I've been a huge fan of using Hypergeometric calculators for deckbuilding, mostly as a tool for establishing likelihood of drawing lands and whatnot. I recently stumbled upon a Binomial Probability Calculator, which calculates the odds of success over a number of trials. I combined the two to calculate some odds of winning games in the long run. I have always found that opening hands with at least 3 lands gives me the most confidence in winning a game. If we assume a 3 land opening hand is a Winning hand then we can calculate the odds of consistently Winning in the long run. Using the Binomial Probablity calculator I ran the numbers for the likelihood of having Winning hands in 200 out of 300 games, which would just be having a record of winning 100 matches over 300 games. Now, this doesn't actually mean anything because the premise is so vague, but I thought it was an interesting exercise in looking at deckbuilding. What I found was pretty surprising. While adding 1+ land to the list had little effect on the likelihood of drawing 3+ lands each opening hand, the cumulative effect on consistently seeing those 3 lands over hundreds of games increased significantly with each land added. What I found , if we're assuming having 3+ lands in our opening hand is the benchmark for winning a game, then it takes running 27 lands in a 60 card deck to consistently see those results in the long run. What I'm wondering now is if anyone else has things they look for in opening hands that give them the most confidence in winning a game, and how we can build to maximize that likelihood.
Number of Lands in a 60 card deck Probability of 3+ lands in 7 (A Winning Hand) Probability of 200+ Winning Hands in 300 games
22 0.509838301 < 0.000001
23 0.549341464 0.0000231
24 0.587929496 0.003075374
25 0.625348194 0.077150495
26 0.661372406 0.449759624
27 0.695807315 0.876423503
I disagree with your conclusion about the results, unless I am misunderstanding things.
I used two different binomial calculators (https://stattrek.com/online-calculator/binomial.aspx and http://statisticshelper.com/binomial...ity-calculator). Both calculators gave me slightly different numbers from yours. I have rounded the percentages in this post, but the numbers should be close enough for our purposes.
For example, with a 23 land deck, the probability is 23/60 =~ .3833333. The trials = 7. The successes = 3. The probability is 0.28510161444. Doing this for each number (1-7) yields:
0: 3.4%
1: 14.8%
2: 27.5%
3: 28.5%
4: 17.7%
5: 6.6%
6: 1.4%
7: 0.1%
For a 27 land deck (27/60 = .45 probability):
0: 1.5%
1: 8.7%
2: 21.4%
3: 29.2%
4: 23.9%
5: 11.7%
6: 3.2%
7: 0.4%
23 land deck probability of 2-4 land (73.7%) and 3-4 land (46.2%) in the opening hand.
27 land deck probability of 2-4 land (74.5%) and 3-4 land (53.1%) in the opening hand.
So far, this agrees with your analysis. The numbers make it seem like the 27 land deck is better. However, extending to look at the first 10 cards (going first + draws for the first 3 turns):
23 Land Deck
0: 0.8%
1: 4.9%
2: 13.8%
3: 22.9%
4: 24.9%
5: 18.6%
6: 9.6%
7: 3.4%
8: 0.8%
9: 0.1%
10: 0%
27 Land Deck
0: 0.3%
1: 2.1%
2: 7.6%
3: 16.6%
4: 23.8%
5: 23.4%
6: 16%
7: 7.5%
8: 2.3%
9: 0.4%
10: 0%
23 land deck probability of 3-5 land in the first 10 cards (66.4%).
27 land deck probability of 3-5 land in the first 10 cards (63.8%).
So a 23 land deck will have a “better” land situation in 2.6% more games. This depends a lot on what kinds of spells are in the deck, both in terms of things like cantrips and the average cmc.
When looking at the 3-4 land situation, it gets even better for the 23 land deck:
23 land deck probability of 3-4 land in the first 10 cards (47.8%).
27 land deck probability of 3-4 land in the first 10 cards (40.4%).
EDIT: None of this takes fetchlands into account. With fetchlands, it is likely that your numbers are better than mine. I will ask some others about the math regarding fetchlands.
Binomial calculators will give less accurate results than hypergeometric calculators, since binomial calculators assume sampling with replacement.
tl;dr - It's possible to optimize this with math, but it's also very easy to get the math wrong depending on assumptions and calculations used. In cases where the math contradicts common sense from real games, consider the possibility that the math is off.
With the two calculators, just have to be sure you're using the right calculator for the right thing. Playing 300 games is sampling with replacement. Each new game you replace the deck, shuffle, and draw from a fresh deck. Binomial makes sense to compare different games. For drawing a certain hand within one game, the cards are not replaced so it's sampling without replacement and a Hypergeometric calculator.
Hypergeometric Calculator - for chance of seeing a certain opening hand (e.g 3 lands in 7 cards)
https://en.wikipedia.org/wiki/Hyperg...c_distribution
Binomial Calculator - for chance of seeing a certain outcome after many games (e.g. 200 of the target hand in 300 games)
https://en.wikipedia.org/wiki/Binomial_distribution
@Whoshim: You're using a Binomial calculator where a Hypergeometric should be used.
@jrw: There's a major flaw in your analysis, not related to that.
Re-run using 59 lands in a 60-card deck. You'll get 100% probability of winning >290 of 300 games. What a winning deck!
Why does it do that? The problem is you've defined anything with 3+ lands as a winning hand. Including 7 lands, 6 lands, or 5 lands with 2 slow cards. Those are obvious mulligans, but the calculation counts everything with at least 3 lands and ignores the possibility of flooding. With a low land count those events are rare, maybe negligible, but as you increase the number of lands they become more common. As you add more lands, of course the chance of having more lands in your hand will go up, so your analysis will always point you towards the highest number of lands in your table. If you keep going past 27, it will tell you to add even more lands. The assumption that "3+lands = win" is too strong and makes the math do things like recommend a 60-land deck.
In a real game there's a tradeoff between not having enough mana and being flooded. 27 seems like too many.
Last edited by FTW; Yesterday at 04:56 PM.
I think the Hypergeometric probability you want is not for 3+ lands in 7 cards but for 3-4 lands in first 7 cards and no more than 5 lands in the first 11 cards (3 or 4 lands in the opening hand but you don't hit more than 5 lands after 4 draws - turn 5 on the play).
That represents seeing the 3-4 lands in your opening hand, but also not flooding over the critical first turns of the game when you need business and interaction. Unlike the condition of "3+ lands = win", this would have diminishing returns as you increase the land count too high. It would not give the unreasonably high win %s for land-heavy decks.
*Hypergeometric probabilities only
Lands 3-4 lands in hand 3-4 lands in hand AND no more than 5 total after 4 draws 18 32.7% 28.7% 19 36.1% 30.8% 20 39.4% 32.6% 21 42.5% 34.0% 22 45.4% 35.0% 23 48.1% 35.6% 24 50.5% 35.8% 25 52.6% 35.6% 26 54.4% 34.9% 27 55.8% 33.9% 28 56.8% 32.5% 29 57.4% 30.8% 30 57.6% 28.8% 31 57.4% 26.6% 32 56.8% 24.3% 33 55.8% 21.9%
If you just try to get 3-4 lands in your opening hand and don't care about flooding, you optimize it with 30 lands in the deck (57.6% of hands will have 3-4 lands). But that will lead to flooding during the game.
If you want to see 3-4 lands in the opening hand but then don't want to flood in the early turns, that's optimized around 24 lands (35.8%), with very close odds for 23 and 25 lands. That's the default land count for many Standard and Casual decks.
Goblins runs 22-23 lands. Should Goblins be running 24? You could test using a card like Mutavault, Ghost Quarter or Teetering Peaks as the 24th land and see how that performs. Goblins has other constraints though too. It needs a high enough Goblin count for Ringleaders, Muxus and Lackey to do their job, which means there is a penalty for having too many lands. Meanwhile the land odds don't increase that much between 22-24 lands.
Accelerants also help cheat on mana costs. Would anyone ship a 2-land hand with Lackey or Vial?? Most 2-land hands are keepable. So what if we expand it to include 2 land hands where you hit your 3rd land in time but don't flood past 5 lands?
Lands 2-4 lands in hand AND 3 lands by 2nd draw AND no more than 5 lands after 4 draws 18 42.0% 19 45.0% 20 47.5% 21 49.4% 22 50.6% 23 51.3% 24 51.3% 25 50.7% 26 49.5% 27 47.8% 28 45.5% 29 42.9% 30 39.9%
This one optimizes around 23-24 lands (51.3% of games). Because of the need for high Goblin count, 23 seems better than 24. And that's what Vial Goblins has already been running for ages.
I would argue that extensive playtesting and tournament results provide better data than a simulation with simplifying assumptions, especially for an interactive aggro-control deck. I think these calculators are most useful when optimizing fast combo decks. Those games are closer to goldfishing and more dependent on opening hand, while an interactive deck like Goblins depends on a lot of factors.
Last edited by FTW; Yesterday at 05:50 PM.
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