[Premium Article] The Grim Long Primer - C-c-combo uncorked!!!

[Klep]

Am I reading this wrong or does your table say with 36 cards and 10 lands left in the deck (40.61%... chance of something?), if you crack a fetch and change the scenario to 35 cards containing 9 lands (40.53%) you increase your odds of drawing less land by 0.08%?  Because that seems intuitively wrong.

Statistics often seems intuitively wrong, but that doesn't make it so.  A classic example is the birthday problem Moxlotus referenced above, which you should read.  Though there are 365 days in a year, it only takes a gathering of about 23 people to make it more likely than not that two of them share the same birthday.  This kind of result is actually not that uncommon.

[Anusien]

Am I reading this wrong or does your table say with 36 cards and 10 lands left in the deck (40.61%... chance of something?), if you crack a fetch and change the scenario to 35 cards containing 9 lands (40.53%) you increase your odds of drawing less land by 0.08%?  Because that seems intuitively wrong.

Yes, you are reading it wrong.  The table appears to say, "For X cards in library and Y lands in library, what is the percentage change in drawing a land?"  According to the second table, going from 36/10 to 35/9, your expected number of lands drawn off Jar drops from 1.94 to 1.8.

[meadbert]

I believe the table is somewhat misleading in that it is only showing us one side of the coin.  I am not sure exactly what it is supposed to tell us, but it appears that it is doing this:

Draw random 7 from original deck and count the number of lands.
Reshuffle.
Removed a land.
Draw a new random 7 from the new deck and count the number of lands.

What is the probability that fewer lands were drawn the second time than the first time.

Note that much of what is going on is just random variations.  For instance no where is there any mention of the probability of actually drawing more lands the second time.

MoxLotus, can you explain in more detail what the table is showing?

[Moxlotus]

I'm not a statistician, so I'm deferring some of these questions to my team's resident math genius

Note that much of what is going on is just random variations.  For instance no where is there any mention of the probability of actually drawing more lands the second time.

Yes, this is how probability goes. Sometimes it works in your favor, sometimes it doesn't. Now matter how many times you fetch, there's always a chance that if you draw two hands before and after, the first hand had 0 lands in it (~18% with 50 cards and 10 lands left), and the second one doesn't. In the latter case, you have a 78% chance of drawing a land in the second hand, the the odds that both happen is 14%.

The thing you need to ask yourself is whether fetching hurt you in that situation. That is, your first hand had 0 land in it, but the second hand had one--did fetching that land out of the deck make it more likely that this would happen? No, because the odds that you draw a land in the second hand decreased a little when you took a land out of the deck. The odds that you draw 0 lands in the first hand remained the same.

Am I reading this wrong or does your table say with 36 cards and 10 lands left in the deck (40.61%... chance of something?), if you crack a fetch and change the scenario to 35 cards containing 9 lands (40.53%) you increase your odds of drawing less land by 0.08%? Because that seems intuitively wrong.

You can't look at it like this. What the (first) table gives you is the odds that you draw less land in a Draw7 hand if you remove one land from the deck before you draw the 7 cards. You look at the number of cards and lands in the deck, and then read the percentage. The table does not give you the odds that you will even draw a land in either hand, or that if you drew two in the first hand that you'll get even one in the second.

Let's suppose you have two fetches, and 36/10 cards left. If you use one fetchland, you'll have a 40.61% chance of drawing less land in the new hand. In that situation, you'll have 35/9 cards left in the deck when you draw the cards. If you had 35/9 the first time, and used a fetchland, then you would have a 40.53% chance of drawing less, having 34/8 lands remaining in the deck when you draw the cards. The reason the numbers are so close is that this is the odds of drawing STRICTLY less land. In both of those situations, you are can draw the same number of land. The odds that you draw the same number of land in that situation:

Note that much of what is going on is just random variations

In one hand, yes, a lot will be just variations.  But if you make this play as a rule all the time, the law of large numbers begins to work. 

Of course, there are still other variables relating to the other cards in the deck.  If there are 25 cards in the deck left in your deck, your will and bargain are in the bin, and you still have ancestral, ponder, brainstorm, timetwister, windfall left (jar, desire, tinker already used), then you really want that Academy.

[Smmenen]

The basic scenario I envisioned could be constructed any number of ways.   The essential idea is that in general, it is a better idea to hold for Academy than to play the Mire and fetch before breaking Jar.   If it would help your math, let’s just say that it is turn one and you have played: Mox Emerald, Sol Ring, Mox Pearl, Black Lotus, Mind’s Desire. 

Simply put, whatever marginal thinning effect there may be, the reality is that unless you are extremely unlucky, you are going to be looking at more cards whether by tutoring or drawing.   And there is a very good chance that you’ll need more than one blue mana.   Even if you draw a black spell, such as Yawgmoth’s Will, you will very likely want to replay Desire.   

Grim Long operates on blue heavily with Windfall, Timetwister, Brainstorm, Ponder, Mystical Tutor, Time Walk, Chain of Vapor, Ancestral Recall and Tinker.   It is certainly likely, although not a given, that you would need more than one blue spell.  Out of memory, I have found that you often need about 1.5 black for every blue mana in Grim Long.   Granted, the restriction of Brainstorm may have diminished that need somewhat, but it also accentuates the importance of other blue spells.   For instance, if your hand reveals Mystical Tutor and Time Walk and just mana, you will really need Academy or another blue source to support it.    And that’s probably the most innocuous scenario.   

[mogz]

Steve what is your opinion on Street Wraith vs. Simians?

Does cycling counts as playing spell?

[Suicideking]

No cycling does not count as playing a spell, since you are not playing anything.

[feyd]

Holding off for academy seems like the wrong play here.  Looking at the numbers it would appear that your chances of drawing academy off a draw seven or flipping it with desire seems minimal.  The slight thinning effect the fetchland has on your deck will, in theory and in practice, increases your chances to flip a ritual effect or tutor effect off desire.  The reality is that you will inevitably draw into land through your draw seven or flip a land at some point, pre-will or post-will, with desire.  I think what people are arguing is that the small benefit you gain from thinning your deck is worth it considering the relatively small chance of drawing into/flipping academy.  
    When you are playing grim long you frequently have to throw away all caution and trust that the fates will provide for you.  Mathematically speaking it is more pertinent to fetch now and increase your chances or drawing less land than it is to wait and possibly draw another fetchland that will be superfluous.

[Purple Hat]

The basic scenario I envisioned could be constructed any number of ways.   The essential idea is that in general, it is a better idea to hold for Academy than to play the Mire and fetch before breaking Jar.   If it would help your math, let’s just say that it is turn one and you have played: Mox Emerald, Sol Ring, Mox Pearl, Black Lotus, Mind’s Desire. 

Simply put, whatever marginal thinning effect there may be, the reality is that unless you are extremely unlucky, you are going to be looking at more cards whether by tutoring or drawing.   And there is a very good chance that you’ll need more than one blue mana.   Even if you draw a black spell, such as Yawgmoth’s Will, you will very likely want to replay Desire.   

er....wouldn't you have access to UUU from lotus in that case thus negating the need for academy?

[Smmenen]

The basic scenario I envisioned could be constructed any number of ways.   The essential idea is that in general, it is a better idea to hold for Academy than to play the Mire and fetch before breaking Jar.   If it would help your math, let’s just say that it is turn one and you have played: Mox Emerald, Sol Ring, Mox Pearl, Black Lotus, Mind’s Desire. 

Simply put, whatever marginal thinning effect there may be, the reality is that unless you are extremely unlucky, you are going to be looking at more cards whether by tutoring or drawing.   And there is a very good chance that you’ll need more than one blue mana.   Even if you draw a black spell, such as Yawgmoth’s Will, you will very likely want to replay Desire.   

er....wouldn't you have access to UUU from lotus in that case thus negating the need for academy?

Yes, but you could easily modify the scenario so you wouldn't.  I am talking about a host of examples. 

Holding off for academy seems like the wrong play here.  Looking at the numbers it would appear that your chances of drawing academy off a draw seven or flipping it with desire seems minimal.  The slight thinning effect the fetchland has on your deck will, in theory and in practice, increases your chances to flip a ritual effect or tutor effect off desire.  The reality is that you will inevitably draw into land through your draw seven or flip a land at some point, pre-will or post-will, with desire.  I think what people are arguing is that the small benefit you gain from thinning your deck is worth it considering the relatively small chance of drawing into/flipping academy. 
    When you are playing grim long you frequently have to throw away all caution and trust that the fates will provide for you.  Mathematically speaking it is more pertinent to fetch now and increase your chances or drawing less land than it is to wait and possibly draw another fetchland that will be superfluous.

I'll tell you what - someone propose a scenario similar to the one I proposed and I'll goldfish it 30 times and report back.   Let's fine a scenario we can all agree on that captures the essence of the hypothetical. 

Steve what is your opinion on Street Wraith vs. Simians?

Does cycling counts as playing spell?

I am not objective about SW, so I couldn't give you an honest opinion.  I feel like that card cost me two matches on the Magic Invitational.   

[Liam-K]

Deciding the "essence of the hypothetical" is going to be a massive arguement in and of itself... a likely Desire hand might go land, mox, DT, pass... lotus, mox, ritual, cantrip, desire.  Your Yawgwill math from here is going to be a lot different than the mox, mox, mox, lotus, desire hand you posit, and is probably a bit more likely (1 of 11 lands + 1 of 4 rituals vs 1 of 3 remaining moxes).

On that note, an important point of the original scenario is that you didn't flip Jar directly, but instead found Tinker.  This will give you a mox and a tinker -> lotus out of the bin that most draw7s wouldn't, as well as reduce the pre-will value of Academy.

[Moxlotus]

For instance, if your hand reveals Mystical Tutor and Time Walk and just mana, you will really need Academy or another blue source to support it.    And that’s probably the most innocuous scenario. 

The second table works for blue cards too.  The list presented has 11 blue cards (including Chain, which isn't business).  You've cast 2 of them, so you're down to 8 blue cards worth casting (although I'm not sure I'd count Walk in there since it's not gonna win you the game this turn).  So you're averaging about 1.2-1.5 blue cards no matter how many cards are left in the deck.  So, again, the need for additional blue mana is probably not needed considering you already have a blue mana.  I'd up my odds for business rather than hope for academy--and then even if you get academy you have to have gotten the multiple blue cards for it to have been useful.

On that note, an important point of the original scenario is that you didn't flip Jar directly, but instead found Tinker.  This will give you a mox and a tinker -> lotus out of the bin that most draw7s wouldn't, as well as reduce the pre-will value of Academy.

This is also true.

[Smmenen]

For instance, if your hand reveals Mystical Tutor and Time Walk and just mana, you will really need Academy or another blue source to support it.    And that’s probably the most innocuous scenario. 

The second table works for blue cards too.  The list presented has 11 blue cards (including Chain, which isn't business).  You've cast 2 of them, so you're down to 8 blue cards worth casting (although I'm not sure I'd count Walk in there since it's not gonna win you the game this turn).  So you're averaging about 1.2-1.5 blue cards no matter how many cards are left in the deck.  So, again, the need for additional blue mana is probably not needed considering you already have a blue mana.  I'd up my odds for business rather than hope for academy--and then even if you get academy you have to have gotten the multiple blue cards for it to have been useful.

I definitely count Walk if it is in combination with another card.   If you draw Walk and Brainstorm/Ponder/A.call/Windfall which finds you Bargain or Necro, that pretty much means you need double blue to win before your opponent gets another turn.

Your raw numbers overlook something very significant: relative importance. 

 Although fetching might thin your deck a tiny bit and increase your chance of seeing a business spell over another land, the reality is that you are very, very likely to see a business spell and more.   The advantage to thinning isn't that important for that reason.  On the other hand, if you draw into multiple blue spells you will very likely need double blue.    So the risk of drawing 2 of the decks remaining blue spells could actually be less than the risk of drawing an additional land thanks to thinning, but the gravity of one is greater.  One could cost you the game, where it is much less liklely that the failure to thin could cost you the game because of failure to draw business.

[Moxlotus]

On the other hand, if you draw into multiple blue spells you will very likely need double blue. 

So the choice is which is more likely to happen--these numbers (which doesn't even take into account that you might draw your black bomb), or risk missing out on a business card (which you might not even get with the thinning).  It depends on the situation.  If the storm count is like 3 or 4, then you might want to go for the academy and hope to up the storm count with blue cards.  But if you're just looking for that grim tutor or black bomb, then I'd fetch.

[Smmenen]

 

So the choice is which is more likely to happen--these numbers (which doesn't even take into account that you might draw your black bomb), or risk missing out on a business card (which you might not even get with the thinning).  It depends on the situation.  If the storm count is like 3 or 4, then you might want to go for the academy and hope to up the storm count with blue cards.  But if you're just looking for that grim tutor or black bomb, then I'd fetch.

It's not simply that you might draw a hand with two blue spells.

What if you draw [Windfall] and Windfall into a [Ponder] and the Ponder finds you a Grim Tutor?   

Substitute [Windfall] and [Ponder] for Ancestral/Ponder/Brainstorm/etc  and the result is the same.   

There are lots of permutations.  You could draw into a hand with just Ponder that shuffles into a Brainstorm that finds the relevant business spell.

You have to calculate the risk of drawing no business what soever (which is going to be very low, the deck is still stacked with business) versus the risk of drawing either two blue spells or a blue spell that draws more blue spells and no more blue mana.   

[Moxlotus]

So the choice is which is more likely to happen--these numbers (which doesn't even take into account that you might draw your black bomb), or risk missing out on a business card (which you might not even get with the thinning).  It depends on the situation.  If the storm count is like 3 or 4, then you might want to go for the academy and hope to up the storm count with blue cards.  But if you're just looking for that grim tutor or black bomb, then I'd fetch.

It's not simply that you might draw a hand with two blue spells.

What if you draw [Windfall] and Windfall into a [Ponder] and the Ponder finds you a Grim Tutor?   

Substitute [Windfall] and [Ponder] for Ancestral/Ponder/Brainstorm/etc  and the result is the same.   

There are lots of permutations.  You could draw into a hand with just Ponder that shuffles into a Brainstorm that finds the relevant business spell.

You coud draw it off one of the blue spells, like Twister or Windfall or Brainstorm, etc, which you play off a Mox Sapphire or a Petal.
 

None of those are going to bump any numbers more than 1-2%.

[Smmenen]

We are dealing in minute percentages.  Each tiny amount counts. 

Also, your math is off. You don't have to have Academy directly in the Jar hand.   You coud draw it off one of the blue spells, like Twister or Windfall or Brainstorm, etc, which you play off a Mox Sapphire or a Petal.

Also, there is a nontrivial chance that you might actually need to have three blue spells or could recur Desire with either Will or Twister.  If you draw Twister, you might then need to Tinker again into Brainstorm into the win, etc.  There are innumerable permutations.   

Grim Long has a voracious appetite for blue mana.   Tolarian Academy is a very important card to fuel that need, as is Black Lotus and even Chain of Vapor.   Unfortunately, with Grim Long, things are rarely straightforward in the tight spots.  The deck will usually make you go through as many hoops as it can set up.   Your conclusion to look for Academy if you generate storm, but fetch if you just want to Grim Tutor really misses the point that to find Grim Tutor ( or a functional tutor for Yawg Will or Tendrils) you might very easily need two blue mana, or more. 

As a side note, if I were to play Grim Long in tournament, I would actually run the 5c mana base with Regrowth, like Kevin Cron did last weekend in San Antonio.   

[Liam-K]

Dude quite frankly, the chances of a dead card in your hand losing you the game are higher than the chances of running into Academy plus it making the difference, and the reduced chances of running into a dead card are bigger than the chances of running into academy to begin with.  You cite the number of permutations where academy is good, but I cite the number of permutations where a land is bad, and frankly I think you're being a little blinded by academy's spectacularly obvious behaviour; obviously drawing juicier hands over time is harder to notice, but the numbers make more sense for the latter.

[Smmenen]

Dude quite frankly, the chances of a dead card in your hand losing you the game are higher than the chances of running into Academy plus it making the difference, and the reduced chances of running into a dead card are bigger than the chances of running into academy to begin with.  You cite the number of permutations where academy is good, but I cite the number of permutations where a land is bad, and frankly I think you're being a little blinded by academy's spectacularly obvious behaviour; obviously drawing juicier hands over time is harder to notice, but the numbers make more sense for the latter.

I think people are missing the essential point: a dead card in hand is not that big of a deal.  It's only a big deal if you have essentially no business whatsoever, which is VERY unlikely.  However, the potential of having Academy is extremely explosive, and potentially absolutely necessary.   Even if you don't see Academy, you have a decent chance of drawing another land which you can just play post-Jar.  So there is really no reason to play a fetch pre-Jar, as I said initially. 

[Shaman]

You are focussing on the chance of drawing a certain card or not, but you miss many other factors that are important as well. When considering the benefits of a certain choice over the opposite, you should be aware of every single detail in the game, not only pure math. Frankly, I cannot disagree with math (nobody can), but here I disagree with the original question that math answered to. You are asking math: "Do I have a greater chance of drawing a non-land card if..." but it is not the same as "Do I have a greater chance of winning if...". It's not simply a fact of land/non-land, so once we discovered that the % difference is so tiny we should realize that in real life there is not a right answer because the influence of other particular factors is greater...by far.

You could have an exact answer only if you know exactly what cards you have in your hand, in your graveyard, and in your deck and in play, and the same for your opponent as well. Obviously, the answer will take you sooooo long, since the math behind will be HUGE. And again, this answer perfect on paper could be wrong in real life because your opponet can react differently from what you expect him to behave (force that card or not).

So I competely miss the relevance of a 0.1%, expecially when it grows from wrong assumptions.

[Moxlotus]

Dude quite frankly, the chances of a dead card in your hand losing you the game are higher than the chances of running into Academy plus it making the difference, and the reduced chances of running into a dead card are bigger than the chances of running into academy to begin with.  You cite the number of permutations where academy is good, but I cite the number of permutations where a land is bad, and frankly I think you're being a little blinded by academy's spectacularly obvious behaviour; obviously drawing juicier hands over time is harder to notice, but the numbers make more sense for the latter.

I think people are missing the essential point: a dead card in hand is not that big of a deal.  It's only a big deal if you have essentially no business whatsoever, which is VERY unlikely.  However, the potential of having Academy is extremely explosive, and potentially absolutely necessary.   Even if you don't see Academy, you have a decent chance of drawing another land which you can just play post-Jar.  So there is really no reason to play a fetch pre-Jar, as I said initially. 

I just simply have to disagree.  In my experience, and the experience of all of my team mates, I have had more hands with 0 business after a draw 7 than hands where I drew the Academy I couldn't play and had multiple blue cards as the only business in my new hand.

[SiegeX]

Yes, you are reading it wrong.  The table appears to say, "For X cards in library and Y lands in library, what is the percentage change in drawing a land?"  According to the second table, going from 36/10 to 35/9, your expected number of lands drawn off Jar drops from 1.94 to 1.8.

As a rule, I like to always double check any numbers (especially probabilities) thrown at me.  I calculated the 2nd table in MoxLotus' first post which shows the average # of lands you would expect to have in an opening grip of 7 with X lands left in a deck of Y cards.   The good news is that these numbers appear to be right, but I'm not seeing how your math guru went from this table to the percentages in the picture-like table above it.  Lets take Anusien's example above.  He interprets the percentage table the same as I do, that is "what is the percentage change in drawing a land if you to compare drawing 7 versus cracking a single fetch then drawing 7"  If we use the 36/10 --> 35/9 example, the table shows an expectation of land to go from 1.94 to 1.8 (which is the same as I calculated.)  But going from 1.94 --> 1.8 is a decrease of 7.43%, yet the percentages table shows something in the 40% range?  Would you please have your team member give more detail on how these percentages were calculated based upon the land-count table, because either its not right or we are still interpreting this percentages table wrong.


*******SKIP BELOW IF MATH SCARES YOU**********
As an aside, I thought I'd share with you how the land-count table was calculated so you're not just taking my word for it either.  This information is very useful (and easy to use) to answer a whole plethora of probabilistic questions with respect to just about any card game.

If you want to know the probability of seeing 'X' of a certain card that has a total of 'Y' copies in a deck of size 'N' in an opening hand size of 'M', you are talking about a hypergeometric distribution.  Fortunately, Excell (and Openoffice) includes the HYPGEOMDIST(X,M,Y,N) function which gives us the answer straight away if you can provide it with the four pieces of information.

Code:

Example:  What is the probability of drawing exactly 2 mana drains in my opening grip of 7 when I have 4 mana drains in my deck of 60 cards?
Answer:   HYPGEOMDIST(2,7,4,60) = 5.93%

Code:

Example:  What is the probability of drawing *at least* one mana drain in my opening grip of 7 when I have 4 mana drains in my deck of 60 cards? 
Answer:   1 - HYPGEOMDIST(0,7,4,60) = 39.95%
*Note that this used a trick in that we find the probability of drawing zero mana drains then subtract that from one (or 100%).  You could have done it the long way by adding up the probabilities of drawing 1, 2, 3, and 4 mana drains and get the same answer.

So now back to the land-count table.  To find the average number of lands you would expect to draw in a hand of 7, we need to first find the probabilities of drawing 0 through 7 of those lands in our opening grip.  If we start with the 36-card deck with 10-land example again, we would calculate the following probability distribution using the HYPGEOMDIST(X,7,9,35) function where X goes from 0 to 7.

Code:

0	1	2	3	4	5	6	7
7.88%	27.58%	35.46%	21.49%	6.54%	0.98%	0.07%	0.00%

We now take the SUMPRODUCT() of the two rows to find the expected number of lands you should draw if you were to repeatably draw 7 cards (with replacement) with 10 lands left in a 36 card deck.  The answer we get is 1.944

Next, we pretend we crack a fetch and now we need to do the same thing but this time with 9 lands left in a 35 card deck.  You get the following probability distribution:

Code:

0	1	2	3	4	5	6	7
9.78%	30.81%	35.22%	18.67%	4.87%	0.61%	0.03%	0.00%

Taking the SUMPRODUCT() of the two rows we get our answer of 1.8

[Smmenen]

Dude quite frankly, the chances of a dead card in your hand losing you the game are higher than the chances of running into Academy plus it making the difference, and the reduced chances of running into a dead card are bigger than the chances of running into academy to begin with.  You cite the number of permutations where academy is good, but I cite the number of permutations where a land is bad, and frankly I think you're being a little blinded by academy's spectacularly obvious behaviour; obviously drawing juicier hands over time is harder to notice, but the numbers make more sense for the latter.

I think people are missing the essential point: a dead card in hand is not that big of a deal.  It's only a big deal if you have essentially no business whatsoever, which is VERY unlikely.  However, the potential of having Academy is extremely explosive, and potentially absolutely necessary.   Even if you don't see Academy, you have a decent chance of drawing another land which you can just play post-Jar.  So there is really no reason to play a fetch pre-Jar, as I said initially. 

I just simply have to disagree.  In my experience, and the experience of all of my team mates, I have had more hands with 0 business after a draw 7 than hands where I drew the Academy I couldn't play and had multiple blue cards as the only business in my new hand.

That may be true, but it may be also because your Long decks rely more heavily on black cards than blue.    In general, Grim Long wants about 1 blue mana for every 1.5 black mana.   GWS Long may be different in that regard. 

[Webster]

Dude quite frankly, the chances of a dead card in your hand losing you the game are higher than the chances of running into Academy plus it making the difference, and the reduced chances of running into a dead card are bigger than the chances of running into academy to begin with.  You cite the number of permutations where academy is good, but I cite the number of permutations where a land is bad, and frankly I think you're being a little blinded by academy's spectacularly obvious behaviour; obviously drawing juicier hands over time is harder to notice, but the numbers make more sense for the latter.

I think people are missing the essential point: a dead card in hand is not that big of a deal.  It's only a big deal if you have essentially no business whatsoever, which is VERY unlikely.  However, the potential of having Academy is extremely explosive, and potentially absolutely necessary.   Even if you don't see Academy, you have a decent chance of drawing another land which you can just play post-Jar.  So there is really no reason to play a fetch pre-Jar, as I said initially. 

I just simply have to disagree.  In my experience, and the experience of all of my team mates, I have had more hands with 0 business after a draw 7 than hands where I drew the Academy I couldn't play and had multiple blue cards as the only business in my new hand.

That may be true, but it may be also because your Long decks rely more heavily on black cards than blue.    In general, Grim Long wants about 1 blue mana for every 1.5 black mana.   GWS Long may be different in that regard. 

The differences between your list, Steve [meandeck], and the list that Eric [GWS] posted are:

[GWS]
1 chain of vapor
1 rebuild
1 pact of negation
1 tendrils of agony
1 swamp
3 street wraith
2 night's whisper

vs.

[meandeck]
3 thoughtseize
1 empty the warrens
1 imperial seal
1 merchant scroll
1 windfall
1 cabal ritual
2 simian spirit guide

The lists run the same number of blue cards eschewing only on the specifics when you compare 1 chain of vapor and 1 rebuild to 1 merchant scroll and 1 windfall. Regarding black cards, it's 2 night's whisper and 1 tendrils of agony to 3 thoughtseize and 1 imperial seal.

I don't see the possible differences in desired mana ratios between the two decks being something relevant to differentiate on. Perhaps you could enlighten us with regard to that point.

[Moxlotus]

Would you please have your team member give more detail on how these percentages were calculated based upon the land-count table, because either its not right or we are still interpreting this percentages table wrong.
 

You want the probability that AF < BF.  Because the occurrences are mutually exclusive (i.e., if we draw exactly 3 lands in a hand, we cannot have drawn 2), mathematically, we have
P(AF < BF) = Sum( P(AF < x AND BF = x), x = 0...7)

Now using independence, we can break up the joint probability into two marginal probabilities:
P(AF < BF) = Sum( P(AF < x)* P(BF = x)), x = 0...7)

Now we use the CDF of a discrete variable, and we get
P(AF < BF) = Sum( P(BF = x)*Sum(P(AF = y), y=0...x-1)), x = 0...7)

and that gives the expression you'll find in the first table.

This is the answer to SeigeX's question.

The first table and the second one have actually nothing to do with each other.  One is not used to compute the other in any way, and the two are completely uncorrelated.

Do note, however, that there is always a chance that you draw less land in the second hand even if you don't fetch.  That's the nature of random events.  I didn't run the numbers, but it's easy to do.

But going from 1.94 --> 1.8 is a decrease of 7.43%, yet the percentages table shows something in the 40% range?

You can't look at it like that.  I didn't give the variances, and a change in expected value doesn't tell you anything without them.  For instance, in the first situation, you could have something like
Odds you draw 0 land - 20%
Odds you draw 1 land - 17%
Odds you draw 2 land - 15%
Odds you draw 3 land - 10%
Odds you draw 4 land - 4%
or whatever

Now, in the second, you can have
Odds you draw 0 land - 23%
Odds you draw 1 land - 20%
Odds you draw 2 land - 16%
Odds you draw 3 land - 5%
or whatever

In the second example, the expected value will be lower because the percentage for drawing 3 lands has dropped significantly.  Expected value is a weighted sum, so the value drops off faster when the values multiplied by the higher weights decrease.

[Smmenen]

Dude quite frankly, the chances of a dead card in your hand losing you the game are higher than the chances of running into Academy plus it making the difference, and the reduced chances of running into a dead card are bigger than the chances of running into academy to begin with.  You cite the number of permutations where academy is good, but I cite the number of permutations where a land is bad, and frankly I think you're being a little blinded by academy's spectacularly obvious behaviour; obviously drawing juicier hands over time is harder to notice, but the numbers make more sense for the latter.

I think people are missing the essential point: a dead card in hand is not that big of a deal.  It's only a big deal if you have essentially no business whatsoever, which is VERY unlikely.  However, the potential of having Academy is extremely explosive, and potentially absolutely necessary.   Even if you don't see Academy, you have a decent chance of drawing another land which you can just play post-Jar.  So there is really no reason to play a fetch pre-Jar, as I said initially. 

I just simply have to disagree.  In my experience, and the experience of all of my team mates, I have had more hands with 0 business after a draw 7 than hands where I drew the Academy I couldn't play and had multiple blue cards as the only business in my new hand.

That may be true, but it may be also because your Long decks rely more heavily on black cards than blue.    In general, Grim Long wants about 1 blue mana for every 1.5 black mana.   GWS Long may be different in that regard. 

The differences between your list, Steve [meandeck], and the list that Eric [GWS] posted are:

[GWS]
1 chain of vapor
1 rebuild
1 pact of negation
1 tendrils of agony
1 swamp
3 street wraith
2 night's whisper

vs.

[meandeck]
3 thoughtseize
1 empty the warrens
1 imperial seal
1 merchant scroll
1 windfall
1 cabal ritual
2 simian spirit guide

The lists run the same number of blue cards eschewing only on the specifics when you compare 1 chain of vapor and 1 rebuild to 1 merchant scroll and 1 windfall. Regarding black cards, it's 2 night's whisper and 1 tendrils of agony to 3 thoughtseize and 1 imperial seal.

I don't see the possible differences in desired mana ratios between the two decks being something relevant to differentiate on. Perhaps you could enlighten us with regard to that point.

I was taking Eric's claim on that point as an article of faith:

I think the variations in our list partially cause this disagreement. Bombs in your list are "less black" (Mystical, ETW, Scroll, Windfall vs. 2x Whisper, Tendrils, 3x Street Wraith). In my list I'd often times be going for that Underground Sea, but in your list there's more potential for a color-hosing problem.

[Liam-K]

Dude quite frankly, the chances of a dead card in your hand losing you the game are higher than the chances of running into Academy plus it making the difference, and the reduced chances of running into a dead card are bigger than the chances of running into academy to begin with.  You cite the number of permutations where academy is good, but I cite the number of permutations where a land is bad, and frankly I think you're being a little blinded by academy's spectacularly obvious behaviour; obviously drawing juicier hands over time is harder to notice, but the numbers make more sense for the latter.

I think people are missing the essential point: a dead card in hand is not that big of a deal.  It's only a big deal if you have essentially no business whatsoever, which is VERY unlikely.  However, the potential of having Academy is extremely explosive, and potentially absolutely necessary.   Even if you don't see Academy, you have a decent chance of drawing another land which you can just play post-Jar.  So there is really no reason to play a fetch pre-Jar, as I said initially. 

...you still have 2 lands available, neither of which will be available next turn.

[Aneurysm]

Even if you don't see Academy, you have a decent chance of drawing another land which you can just play post-Jar.

Smennen, you're proving our point. We don't want to see land on the draw seven. And I have to agree with Phil and say that I've had way too many draw 7s fizzle because lack of bombs.

[Smmenen]

Even if you don't see Academy, you have a decent chance of drawing another land which you can just play post-Jar.

Smennen, you're proving our point. We don't want to see land on the draw seven. And I have to agree with Phil and say that I've had way too many draw 7s fizzle because lack of bombs.

I haven't.   Grim Long is stacked with bombs.  On the other hand, I have lost games, specifically in major tournaments, for not making precisely this play.   Waiting for Academy is the correct play.  The advantage of thinning one card is inconsequential compared to the  much greater chance you'll need Academy for multiple blue or could use the additional mana in general.  If you end up not needing more than one blue, you shouldn't have a hard time finding that blue.  But if you can't win without multiple blue, the Academy may be needed.   You may even need to tutor for it.   On the other hand, if GWS wants to go into battle with Grim Long and play it the other way, the advantage goes to me, I suppose. 

[Aneurysm]

Even if you don't see Academy, you have a decent chance of drawing another land which you can just play post-Jar.

Smennen, you're proving our point. We don't want to see land on the draw seven. And I have to agree with Phil and say that I've had way too many draw 7s fizzle because lack of bombs.

I haven't.   Grim Long is stacked with bombs.  On the other hand, I have lost games, specifically in major tournaments, for not making precisely this play.   Waiting for Academy is the correct play.  The advantage of thinning one card is inconsequential compared to the  much greater chance you'll need Academy for multiple blue or could use the additional mana in general.  If you end up not needing more than one blue, you shouldn't have a hard time finding that blue.  But if you can't win without multiple blue, the Academy may be needed.   You may even need to tutor for it.   On the other hand, if GWS wants to go into battle with Grim Long and play it the other way, the advantage goes to me, I suppose. 

I'll see you at the next SCG P9 tournament.

(If that ever happens? Help me out here; are they having any more?)