three thought hemorage seems pretty good. You can also run Maelstrom pulse as well as mind shatter.

#6 admin on 05.29.09 at 11:12 am

@BitterSting
I don’t see how it is bad or misleading at all.

You can only measure to the extent that you can take action. Well, I suppose you can measure beyond that point, but such measurements become meaningless. The only question is whether or not to keep the hand. If you go back and look at the comments from other readers, you will probably come to understand that anyone considering this build of this strategy wants a lot of hands that look almost exactly like this. With one more mana source, this is the cream dream hand for many Jund players.

Are you going to outright lose if you miss mana source #3 on turn three? Of course not. There are too many factors in an unknown matchup. The opponent could stop on one and discard fourteen times. By the same token, hitting your own third land on turn three is no guarantee of success. If you are playing against a combo deck, you’re probably dead in the water.

We make decisions based on completely arbitrary points all the time. 1/3 of my readers thought the hand wasn’t good at all, whereas I would happily keep it against any of the “reasonable” and known decks in the format (and I did). I was trying to explain how to make an informed decision… based on data we had or could reasonably approximate.

And again, the only decision we can make is mulligan or no for this hand. There is no reason to complicate the hypothetical beyond the parameters of the question at hand (in my opinion).

If I had a sub-50% chance of drawing the lands I wanted, I would have asked a different question (like how are my chances going to be if I mulligan). But in this case we didn’t have to ask that question. Also, if I knew my matchup I could do things like cross off cards I expect to be useless (Broodmate Dragon is poor against Fog, for instance). But again, we can’t act with knowledge we don’t have.

Thanks for your post!

–m

#7 ProdigalT on 05.29.09 at 2:11 pm

Seems silly to add to Mike’s comprehensive post, but I agree. With a deck like this, there’s no way to know whether or not he will definitely win if he gets that land because this isn’t that kind of deck. At most, all he can say is that his deck will execute according to plan. If he were playing a combo deck, that plan tends to be “win right now,” in which case, you can make an assessment like “if I draw the land, I win.” But this deck isn’t winning until many turns down the road, and even if it goes according to plan, the opponent may get lucky. Virtually no interactive deck can look at its opening hand and know whether it’s winning or not. Swamp, Ritual, Negator is only good if your opponent doesn’t go Mountain, Shock.

#8 rpitcher on 05.29.09 at 3:01 pm

I can’t help but think that the naysayers simply weren’t experienced with this version of Jund Ramp.

As I said in my reply to the blog presenting this mulligan issue, the hand shown is a great hand for the deck. You’ve heard/read people whine about how Jund Ramp sometimes just fizzles by drawing lands and fixers with no goodies. They ARE correct. If you have played this deck often, you know it can flood with fixers and not find gas. Well, THIS hand doesn’t walk that path too often. Gift of the Gargantuan in that hand is terrific.

#9 BitterSting on 05.29.09 at 6:33 pm

Let me explain a little more then.

First, I want to be clear that I don’t think your conclusions are wrong. Good hand to keep. The problem I have is how you arrive at that conclusion. More importantly, what your article is emphasising as important. I don’t think the effort to do the math to 1% is worthwile, and at the same time the article ignores or plays lipservice to the things that are important. That’s what I feel is misleading.

I’d propose the simple rule:

You should mulligan if:
In the game you are expecting, a random smaller hand is likely to be better than your current hand.

Let’s look at the things you aren’t paying real service to:
- you don’t discuss what you expect your opponent to be playing at all. You don’t say in this metagame I expect… or knowing my opponent I expect him to be playing…
- you hint that this hand is better than random hands if it draws out properly, but I’m not reading this as being a clear “this is probably a great hand if I draw out”.

Maybe I’m just not aware of the greater context of this article. But it seems to me that you barely noticed two really important parts of the background and the mulligan decision.Instead this article focuses on math.

A fair bit of math actually. You work out the percentages of drawing the right mana help to within a percentage point. And I have to ask if that level of precision is meaningful. Here’s a question you don’t answer: at what percentage do you throw back the hand. If you have a 78% percent chance of drawing out is that keepable? Maybe 77%? What about 76%?

I don’t think you can answer that question because I don’t think there is a good answer. The stats just don’t matter to the precision you are discussing. The bulk of the article (the mathy parts) should reasonably be replaced by your conclusion “(Keeping) is a good bet, so I kept.”

Unless you present an argument that is based on being that precise by saying something like “I keep if the odds of drawing the mana I need are 57% or better” then the exercise of narrowing down to a specific number is pretty much meaningless. Its a distraction, it is not the important part of the argument. You are going down a rabit hole to a depth that really doesn’t matter.

And that’s the fault I have with this article. You skip lightly over some really important things (expected matchup, what a random smaller hand is likely to look like) and are very focused on a level of precision that doesn’t matter.

Which I think is misleading.

The difference between the exact 79-80% number and my rough estimate that you’ll draw out somewhere between 7/8 (1-1/2*1/2*1/2) and 2/3rds (1-1/2*1/2*2/3) of the time isn’t meaningful. Its a good bet, keep the hand.

#10 hudnall56 on 05.29.09 at 8:59 pm

@ BitterSting:

I’m pretty sure that concerns of what deck we might be playing against (without any information about our opponent’s deck) are null. At this point we have to consider the hand on it’s own merits. If the deck isn’t good against the metagame as a whole, then we probably shouldn’t even be sleeving it up.

And the math is important. It tells us how often this hand will do what we need it to do. Sure, ideally we would measure the probability of making this hand against the probability of making a better hand with a mulligan, but the calculations for that would likely be complicated enough as to be impractical.

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