Preliminary results for ideal number of lands and ramp in a deck (warning: math)

[Bonethor]

As mentioned being able to see if including mdfc spells in the sense that one could allow for hands that contain more "lands" to be considered keepable - if only to see what kind of an effect they have compared to not running any.

One must be really specific when asking questions from the pantheon of statistics. I think what you're asking is "what if you can keep the fourth land in the case it's a MDFC"? That is an interesting question. I'll see if I can answer that.

This was the earlier way of alluding to it:

and accounting for mdfc spells somehow - since having those would also increase the keepability of the hand if the 4th "land" was a mdfc one.

So that is indeed the question, I guess I'll have to try to be clearer instead of assuming (which is a no-no in general) that people can easily see inside my head and grab onto what I mean. I too think it's an interesting question and it'd be awesome to see the results!

Try again. Instead of attacking the person behind the argument, attack the argument itself. Explain why it's bad advice, and what should be done instead.

I'm not sure I was attacking the person there, the last bit was about genuinely wondering why you'd reply with that.

I didn't bother explaining further since I figured it'd be pretty clear from the context if a person read what was being written. I'm noticing I can't just use stream of thought for these posts.

Counting mdfc as lands is pretty much an established way of looking at them, however in relation to generating the chart they can enable a starting hand to be keepable that would've otherwise "flooded" with too many lands to fit the success -category. The chart just doesn't account for that possibility so one can't really ballpark guess, the math needs to be done if that is to be figured out.

[Aetherium Slinky]

This is still work in progress but I wanted to publish the intermediate findings for cEDH, too.

This chart is optimised for cEDH. Changes: we prefer hands with more ramp and fewer lands. We also assume there are 8 ramp spells at mana value 0 (Moxen, Mana Crypt, etc.). Additionally this chart assumes 4 hands (=3 mulligans), same as before. Here are the keepable hands by our definition:

The "keepable" column describes whether the hand is keepable or not. The "lands" column is for how many lands we accept in a hand in addition to the ramp spells: column "ramp 0" means ramp at mana value 0 (Moxen, Mana Crypt, etc.) and column "ramp 1,2" means ramp at mana value 1 or 2. The column "other" is simply the rest of the cards in the hand. There's a slight problem with Ancient Tomb and Gemstone Caverns because they're both technically lands and ramp but we chose to classify them as ramp this time.

Here's now the chart:

Lands is probably pretty self-explanatory but the two top rows are not. So "total ramp" refers to the total amount of ramp spells in the deck of which the row below called "ramp mv 1 or 2" describes the number of the ramp spells at mana value 1 or 2 (that is the variable). Remember we established that a cEDH deck can have a maximum of 8 ramp spells and mana value 0 and one should always play those. The optimum now shifts to prefer way fewer lands and way more ramp spells. Best probability is found at 32 lands and 17 ramp spells (of which 8 are mana value 0) at a 78.8% chance of success within 3 mulligans. This is because we refuse to take hands with no ramp in them.

The only debatable hand is the one with 3 lands and 0 ramp which in theory is keepable in most situations but optimising for it would cause the probabilities to prefer lands way too much.

[Aetherium Slinky]

EDIT: The following math is correct but it probably doesn't answer the question I'm trying to ask...so take it with a pinch of salt.

Posted this here earlier but noticed a couple of mistakes so took the post down... Anyway, here's some interesting data about Gilded Lotus! Is it worth it? Assumptions:

• When we draw a Gilded Lotus in our opening hand and it's a keepable hand what's the probability that Gilded Lotus is a dead card?
• We consider the opening hand and five draws.
• No mulligans are considered. Doesn't change the math, really, unless you mulligan really low.
• A "keepable hand" is one of the following four options: 2 lands and 1 ramp spell, 2 lands and 2 ramp spells, 3 lands and no ramp spells, 3 lands and 1 ramp spell.

The leftmost column is the number of lands in the deck. The topmost row is the number of ramp spells in the deck. For the previously found "ideal deck" - 36 lands and 12 ramp spells - the chance of us hitting 3 or 4 mana only by turn 5 when starting with a keepable hand is 6%. Sorry for the duplicate 0-columns, they're there for a reason and I can't delete them (yet).


Math: P(good starting hand) x P(not enough mana sources in the subsequent draws). That's it.


Some of you may have noticed some funny business going on in the top left corner. It's a bit hard to explain but let's look at an example: the 0-column does not consider ramp spells at all. This makes the calculation simple. The full formula used there is P(3 lands in opening hand) x P(0 lands in 5 draws) + P(3 lands in opening hand) x P(1 land in 5 draws) = P(3 or 4 lands by turn 5 given that we kept a 3 land hand). The problem here is that when your land count is significant and goes even higher everything works as expected: you're more likely to draw that 3 land opening hand and that 4th land. However... When the land count approaches zero your chance of drawing a 3 land hand goes down faster than the chance of getting 0 or 1 lands goes up. This results in a weird situation where the question we pose forces you to draw that good hand despite the low chance of that happening after which you're "stuck" on the low amount of mana with a very high percentage of that happening. I'm open to solutions to this problem.

Having fewer mana sources does not make casting Gilded Lotus easier. It makes it easier to get stuck on low mana.


I found another approach to this problem that solves the "funny business" problem. Assumptions:

• We draw an opening hand and do not take mulligans or subsequent draws into account.
• A "keepable hand" is one of the following: 2 lands and 1 ramp spell, 2 lands and 2 ramp spells, 3 lands and 0 ramp spells, 3 lands and 1 ramp spell. (Same as before.)
• Gilded Lotus replaces either a land or a mana rock.

Here the leftmost column represents the number of lands in the deck. The top row represents the number of ramp spells in the deck. Gilded Lotus replaces a land. The chart answers the question: "What percentage of hands would have been keepable if they did not contain Gilded Lotus?" From the chart we see that for the ideal deck of 36 lands and 12 ramp spells the failure rate of Gilded Lotus is around 4%.

Math: P(hands ruined by Gilded Lotus)/P(all keepable hands). The resulting percentage is not a probability, it's just a ratio.



Here the leftmost column represents the number of lands in the deck. The top row represents the number of ramp spells in the deck. Gilded Lotus replaces a mana rock. The chart answers the question: "What percentage of hands would have been keepable if they did not contain Gilded Lotus?" From the chart we see that for the ideal deck of 36 lands and 12 ramp spells the failure rate of Gilded Lotus is around 2%.

Math: P(hands ruined by Gilded Lotus)/P(all keepable hands). The resulting percentage is not a probability, it's just a ratio.


All in all these exercises reveal that depending on how you calculate it Gilded Lotus ruins your opening hand 2%-6% of time. That's not horrible but it's not good either. The latter exercises don't even guarantee you can cast Gilded Lotus at all, they simply compare keepable opening hands to non-keepable opening hands. I will try and see if I can add the subsequent draws to the latter charts somehow.

[anjinsan]

Really interesting stuff. "How many lands" is an age-old question in Magic, especially EDH, but it's one I've been revisiting recently as I've found myself wanting to cut lands in favour of more rocks and ramp.

• Example 3: you have no idea what you're doing. You just want to know what is the ideal composition of lands and ramp. First find the highest value in the chart (for your convenience I've highlighted it in orange) and then look up the number of lands from the left and number of ramp spells from the top. The highest probability of a good hand in the chart is 89.1% at 36 lands and 12 ramp spells.
• Example 4: you're happy if you get screwed in 12.5% of the games (one in eight) you play and you want to know what is the bare minimum number of lands and ramp you should have. You should assume that the number of ramp spells is the more important factor here because that propels you ahead of curve. Lands would not. You prioritise the number of ramp spells and try to find a cell with a chance of 100-12.5=87.5 with the highest number of ramp spells and minimum number of lands. There's a pretty good match at 31 lands and 16 ramp spells (with a probability of 87.6%) so you pick that.

I think though that it's not quite accurate to call this the "best" distribution, necessarily. The 89.1% is the highest chance of getting a good starting hand, but it doesn't say anything about the rest of the game. You've partially built this in in that a "good" starting hand includes no more more than four land/ramp cards, too, which forces the decks not just to be all ramp and lands, but it still doesn't really tell us what subsequent draws will look like. I might prefer to have a slightly larger chance of getting a sucky opening hand if it lowers my chance of getting flooded with lands later.

This applies to your example 4 as well: it's not a 12.5% chance of getting completely screwed. It's a 12.5% chance of having a hand which looks like you're screwed - but in some number of those you will just topdeck another land or piece of cheap ramp on your first turn. I never want to keep a two-land hand because it's just too risky but, actually, if ~1/3 of your deck is lands and ramp, you'd expect to get your third piece by T3 anyway. I wouldn't rely on it, but it will turn some percentage of those bad hands back into good ones.

Strictly speaking they did not optimise. They couldn't have because they did the wrong math. They just said "yeah I'm fine with X percentage" which isn't the same as figuring out "what's the best percentage". I know why, though. They chose to include 4 and 5 land hands which probably makes the ideal land count something like 50+. Vaasa and I determined that if you include the 4 land hands your "best option" is to run 49 lands and no ramp. With 5 land hands it must be even worse.

This is possibly because the model doesn't weight ramp and lands right?

Typically in EDH, three lands and a ramp is way better than four lands, for example; ramp is "better" in that it lets you go faster rather (of course, as I think many of us forget sometimes, ramping and missing a land drop is just a waste of mana). It looks like your cEDH version sort-of favours this, but I think it boils down to the fact that your hands are simply acceptable or not, there's no way to distinguish between them. What this means is that if you include four-land hands, you're also saying that four-land hands are exactly as good as every other hand, so the model will happily try to get as many of those as it can by amping up your land count (also - see above).

On a related note, I don't think this is an optimal mulliganning strategy (though it's probably close enough, and an optimal one would be rather too headache-inducing I fancy); you stop when you have a hand that's "good enough", but whether it's wise to gamble on a better hand or not depends a bit on the deck too (giving you a bit of a feedback loop, unfortunately). For example, if I only included 10 lands in my deck and no ramp, a two-land opening hand is frankly probably the best I could hope for, so I wouldn't want to mulligan. I wouldn't suggest changing your model for this but it possibly explains some of the weirdness around the Gilded Lotus thing (I only skimmed that, I feel that needs separate consideration  .

[Aetherium Slinky]

I've explored a few angles to Gilded Lotus (or any 5-drop) already but here's another one: assuming in all cases we're going to keep a "good hand" we count the hands that contained Gilded Lotus and could cast it as well by turn 5. Then compare that to all the potential good hands that contained Gilded Lotus. We could call this "success rate" for being able to cast Gilded Lotus if your opening hand contains it and it's a reasonable hand in general.

Assumptions:

• The deck contains 99 cards.
• A "good hand" is one with 2 lands & 1 ramp, 2 lands & 2 ramp, 3 lands & no ramp, or 3 lands & 1 ramp.
• We do not consider mulligans.
• We consider "by turn 5" to mean exactly 5 extra draws.
• Each hand contains Gilded Lotus.
• A "success" is when you have enough mana on turn 5 to cast Gilded Lotus.
• Unfortunately I wasn't able to eliminate the hands that had a total of 4 mana sources on the battlefield plus in hand on turn 4 and drew a ramp spell instead of a land. Hence in reality the success rate is a little lower than the chart's indicated value.

The leftmost column is the number of lands in your deck and the topmost row is the number of ramp spells in the deck. For a reasonable 36-land, 12-ramp deck the success rate is about 86%. This isn't a probability, rather, it's a rate or a ratio of probabilities.


Math: we take all hands that are "good hands", contain Gilded Lotus and are able to cast it by turn 5 after which we divide that by the number of all "good hands" that contain Gilded Lotus. This way we get a ratio that most likely answers the question "if my opening hand is otherwise keepable how likely is it that I'm able to cast my Gilded Lotus on curve (or a little ahead)?" I think this approach is the best out of the ones I've explored thus far because it clearly demonstrates how castable Gilded Lotus is in a reasonable hand.


A poor laptop was harmed during the making of this chart. Okay, it didn't die but it crashed my spreadsheet four times before it calculated the results. Fun fact: I had to learn some javascript for this because I had to make a custom function that is not available in any of the spreadsheet programs as a built-in function.

[Aetherium Slinky]

I think though that it's not quite accurate to call this the "best" distribution, necessarily. The 89.1% is the highest chance of getting a good starting hand, but it doesn't say anything about the rest of the game. You've partially built this in in that a "good" starting hand includes no more more than four land/ramp cards, too, which forces the decks not just to be all ramp and lands, but it still doesn't really tell us what subsequent draws will look like. I might prefer to have a slightly larger chance of getting a sucky opening hand if it lowers my chance of getting flooded with lands later.

What's your exact suggestion? I can simulate extra draws but going beyond ~5 draws is in my opinion pointless because people will start casting draw spells at that point already which makes the data incredibly hard to handle.

This applies to your example 4 as well: it's not a 12.5% chance of getting completely screwed. It's a 12.5% chance of having a hand which looks like you're screwed - but in some number of those you will just topdeck another land or piece of cheap ramp on your first turn. I never want to keep a two-land hand because it's just too risky but, actually, if ~1/3 of your deck is lands and ramp, you'd expect to get your third piece by T3 anyway. I wouldn't rely on it, but it will turn some percentage of those bad hands back into good ones.

Yes and no. You said it yourself: you never want to keep a two land hand unless there's something really special going on (such as Sol Ring or something). You are absolutely right about the subsequent draws providing some help in a situation like that but we already established that such a hand is not a keeper. Note that I didn't mulligan like crazy so there's still some room to go down a card or two if you're getting abysmal hands.

Typically in EDH, three lands and a ramp is way better than four lands, for example; ramp is "better" in that it lets you go faster rather (of course, as I think many of us forget sometimes, ramping and missing a land drop is just a waste of mana). It looks like your cEDH version sort-of favours this, but I think it boils down to the fact that your hands are simply acceptable or not, there's no way to distinguish between them. What this means is that if you include four-land hands, you're also saying that four-land hands are exactly as good as every other hand, so the model will happily try to get as many of those as it can by amping up your land count (also - see above).

I agree with your statement about four lands. At some point you do need to make a decision, though. There must be some criteria you're following when you decide to keep a hand. You can't Partial Paris mulligan anymore so each hand is either acceptable or not acceptable. There's no in between. My model may not catch every scenario and in fact I'm interested in hearing what kind of improvements you'd suggest!

On a related note, I don't think this is an optimal mulliganning strategy (though it's probably close enough, and an optimal one would be rather too headache-inducing I fancy); you stop when you have a hand that's "good enough", but whether it's wise to gamble on a better hand or not depends a bit on the deck too (giving you a bit of a feedback loop, unfortunately). For example, if I only included 10 lands in my deck and no ramp, a two-land opening hand is frankly probably the best I could hope for, so I wouldn't want to mulligan. I wouldn't suggest changing your model for this but it possibly explains some of the weirdness around the Gilded Lotus thing (I only skimmed that, I feel that needs separate consideration  .

What is the optimal mulliganing strategy?

[Slyvester12]

A poor laptop was harmed during the making of this chart. Okay, it didn't die but it crashed my spreadsheet four times before it calculated the results. Fun fact: I had to learn some javascript for this because I had to make a custom function that is not available in any of the spreadsheet programs as a built-in function.

Couldn't you use excel's lambda function to write a new function? I've never tried it, but I know a lot of people were excited when it released. It's also pretty easy to learn some basic R for data wrangling and excel manipulation.

[Aetherium Slinky]

A poor laptop was harmed during the making of this chart. Okay, it didn't die but it crashed my spreadsheet four times before it calculated the results. Fun fact: I had to learn some javascript for this because I had to make a custom function that is not available in any of the spreadsheet programs as a built-in function.

Couldn't you use excel's lambda function to write a new function? I've never tried it, but I know a lot of people were excited when it released. It's also pretty easy to learn some basic R for data wrangling and excel manipulation.

I did not know that exists. Also I don't have a license for Excel available at home - only Google Sheets and LibreOffice and they don't seem to have that function. I could have used the lambda function at work. But I did get the job done with the javascript mess so mission accomplished I guess?

[Slyvester12]

A poor laptop was harmed during the making of this chart. Okay, it didn't die but it crashed my spreadsheet four times before it calculated the results. Fun fact: I had to learn some javascript for this because I had to make a custom function that is not available in any of the spreadsheet programs as a built-in function.

Couldn't you use excel's lambda function to write a new function? I've never tried it, but I know a lot of people were excited when it released. It's also pretty easy to learn some basic R for data wrangling and excel manipulation.

I did not know that exists. Also I don't have a license for Excel available at home - only Google Sheets and LibreOffice and they don't seem to have that function. I could have used the lambda function at work. But I did get the job done with the javascript mess so mission accomplished I guess?

Yeah, I wasn't try to say using JS was a bad decision. Just offering some other routes if you had a similar situation in the future. Personally, I prefer doing data stuff in R (which is free), but whatever you're comfortable with works.

Also, cool analysis. How did you make the chart? Is it just a spreadsheet and the formulas use row/column number as variables?

[Aetherium Slinky]

How did you make the chart? Is it just a spreadsheet and the formulas use row/column number as variables?

It is just a spreadsheet indeed and the row and column are the variables, yes. G Sheets has this neat feature "conditional formatting" which basically turns the table into a heat map. I tried a few 3D plots but they didn't either work with this much data or they didn't look nice. The simplest solution turned out to be able to visualise the data rather nicely.

[anjinsan]

Yes and no. You said it yourself: you never want to keep a two land hand unless there's something really special going on (such as Sol Ring or something). You are absolutely right about the subsequent draws providing some help in a situation like that but we already established that such a hand is not a keeper. Note that I didn't mulligan like crazy so there's still some room to go down a card or two if you're getting abysmal hands.

What I mean is, whilst I don't want to risk keeping e.g. a two-land hand, some percentage of those hands will actually turn out to be OK. That means that if though such a hand might not be a keeper if I can still mulligan, running out of mulligans and being stuck with one isn't quite as bad as we're saying - some percentage of the time it will be fine.

This also means that we don't actually have a one in ten chance of throwing the game, or whatever. We have a one in ten chance of keeping a bad hand, which means a one in twenty (say) chance of keeping a bad hand and not then topdecking your way out of it (also a chance of being manascrewed for a turn or two then making a late-game comeback, for example). You might be willing to put your proportion of bad starting hands a little higher once you take that into account.

That also plays into the mulligan strategy a bit - you might decide that you're more likely to topdeck your third land than get a decent hand if you go down to 5 or 4 cards, for example.

What is the optimal mulliganing strategy?

I don't know - that feels like a whole topic in and of itself.

In fact, it's actually a little subjective, since I'm not sure what the optimal outcome is. Do you want to maximise your chances of getting an acceptable hand (as you are), or try to get the best hand? They're slightly different.

[Aetherium Slinky]

What I mean is, whilst I don't want to risk keeping e.g. a two-land hand, some percentage of those hands will actually turn out to be OK. That means that if though such a hand might not be a keeper if I can still mulligan, running out of mulligans and being stuck with one isn't quite as bad as we're saying - some percentage of the time it will be fine.

This also means that we don't actually have a one in ten chance of throwing the game, or whatever. We have a one in ten chance of keeping a bad hand, which means a one in twenty (say) chance of keeping a bad hand and not then topdecking your way out of it (also a chance of being manascrewed for a turn or two then making a late-game comeback, for example). You might be willing to put your proportion of bad starting hands a little higher once you take that into account.

That also plays into the mulligan strategy a bit - you might decide that you're more likely to topdeck your third land than get a decent hand if you go down to 5 or 4 cards, for example.

Some of the things you suggest mean I should consider extra draws, too. I can do that but the problem is that then I need a "target" for what is a good overall outcome.

In order to know how likely you're to "draw yourself out" of a bad starting hand I need to know how many lands and ramp spells I need to draw in the subsequent turns in order for the outcome to be deemed as acceptable.

There is a way to approximate that: average value of the deck. Most folks want to be able to cast two spells per turn or cast the highest mana cost of the deck. We could assume that the highest costing spells cost twice the amount of mana (rounded up) as the average mana value of spells of the deck. For two spells we could assume two times the average mana value (rounded up) which is a higher number. So one of those could be a target, for example.

Then accounting for draw spells becomes a problem. There's probably a way to do a (Monte Carlo) simulation that takes this into account but it is not doable with a spreadsheet as that would bring the chart into four dimensions and visualising that is incredibly hard, at least in a way that is easily readable. There's always an option for calculating just a single set of numbers out of all this but that reduces the usefulness of those numbers because you can't really see what is happening in the data. I mean we'd lose the ability to answer the question "how bad is it to replace a card of <a card type> with an actual game piece that advances your game plan directly".

All in all I think the best course of action is to stick to the opening hand or a couple of draws at most when draw spells aren't a consideration yet.

I also want to remind you that the event of having to keep a bad opening hand is really small and the possibility of drawing back into a decent situation is also small. These numbers need to be multiplied which means the overall probability is really small. I understand that it's there but taking it into account may not change the visible numbers at all due to rounding, for example. Your guesstimate:

We have a one in ten chance of keeping a bad hand, which means a one in twenty (say) chance of keeping a bad hand and not then topdecking your way out of it (also a chance of being manascrewed for a turn or two then making a late-game comeback, for example).

My guesstimate: not that high of a chance. The chance of seeing four low mana hands in a row is less than 1%. Multiply that by any number and your overall chance of having to keep a crap hand and drawing anything on top of that is still less than 1% (which is really low). That one in ten number also includes mana flooded hands so if we exclude those and consider yourself being "totally screwed" because you didn't draw a single good hand in three mulligans the chance is around the aforementioned 1% of that happening. Mulliganing to 4 and keeping any hand that comes up is still less than 1% and it doesn't really matter if we consider the future draws or not because that's not going to affect the overall percentages I displayed in any meaningful way.

[anjinsan]

Honestly, I'm not suggesting you change your analysis in that respect - just that the conclusion that if you only put this many lands in, you're screwed that many times, isn't quite right.

The difficult question is, how much chance of getting a "bad" hand is too much? The problem is that this depends upon the rest of your deck, too. To me, anything less than, say, 95% sounds pretty low... but firstly, as I said, the starting hand isn't everything as you still have a chance to draw out of it, and also this has to be balanced against the rest of the deck. Rejecting starting hands with too many lands keeps the deck from being all lands and ramp, but it still doesn't measure the actual chances of, say, getting mana flooded later on.

Is there also something we can say here about Magic as a game? If the best we can hope for is a ~80% or so chance of a decent starting hand, and we're still potentially facing mana flood later, is the actual conclusion just that EDH is a hella random format and pure chance will just ruin you plenty of the time? 

[Aetherium Slinky]

I've done some more math. I finally figured out the Gilded Lotus thing and I can tell you it wasn't easy. Ultimately I wasn't able to answer the question I set out to but that's okay. I found other answers. I tried to ask "How often will Gilded Lotus screw up your starting hand?" but that was really complicated to formulate into probabilities. Instead I got the answers to the following questions:

• How much does replacing a land with Gilded Lotus help you get a good hand?
• How much does replacing a ramp spell with Gilded Lotus help you get a good hand?
• What's the probability of being able to cast Gilded Lotus on curve given that your hand was reasonable otherwise?
• What's the proportion of games where you're able to cast Gilded Lotus on curve when you kept a reasonable hand?

Let's get into it. I've laid out the basic assumptions before already so I'm not going to through them again.

How much does replacing a land with Gilded Lotus help you get a good opening hand?

The question formatting is very deliberate. The math is P(good hands with Gilded Lotus)-P(generically good hand). So for example if you've got 35 lands and 9 ramp spells the net effect of Gilded Lotus on drawing a good hand is about -1% i.e. replacing a land with Gilded Lotus decreases your chance of drawing a good hand. Having 37 lands and 15 ramp spells on the other hand is excessive from the perspective of drawing an ideal hand so replacing a land with Gilded Lotus actually improves your chances of drawing a good hand by 0.5%. You're less likely to draw too many lands in your opening hand.

How much does replacing a ramp spell with Gilded Lotus help you get a good opening hand?

The question formatting is very deliberate. The math is P(good hands with Gilded Lotus)-P(generically good hand). So for example if you've got 36 lands and 8 ramp spells the net effect of Gilded Lotus on drawing a good hand is about -2% i.e. replacing a ramp spell with Gilded Lotus decreases your chance of drawing a good hand. Having 37 lands and 15 ramp spells on the other hand is excessive from the perspective of drawing an ideal hand so replacing a ramp spell with Gilded Lotus actually improves your chances of drawing a good hand by 1%. You're less likely to draw too many ramp spells in your opening hand.

What's the probability of being able to cast Gilded Lotus on curve given that your hand was reasonable otherwise?

Here Gilded Lotus doesn't replace anything. This chart assumes you always draw Gilded Lotus in your opening hand. No mulligans. We do 5 extra draws to see if we can cast Gilded Lotus on turn 5. The math is P(good hand) x (1-P(not drawing enough mana)). I actually went hand by hand so it takes into account the fact that 3 lands and 0 ramp needs to draw 2 additional mana sources whereas 3 lands and 1 ramp only needs to draw 1 additional mana source. For example the "ideal composition" of 36 lands and 12 ramp spells produces a 36% chance to draw a good hand and into a situation where you can cast Gilded Lotus on curve. This also means that if you've got a really high mana curve (say average mana cost over 4) you might want to optimise your deck differently and add a lot more lands and ramp. The ideal amount of lands and ramp based on this chart would seem to be 43 lands and 16 ramp spells.

What's the proportion of games where you're able to cast Gilded Lotus on curve when you kept a reasonable hand?

Here it's kind of the same math as the previous example except we divide out the probability of drawing into a good hand. So this here is the proportion games where you were able to cast Gilded Lotus on curve given that you kept a good hand. For example with 36 lands and 12 ramp spells you've got an 86% chance to cast a 5-drop on curve. This actually applies to all 5-drops so if your commander costs 5 mana 1 game in 7 games you're not able to cast your commander on curve given that you kept a reasonable opening hand. I think this chart is the best one at describing how important it is to run enough lands and ramp in your deck. The differences are significant and even one land / ramp spell makes a difference in your ability to cast bigger spells.

Summary of results

• If you're playing reasonable amounts of lands and ramp replacing a land with Gilded Lotus doesn't make much of a difference in terms of drawing a good hand. Mulliganing away a Gilded Lotus hand is always an option.
• Replacing a ramp spell makes a bigger difference but it's still not much. One could argue that 0.5%-1% difference is enough of a difference not to replace a ramp spell with a land but that's up to personal preferences. Mulliganing away a Gilded Lotus hand is always an option.
• The chance of being able to cast Gilded Lotus on turn 5 is less than half so if you do find a hand with Gilded Lotus in it it would make sense to keep it only if it has an abnormally large amount of mana sources.
• If your average mana cost is really high or your commander costs a lot you may want to up your mana source count significantly.

[Akira Foxmind]

I need to rest my browser so I'll edit this message in a bit.

Just offtopic and out of curiousity, what do you mean by 'resting your browser'?