Introduction

Hello GB friends, I’m Seven. I wanted to write an article on probabilities for certain decisions that are common in Green Blue. I’ve been playing ramp decks (mainly DKP RB), a couple of sets, and with these kind of decks relying so much on chance to draw ramp and some key cards, I believe that is key to delegate certain decisions to probability.

I like to think about a card game as you would for a chess game for the most part. There is an opening phase, in which you have to already have the lines memorized, because if you start improvising and your oponent knows the correct line you’re probably already lost, or at least on a big disadvantage; A midgame, in which you try to develop your game plan; and a late game, in which a single decision is game deciding. In card games you don’t always have the same pieces in hand, nor does your oponent, so a chess opening would more correctly relate to a matchup opening. This is really important as we want to consistently have the correct pieces for the right matchups. What I want to say with this is the less decisions you have to make before entering midgame, the most consistent you can make your gameplan. This is why the correct Altering decisions can make us have better hands and convert more games into victories.

For the most part I just want to present the data and then my conclusions as to why I decide to Alter a certain way based on the data in the end. This way you can see my interpretation of the data, which can obviously be wrong, and I invite anybody that can refute it with data to do so. I tried to double check everything that I am writing here.

Concept glosary

During this article I will use a shorthand for certain concepts that encompass groups of cards, I want to use this section to explain what those mean.

  • Ramp: Ramp would include tipo and sail. We would have 8 targets for a ramp card.
  • DyB: Develop your Brain.
  • Clara7: Just big Clarabelle.
  • Clara3: 3 cost Clarabelle. 4 targets.
  • Clara1: 1 cost Clarabelle. 4 targets.
  • SmolClara: The combination of both Clara3 and Clara1, for which we would have 8 targets.
  • ClaraEnablers: This would include both clarabelle shift targets and donald5. As this cards enable lines where you can cheat Clara7. We have 12 targets for this group. This group will serve as a baseline to calculate the probability of at least being able to cheat out clarabelle earlier than you would normally. Which IMO is the baseline to have a decent game. Not being able to do this normally means youre at a disadvantage, so we want to at least hit one of these.
  • 3UtS: When I refer to running 3 targets for Under the Sea.
  • 4UtS: When I refer to running 4 targets for Under the Sea.

Altering for ramp

This has long been a discussion in the community, also for other ramp dependant decks. You normally throw everything away to have the best chance to draw back ramp after 7 new cards. The two cards that generated discussion as to why to Alter a different way were Pawpsicle and Develop your Brain. Which change the math around being able to draw the ramp we need after the Alter phase. Taking pawp out of the ecuation, there is another card that people run sometimes in GB, which is Bobby. Bobby doesn’t exactly draw you a card, but it lets you dig one card deeper. Develop allows us to dig two cards deeper.

Keeping DyB with no ramp has been the main question for x8 ramp blue decks for a while, and I was on the boat for throwing it because I thought the variance of not seeing that last extra card didn’t compensate for the two extra cards you’re seeing later, but I did the math, and it turns out it does. It’s kinda stupid to argue about this decision, as the difference is so minuscule that most of the time it won’t matter. But nonetheless, knowing that seeing two cards after shuffling is better than the extra chance that seeing 14 cards deep before shuffling gives is something that applies to most decks and is information that I will use looking forward.

This data is taken from a script that a fellow GB player made for making simulations on the Alter phase and then two turns of play, to calculate the mean for how many times depending on the situation you’re in you will draw ramp. I modified the script to run several simulations so that I could calculate Standard Deviation, to get a bit of context. The mean doesn’t always tell the full story, but variance, in this case stdev gives us a look at the inconsistency over all those simulations.

This stats are the result of a simulation where 100.000 players “played” 500 games with a determined way of playing if not finding ramp in the first seven cards. This takes into account all the different nuances that can occur, like throwing 7 can sometimes result in having a DyB, or throwing 6 but drawing into another DyB. Even with all that the stats are clear. Even though that it’s a stupidly small number, keeping DyB is always the best, it even nets smaller variance. So in about 0.55 games every 500 OTP and 0.4 games OTD, you will not find ramp if you throw away your develop as opposed to keeping it, while also being a tiny bit less consistent. Without taking variance into consideration, which only improves the odds for keeping DyB.

(script will be provided but not present in this article)

Situation Mean Standard Deviation
OTP Throw 7 92.46% 1.187%
OTP Keep DyB 92.57% 1.177%
OTD Throw 7 93.73% 1.084%
OTD Keep DyB 93.81% 1.081%

Statistics of the simulation in which we calculate how many times over the simulations we’re getting ramp back with each strategy

Cow math

Having Clarabelle 7 on board, specially shifted on 5 ink is really important for GB, for this reason we do want to maximize the chances of enabling the shiftline, or at least, being able to play donald5 to be able to play Clara next and get value with maybe MKB or YW. This decision is critical, as we’re going to see, taking an incorrect strategy in the Alter phase means you’re not going to get Clara7 on board as much as you want to. For this section and onwards, I will use hypergeometry to analyze the decisions. So we’re talking probability, not statistics.

Pre alter situations

We do not have ramp in hand

We’re going to start with a hand where we do not have ramp but we do see a Clara7. In this situation, we could think throwing 7 cards is the correct choice. Here we have to think if having Clara7 on board is critical to the game plan, which on most matchups is. So we have two options, we throw everything, and hope to find ramp and Clara7 back, or we keep the Clara7 and throw everything else. Let’s see how the probability changes with this two scenarios in mind.

Situation Odds of getting ramp
Keep Clara7 70.6%
Throw Clara7 64.5%

Probability of drawing ramp if we keep Clara7

Now, the probability of getting a Clara7 back in hand if we throw one away is 35.2%. So the question is, do we think that even if we find ramp by throwing 7, which will make a difference in 6.1% of games, will result in a game worth playing, or will it be a nongame anyway in which we ramp into nothing and lose anyway. For me right now, I think 64.5% is still decent odds that I prefer to Keep the Clara7 and make sure we have it. As we will see later, it’s much easier to find a ClaraEnabler and at 64.5%, we have decent odds to have a normal gameplan if the ramp comes back, as opposed to possibly never being able to find a Clara7 in time until it’s too late after shuffling them back into the deck.

====

We do have ramp in hand

In this situation we already have the most important piece in hand, so let’s see all the situations we can find ourselves in and see what probabilities we’re playing with.

Situation Odds
Clara7 alone 39.1%
Clara7 + SmolClara 23.2%
Clara7 + ClaraEnabler 29.6%

Probability of finding Clara7 in combination with other cards while keeping ramp and throwing everything else

Situation Odds
Keep ramp + 1 SmolClara 33.1%
Keep ramp + 2 SmolClara 27.6%

Probability of finding Clara7 while keeping ramp and some SmolClaras

Situation Odds
Any shift target 56.7%
A specific shift target 33.6%
ClaraEnabler 73.1%

Probability of finding Clara7 enablers if we keep ramp and Clara7 in hand

====

Post alter situations

We have no Clara7 in hand

Here we’re analyzing the probability of drawing into Clara7 before the shift turn with five inks. Having a minimum of three draws OTP and a maximum of 8 with a combination of being OTD while playing Bobby, Sail and DyB.

Number of draws Odds
3 21.4%
4 27.6%
5 33.6%
6 39.1%
7 44.0%
8 49.1%

Probability of drawing Clara7 with each consecutive draw

We have Clara7 in hand

In this situation we’re looking to, while having Clara7 in hand, being able to play her before turn 6.

Number of draws Odds
2 28.2%
3 39.4%
4 49.1%
5 57.4%
6 64.5%
7 70.6%
8 75.7%

Probability of drawing a SmolClara before being able to play her to be shifted with 5 ink

Number of draws Odds
3 54.5%
4 65.4%
5 73.9%
6 80.4%
7 85.4%
8 89.2%

Probability of drawing a ClaraEnabler to at least be able to play Clara7 earlier than usual

Under the Sea after the alter phase

In this section I want to analyze some common lines in which we don’t keep UtS against Dogs while still being able to sing it later. Ranging from some of the least common to a bit less optimal one. There are a million situations we could calculate here, but I decided on this 5 to have a general idea of the differences between 3UtS and 4UtS, which I think are enough to make conclusions.

Situation 3UtS/4UtS
Line 1* 11.3%/14.7%
Line 2* 16.6%/21.4%
Line 3* 40.1%/49.1%
Line 4* 48.1%/57.9%
Line 5* 31.3%/39.1%

Line 1: Clara>Tipo>Baymax 2 cards deep OTP

Line 2: Bobby>Tipo>Baymax 3 cards deep OTP with Bobby or OTD organically

Line 3: Clara>Tipo>Baymax>Visions 8 cards deep OTP

Line 4: Clara>Tipo>Baymax>Visions>DyB 10 cards deep OTP

Line 5: Tipo>Donald4>Donald5 6 cards deep OTP

Conclusions

After presenting all this data, I want to elaborate on the decisions I am making based on it.

First of all, talking Develop your Brain, we know that looking at two cards after shuffling is better in terms of consistency and variance to seeing that 13th or 14th card. So what I conclude is that I am always going to keep a DyB in hand, either without ramp, or in combination with ramp, to get closer to that Clara7. It also improves your chances of drawing into your next wanted card which could be UtS or the Basil7, Hades and the likes.

When looking into getting Clara7 and her enablers, I conclude that If I already have a Clara7 in hand, even with no ramp, I prefer to keep her over the possibility of never getting her back again. There are alternative lines where you can save a matchup with no Clara7 on board, but in general I want to avoid those at all costs as when you don’t have the Clara7, the shiftlines can become completely useless, essentially having 12x Tipo in your deck which you don’t want to draw into but you’re a lot more likely to do so.

Having no Clara7 in hand leaves me with no options but never keeping shiftlines in hand unless it’s Clara3 OTP, which trades a bit of variance of having less chances to get Clara7, with the option of having the best shiftline and also drawing that extra card with her, and possibly getting that shift anyway. I would never do this if I’m OTD or in a matchup where I know Clara3 is not drawing a card. The chances of getting a ClaraEnabler after shuffling them back in is still much much better than the chances of getting Clara7 in hand, so I think based on the data that it’s always better to throw them away.

With UtS numbers, in metas where singing it is a necessity to win a big amount of your expected matchups, having 4 copies in your deck is a must. Specially because having to ink one with Tipo or losing it with an opposing Ursula2 can be game losing.

Also in general, being able to sneak in a couple of DyB to fill the curve over the game makes your wanted draws much more likely and I will want to run 4 copies of them always, specially in decks with lower card quality and almost no cards that replace themselves, in this case GB.

And with this, this article comes to an end. After paying much more attention to probability and Alters, I’ve felt like I’m having much more consistent games. Now I’m not a person that likes to grind 500 games per set, but I’ve getting good records with my last 13 games being 11-2 in the site we must not name in the Bo3 ladder, with lost games being nongames with no ramp or having unlucky games against aggro decks. I’ve been able to climb up to 1480MMR pre Bologna which sadly I was not able to attend. I definitely feel like this is THE deck for me going forward. I have no doubt that it has a huge floor to pilot it correctly and a huge ceiling which we can try to maximize paying attention to probability and statistics, and it can only improve if we get better cards printed.

I hope you have enjoyed this read and have a great Lorcana Set Championship season,

Seven.