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Very spikey! Thanks for such a great article.
And I think everyone has already arrived at the consensus that Theros will be an 18-land deck for limited games because of all the things to do with one’s mana at the high end of the game (bestow and monstrous and activated abilities of gods and equipment)! So this article is another reason to go with 18 land for Theros!
You should be boycotting math, the payout for knowing it is way too low.
In the end you say: ” Decreasing the probability for the “not-enough-land” scenario by 6% seems to be good enough for me to accept that I might have too many lands from time to time; in fact, these chances are increased by about only 8%.”
By running 18 lands instead of 17, you decrease the land-screw situation by 6% and increase the flood situation by 8%. Does that not make your deck stumble 2% of the time more often overall?
Not if you have things to sink your mana Into. Just because you flood does not mean you loose. That normally is the opposite when you are short on mana.
If you have mana sinks you are not flooding. The definition of flooding is “Having noticeably more mana than ways of using it,” not “having lots of mana and things to do with it”.
Great article! It’s great to read about probability and simulations for magic games. I just can’t get enough.
I have one comment about the first graph. I think that graph can also be generated from the hypergeometric distribution. Please correct me if I’m wrong (I’m much better at being wrong than at probability).
Here’s what I was thinking. A player wants 3 lands by turn 3. If on the play, the player gets two draws meaning we can just look at the probability of drawing three lands in a “starting hand” of 9 cards. That means the probability of drawing 3 lands (and thereby playing 3 lands) is ncoosek(L,3)*nchoosek(40-L, S+D-3)/nchoosek(40,S+T), where L is the number of lands in the deck, S is the number of cards in a starting hand (7), and D is the number of draws (2). For L=16, this comes out to be 25.6%. To obtain the final result, we have to multiply this result by the probability that the player draws a nonland card next, which, for these numbers, is 18/31.
To completely generalize with a bunch of confusing letters, the probability of playing exactly X lands in a row (i.e. X lands by turn X) is :
where L is the number of lands in the deck, C is the number of cards in the deck, and S is the size of the starting hand. This might be able to be further simplified… (with logic or with identities).
Fascinating analysis — thanks very much! I’m about as far from being a statistician as possible, but I rather love consuming this sort of information when it’s produced by people such as yourself who not only understand it but who are able to render it readily digestible to people like me. Bravo! I thoroughly enjoyed this article, and I eagerly await its sequel.
By the by, here’s a plug for MTGO Academy contributor Marshall Sutcliffe’s podcast. In an episode from a few months ago, Marshall’s co-host, Brian Wong, does a fine job treating probabilities and the question of splashing cards. How to optimize colored source count for a one card splash, two card splash, etc.? http://lrcast.com/limited-resources-189-mana-bases/
Hey, thanks for all that comments!
@NoTeef: You are very right. For some reason I thought to complicated and I am so used to solve problems like that by simulation.
@premiersoupir: Indeed this episode inspired me to write this article!
Did you seriously ignore my questions? Extremely unprofessional.
Thanks for the graphs and probabilities. This may be a necro type comment, but we’re still in the Theros block and I’m brand new to MTGO so I’m sponging as much info as I can in my free time at work
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