I'm playing a deck that needs to hit that 2nd land quite consistently. Any lands past 4 turn into more or less dead, weak draws. If I'm playing 4 Serum Visions and no other mana producing things, how many lands should I be running?
Some math:
If I'm running 21 lands, then the deck consists of 35% lands (21/60), meaning 65% of the deck are non-lands.
Serum Visions digs 3 deep. To work out the probability of Serum Visions missing a land: 0.65*0.65*0.65=0.275.
Therefore, Serum Visions equates to 0.725 of a land when running 21 lands. 4 Serum Visions is approximately equivalent to 3 lands. So when running 21 lands with 4 Serum Visions, one is almost playing essentially 24 lands.
This is a rather useful tool for this sort of thing:
http://stattrek.com/online-calculato...geometric.aspx
Using a population of 59, number of successes in population of 24, sample size of 8 and number of successes of 2, it tells me that one will hit 2 or greater 92% of the time. 23 lands is 90%.
I quickly did the math and found when running 20 lands, Serum Visions equates to 0.7 of a land, compared to 0.75.
Now I'm curious, adding that extra land gives one about an extra 2% chance of hitting that 2nd land on time, but what sort of impact will it have on flooding? This is more difficult to calculate, but I can do a rough calculation:
Population: 59
Number of successes: 24
Sample size: 10 (T4 on the play without drawing)
Number of successes: 4
4 or greater equates to a 65% chance of occurring, pretty friggin huge. 23 lands instead equates to a 60% chance of occuring.
So when comparing 23 lands to 24, the extra land drop causes one to hit the 2nd land 2% more often, and also causes one to flood 5% more often. I also play Remands in the deck, which will make this later effect worse.
So because of this, could I reason that 20 lands, with 4 serum visions, is where I want to be?
That's...definitely some math. I think your figures are pretty solid and 20 lands seems good for your purposes. That being said, are some of your lands fetch lands? If so, playing a fetch takes an additional land out of your deck, reducing the probability of a potential future land draw. I'm sure the percentage reduction is very small, but it may be there deciding factor between running 8 vs 10 fetches or something along those lines.
I'm just running 4 fetches, which I'm pretty sure is optimal here. I read some article awhile ago that simulated the effect of running fetches and the reduced chance of drawing lands, and it was rather minimal. My manabase is rather solid, its:
4 Flooded Strand
4 Island
2 Plains
2 Hallowed Fountain
4 Seachrome Coast
1 Eiganjo Castle
2 Tectonic Edge
1 Mutavault
Something to note is if I was to include an extra land to bump up to 21, it'd be a utility land. It'd either be a 3rd Tectonic Edge, maybe even a Ghost Quarter, or 2nd Mutavault or maybe even a Moorland Haunt. So even though I'd have a higher propensity to flood with it, at least the extra land performs some sort of utility function other than tapping for mana.
If you like math...
http://www.channelfireball.com/artic...t-your-spells/
-rob
I dislike Karsten's methodology because it does not take into account the potential to flood; it is an at least, not an exactly.
For a balance between Flood and Screw, I suggest you use the following formula (a simplified version of the sum of each hypergeometric distribution probability for a desired multiplied by that desired result):
S = s / p * P, where S is the 'sweet spot' number of lands you want to run in your library, s is exactly the number of lands you want, p is the number of cards by that turn (most likely 6 + s), and P is the number of cards in your library before opening hand (typically 60, 100, or 40).
So for an example (and let's assume on the play):
S = 3 / 9 * 60 = 20
EDIT: Hmm ... I missed the point if the thread it seems ... I shall return with further insight.
EDIT 2: Alright so, concerning card selection via Serum Visions ... I think OP is pretty spot on in terms of evaluating its effectiveness. Only thing I might be able to contribute is that with 20 lands, each Serum Vision would 'whiff' ( 2 / 3 ) ^ 3 or 8 / 27 times. This would mean each Serum Vision is worth 19/27 lands ... a little over 2/3 ...
Ok ... Table Time!
Lands / Serum Visions / Total
10 / .40 / 11
15 / .55 / 17
20 / .70 / 22
24 / .80 / 27
That is starting with a sweet spot of lands for 1 to 4 lands.
Lands / Serum Visions / Total
9 / .40 / 10
13 / .50 / 15
18 / .60 / 20
21 / .70 / 24
That is after a bit of trial and error (and some fairly oddball hypotheses).
However this is reliant on having mana to cast said cantrip in the first place (same could be said about Mana Dorks). So that first entry in each table? Ignore that.
It's more complicated than that, really. Serum Visions only count as a land if you already have a land, otherwise they don't really count as anything since you can't play them. It's also similar to how Gitaxian Probe used to mess with things, you can kind of pretend you have a 56 card deck but if you get a no land + X Gitaxian Probe hand you'd always have to mulligan it (there might even be a land on top of your library, you just don't know), so it isn't the same as just having 56 cards.
You'd have to run the hypergeometric distribution to get *at least* one land and then a Serum Visions in hand, and even then there isn't really a definitive best solution without knowing more. 2% increase in hitting your 2nd land on time might win you the game whereas 5% increase to hitting your 3rd land likely doesn't straight up lose you the game.
I dislike the coloured mana source approach here as well. I've actually already used the sort of thought in that article to determine how many coloured sources, but what I'm interested in is getting mana screwed/flooded, actual total land count.
I must confess EpicLevelCommoner that your math and sweet spot number has rather perplexed me.
I did make the fallacious assumption that I'd always have the U source to cast Serum Visions. Couldn't I just factor in the times where I don't have a U source into the calculation? So I know by using that calculator that I'll have a U source 87% of the time by T1. Therefore, 13% of the time I won't have the U to cast Serum Visions.
So at 20 lands, a Serum Visions, according to previous calculations is worth 0.7 lands. So:
0.7*0.13=0.091
0.7-0.091=0.609
So they're more like 0.6 of a land.
One thing to note is, the detriment of not hitting the 2nd land is a lot worst than going to 4 lands. At 1 land, my capacity to interact and advance my board state is rather limited, hitting 2 lands to play Remand is particularly important, whereas I still can use that 4th land by casting 2 2cmc spells.
Not exactly. A 13% chance in your calculations there is that you don't hit a land at all, and this is with drawing 7 cards. The chance that you won't draw a land but will draw a Serum Visions is different because your hand contains (at least one)Serum Visions, which means that you'd only have 6 (or less, if you had more Serum Visions) possible cards that are potentially lands in your hand.
You should also be careful to factor mulligans into your decision. It doesn't matter how good the 6 cards are that you draw in your mulligan if you don't get a land.
How to put this ... I'm not as well-versed in mathematical proofs as I should be, but the S = s * p / P holds true for relevant products hypergeometric distribution probabilities and their respective results. It just doesn't give probabilities.
For reference, here is a sheet I've been using for Modern:
https://docs.google.com/spreadsheets...it?usp=sharing
Also, for compound Hypergeometric Distributions (like having Serum Visions and a blue source in opening hand), the formula for exactly one probability goes COMBIN(S1,s1)*COMBIN(S2,s2)* .... COMBIN(Sn,sn) / COMBIN(P,p), where the sum of S1 through Sn is P and the sum of s1 through sn is p.
So, for that example: COMBIN(4,1)*COMBIN(20,1)*COMBIN(36,5)/COMBIN(60,7) ~ 7.8%
That may seem low, but with probabilities, you add when you want either result and multiply when you want both results. You just want two lands, so definitely this result AND having the second land be in the top 3 (also a HGD), OR two lands.
1) T1 Serum Visions with exactly 1 of 4 Visions and exactly 1 of 20 Land: 7.8%
2) Serum Visions hitting 1 or more lands given the above: 71.1%
3) 1 AND 2: 5.6%
4) Exactly 2 of 20 Land: 32.4%
5) 3 OR 4: 37.9%
Of course, you'd probably be happy with more than one Visions T1 if you only had 1 land, as well as having at least two lands.
1) T1 Serum Visions with at least 1 of 4 Visions and exactly 1 of 20 Land: 9.8%
2) Serum Visions hitting 1 or more lands given the above: 71.1%
3) 1 AND 2: 7.0%
4) At least 2 of 20 Land: 75.3%
5) 3 OR 4: 82.3%
However, having all lands means very little without business spells:
1) T1 Serum Visions with at least 1 of 4 Visions and exactly 1 of 20 Land: 9.8%
2) Serum Visions hitting 1 or more lands given the above: 71.1%
3) 1 AND 2: 7.0%
4) Between 2 and 4 of 20 Land: 71.7%
5) 3 OR 4: 78.7%
To be fair, the impact Visions (or any card quality tool for that matter) has on card selection seems to be negligible in and of itself the more cards with the desired trait you have (for instance, exactly 1 of 24 land and at least 1 of 4 serum visions hitting that 2nd land drop is 5.1% as opposed to 7.0%).
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