In game one of a match on the play, Kanye drew an opening six with one land and a relatively powerful curve, including at least one explore creature to find more lands (this was during the Vancouver mulligan). Since there was a reasonably high chance he would brick on a second land and just not get to play Magic, he was leaning toward taking a mulligan. I told him to keep. He kept, found a second land by turn two, and suddenly had a good hand on six cards.
Here's the thing about keeping hands like that: Under the Vancouver mulligan, the odds of drawing a land with one scry and one draw were not actually that low. With 23 lands in 53 cards, the odds of missing are (30/53)*(29/52), or 31.57%, which means the odds of hitting a land are 68.43%. The odds of winning the game in that case are lower, of course, and the odds of winning if it bricks on a second land are extremely low - I don't estimate more than three to five percentage points to be added to my overall odds of winning, almost all of which come from drawing a land on turn three. This equation doesn't even account for factors that can skew the math like fetchlands, which I know is going to come up here because my turn-one play is fetchland, end step fetch Overgrown Tomb (unless I suspect my opponent might be on a Blood Moon deck, in which case I still fetch before my next draw but I get a Forest instead). With that said, the equation [1 - (misses in deck/cards in deck)*(misses-1/cards-1)] does a reasonable enough job of estimating likelihood, even if it isn't exactly right here.
In this case, the hand was good enough that I felt it would win more than half of the scenarios where it hit a land, which would be a 34.2% winrate overall - let's say 40% of the games with this hand are ones where it hits a land on turn two and wins. Adding 4% from the games where it doesn't draw a land on turn two, that's an overall winrate of 44%.
While 44% is obviously not an ideal winrate, I feel better about this than about a mulligan to five on the play, even a large part of the fail rate being that I don't get to cast a single spell before I die means the mull to five will probably be more fun.
I had a lot of success with Mono-Green Ramp on Magic Online - the most wins I've ever had over four leagues was 18 with that deck (4-1 5-0 5-0 4-1). In addition to better than usual technical play and making mostly reasonable decisions with Karn and sideboarding, I was doing two things to gain percentage points: aggressively activating Field of Ruin in the early turns against opponents who might have one or even zero basics, and keeping hands that had a very real risk of not playing Magic.
Yesterday, I was goldfishing with Jund and sent the following hand to a friend:
"This is a pretty good keep, right?" "Ehh. I'm off it," he replied. I liked the hand, but was willing to defer to him. He didn't like throwing away a card with the second Scavenging Ooze, which is unlikely to do anything. He also didn't like the risk of drawing badly and being unable to leverage play skill at all, especially when there are certain cards and even entire decks like Tron that this hand is awful against. I liked how strong the curve of double Tarmogoyf into Scavenging Ooze is, assuming I draw at least one relevant piece of interaction that can help me win the game within the first two turns (this is not an exact calculation, and I calculated the chance of drawing interaction within the first two at up to 74%. As with the Mono-Green Ramp hand, that number might be lower due to a few specific pieces of interaction being dead against most decks.
The two people I asked about the hand, and the two people they asked, were split 2/2 on whether they would keep or mulligan on the play. Both people who said they would mulligan thought it was close.
What interested me when I posted the hand on Twitter was that so many people thought this hand was an easy mulligan. Yes, there is a very decent chance it does nothing or gets run over. The games it loses won't be close. But the chance of drawing relevant interaction is high, and taking a mulligan is not free.
Even if it's justifiable to keep, it's probably close enough that a mulligan is also justifiable by personal preference, but I am certain that the difference between this hand and the winrate of an unknown six-card hand is not high.
David Inglis, also known as tangrams, made light of the post by asking if I'd keep this hand:
Tangrams is good at Magic. If I could only get his opinion on whether to keep or mull this hand and I didn't trust my own opinion, I'd snap off a mulligan. I thought the meme was funny, and it hits on the main reason to mulligan this hand (a pile of two-mana creatures and nothing else is awful against most Modern decks).
Aside from the lands being on the right instead of on the left, my problem with the meme is that it misses two key points in the counterargument for keeping the hand. The first is that we are playing Jund, a deck with a lot of interaction. If I keep the Grizzly Bears hand and my draws are disruption for my opponent's combo, I feel less bad about it. The second is that Tarmogoyf-Tarmogoyf-Scavenging Ooze goldfishes a turn faster than the quadruple Grizzly Bears hand and can more effectively deal with interaction or decks that rely on the graveyard.
This is a fundamental point I think a lot of people miss: They consider what a hand is capable of doing on its own, but undervalue how the likelihood of winning is affected by drawing certain cards (like lands or interaction) with a lot of copies in the deck.
When is it wrong to keep hands that might do nothing?
Tron hands that can only find two of the three Tron lands are often mulligans. Tron's odds of winning on a mulligan are high compared to other Modern decks, and the odds of drawing the third Tron piece are probably a bit lower than the odds of finding relevant interaction with the Jund hand. Since the Jund hand was probably at most a few percentage points better than a mulligan, tilting the odds of winning by, say, 8% in the direction of a mulligan with a Tron hand makes it a fairly clear decision to mulligan.
If a hand is missing more than card it needs to win the game and doesn't have many ways to speed up its draws, I almost always mulligan. This often comes up with one-land hands in decks that need to hit their first three land drops - whereas the odds of hitting one land in your first two draws in a 25-land deck are 1 - (29/53)*(28/52), or 70.54%, the odds of hitting two lands in your first two draws are (24/53)*(23/52), or 20.03%. The actual difference in winrate between these two hands is much smaller, but this is mostly because the 20% hand has so few percentage points to start with that it can't lose many by the opponent's hand doing better things.
Being on the play or on the draw can also make a difference, like with the Jund hand. Stealing wins on the draw often involves being proactive to minimize the tempo advantage the opponent gains from starting on the play. The Tarmogoyf plan is much less effective if it's a turn slower.
Thinking about odds in more concrete terms than guesstimated percentages (like 8% for Tron vs Jund) is optimal for informing decisions in competitive Magic, because it eliminates the chance of making a mathematically incorrect plays based on rough estimates when non-mathematical factors like bluffing or the opponent's play pattern don't matter much. Even so, for someone who doesn't know the exact percentages or how to use them, thinking about the math in rough estimates generally leads to better decisions than not thinking about it at all.
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