Just a quick reference to some of the mathematics involved in Commander. I use these tables a lot when figuring how many of certain types of cards to put in my deck and thought I'd share.
*** These figures are based on a 99-card deck (since your commander usually isn't in the library) and calculated using MWS.
Table 1 - How many copies of a particular card type (like a counterspell or a sac outlet or removal) do I need to insure that I see one by turn "x". (obviously these can be enhanced by tutors, card draw, scrying, etc)
***This can also be used to figure mana sources needed for a reliable color splash.
Table 2 - 2 and 3 card combos where all cards are tutorable ("chance by turn" is "chance of having either the combo pieces or the tutors to go get them in hand by"):
Cards in combo........Total # of cards and tutors......% Chance by turn 0.....turn 4......turn 6......turn 10
2.........................................2...................................<1......................1............2.............3
2.........................................3....................................1.......................3............4.............8
2.........................................4....................................2.......................6............8.............14
2.........................................5....................................4.......................9............13...........20
2.........................................6....................................6.......................13..........18............27
2.........................................8....................................10......................22..........28............41
2........................................10....................................15......................31..........39............54
2........................................12....................................20......................40...........49...........65
Table 3 - 2-card combos where only one is tutorable and 3-card combos where two of them are tutorable.
What are my chances of seeing (by turn x) 2 or 3 specific cards for a combo, and then what are my chances of seeing one of those and the other(s) or a tutor for it given "x" number of tutors.
Table 4 - Manabase - This checks how many mana sources in the deck against the % chance of having "x" of them in your opening hand
Sources........# in opening 7 cards..... 2 or more...3 or more...4 or more
34...................................................76%..........45%...........18%
35...................................................78............48..............20
36...................................................80............50..............22
37...................................................81............52..............23
38...................................................83............55..............25
39...................................................84............57..............27
40...................................................86............59..............29
41...................................................87............62..............31
It should be noted that with all the number of mana sources listed, the chance of getting 6 or more in your opening hand was no more than 2%.
I'll add on to the above. For the more mathematically inclined players: -
Drawing a discrete number of card of a particular characteristic from a finite population (without replacement) closely describes a hypergeometric distribution where drawing card(s) of the particular characteristic is described as a "success" while the inverse -- not drawing the card(s) of particular characteristic should be deemed as a failure.
As such, the probability of drawing k cards of a particular characteristic can be found from the hypergeometric distribution with: -
Population Size (N) = number of cards in the deck = 99
Sample Size (n) = number of cards drawn. Assuming that you start first, this is equals to: -
7 if you are concerned with the probability of drawing a particular card in your starting 7
8 if you are concerned with the probability of drawing a card by turn 1
9 if you are concerned with the probability of drawing a card by turn 2
10 if you are concerned with the probabilty of drawing a card by turn 3, etc.
Number of success in the population (m) = This is equals to the number of cards with the particular characteristic in your deck
Number of success in the sample (k) = This is equals to the number of those cards in your sample (i.e. in your hand)
To put it into practice, say there are 37 lands in your deck and you want to know what is the probability that you draw the following amount of lands in your starting 7 (i.e. N = 99, n = 7, m = 37): -
P(k = 0) = 3.3%
P(k = 1) = 15.3%
P(k = 2) = 28.9%
P(k = 3) = 29.1%
P(k = 4) = 16.8%
P(k = 5) = 5.5%
P(k = 6) = 1.0%
P(k = 7) = 0.1%
Rounding error may explain why the above figures do not add up to 100% (not that I checked if they did or not). How did I get the above figures? I used a software to calculate the above but you could easily find a hypergeometric table somewhere or an online calculator to get more precise figures (I rounded the above figures, in case it isn't obvious). You will note that my answers approximates the numbers generated by MWS (MWS figures are rounded to percentages in whole numbers).
Here's hoping that my rambling helped (or at the very least, it isn't misguided, inaccurate, wrong or misleading)
Thanks for posting this guys. I'm a middle school math teacher, I took a lot of calculus-type classes when I was studying engineering, but not a ton of probability and statistics.
This is a beautiful thing. Thank you so much!
Here is a real puzzle for you:
a) A deck has a 5 card combo, and 7 tutors.
b) Beginning anywhere between turns 4- 7 (6 is usually the latest), the deck loses life every turn, beginning with 2, and doubling every turn. (Glacial Chasm)
c) The deck can reliably draw at least 1 extra card per turn beginning, generally, a few turns after Glacial Chasm. (Say, 2 turns)
d) By the time Glacial hits, life total averages 34.
e) Glacial can be reset once.
f) When the combo is assembled, you win.
EDIT: g) Glacial is NOT part of the combo.
What are the odds of winning by turns: 5, 10, 13, 15, 17, 20?
More power to you if you can do this one....b t dubs if you explain the math you use, the explanation will be read and appreciated, at least by myself:)
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Stop wasting your time here and come to a forum that actually fosters intelligent discussion: Fires Rf Salvation
This is a beautiful thing. Thank you so much!
Here is a real puzzle for you:
a) A deck has a 5 card combo, and 7 tutors.
b) Beginning anywhere between turns 4- 7 (6 is usually the latest), the deck loses life every turn, beginning with 2, and doubling every turn. (Glacial Chasm)
c) The deck can reliably draw at least 1 extra card per turn beginning, generally, a few turns after Glacial Chasm. (Say, 2 turns)
d) By the time Glacial hits, life total averages 34.
e) Glacial can be reset once.
f) When the combo is assembled, you win.
EDIT: g) Glacial is NOT part of the combo.
What are the odds of winning by turns: 5, 10, 13, 15, 17, 20?
More power to you if you can do this one....b t dubs if you explain the math you use, the explanation will be read and appreciated, at least by myself:)
I LOVE YOU! Hi, I'm a math freak.
I'll get you your answer as soon as possible. Though do keep in mind that I can only work with pencil, paper, and calculator so it'll take a while. Thanks for the opportunity.
I kinda have to wonder how people end up bringing threads this old up sometimes, do they search for them, do they just go pages and pages back and ignore the dates...?
I kinda have to wonder how people end up bringing threads this old up sometimes, do they search for them, do they just go pages and pages back and ignore the dates...?
I was actually just looking for an Orzhov EDH deck and I found this in Blackjack's Sig. Also, ya I do just ignore how old it is since math is one of those things that just can't change.
If you guys have any Math problems I'd be more than a little excited to work on them (after the first one of course).
Is there any way to get this information stickied, or otherwise added-in with a pre-existing sticky? This would have been invaluable when I was crafting my UB Storm list.
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Rollback Post to RevisionRollBack
Thanks, Heroes of The Planes! You guys are great!
Actual Truth:
"You heard it here folks:
Anyone who disagrees with "Jack from NC" is an idiot."-The Dead Weatherman
Table 4 - Manabase - This checks how many mana sources in the deck against the % chance of having "x" of them in your opening hand
Sources........# in opening 7 cards..... 2 or more...3 or more...4 or more
34...................................................76%..........45%...........18%
35...................................................78............48..............20
36...................................................80............50..............22
37...................................................81............52..............23
38...................................................83............55..............25
39...................................................84............57..............27
40...................................................86............59..............29
41...................................................87............62..............31
I was actually just looking for an Orzhov EDH deck and I found this in Blackjack's Sig. Also, ya I do just ignore how old it is since math is one of those things that just can't change.
If you guys have any Math problems I'd be more than a little excited to work on them (after the first one of course).
Always good to know where the math guys are, just in case. Math hurts my brain.
BTW, if you (or anyone) come up with any tables that would be useful for deckbuilding, post it, and I'll copy it to the op.
The problem with using mathematics / statistics to estimate probabilities is the small caveat that these figures are going to overestimate the probability of mana flooding or mana screwing because hand shuffled decks are almost never totally random (unless you are playing digitally, in which case depending on the algorithm used, you will see something that reflects the statistical tables a lot more).
Thanks for this thread. I've used it for all my EDH decks. I'm also a fan of your Orzhov lists, and I'm brewing on my own super-secret Teysa Combo list. Your contributions are appreciated.
it's all well and good for monocolored, but, wouldn't it be moar awesomez to be able to calc stats for individual colors in a multi-colored deck? this becomes important when packing low cost spells that still got double or even triple mana symbols as their casing cost. have fun and thanks.
*** These figures are based on a 99-card deck (since your commander usually isn't in the library) and calculated using MWS.
Table 1 - How many copies of a particular card type (like a counterspell or a sac outlet or removal) do I need to insure that I see one by turn "x". (obviously these can be enhanced by tutors, card draw, scrying, etc)
***This can also be used to figure mana sources needed for a reliable color splash.
# of copies -- % chance by (first 7) - turn 4 - turn 6 - turn 10
1.......................... 7.......................11..........13.........17
2.......................... 14......................21..........25.........32
3...........................20......................30..........35.........44
4...........................26......................38..........44.........54
5...........................31......................45..........51.........62
6...........................36......................52..........58.........69
7...........................41......................57..........64.........74
8...........................46......................62..........69.........79
9...........................50......................67..........73.........83
10.........................54......................71..........77.........86
12.........................61......................78..........83.........91
Table 2 - 2 and 3 card combos where all cards are tutorable ("chance by turn" is "chance of having either the combo pieces or the tutors to go get them in hand by"):
Cards in combo........Total # of cards and tutors......% Chance by turn 0.....turn 4......turn 6......turn 10
2.........................................2...................................<1......................1............2.............3
2.........................................3....................................1.......................3............4.............8
2.........................................4....................................2.......................6............8.............14
2.........................................5....................................4.......................9............13...........20
2.........................................6....................................6.......................13..........18............27
2.........................................8....................................10......................22..........28............41
2........................................10....................................15......................31..........39............54
2........................................12....................................20......................40...........49...........65
3.........................................4..................................<1......................<1............1.............2
3.........................................6..................................<1.......................2............3.............6
3.........................................8....................................1.......................4............7............14
3.........................................10..................................2.......................8............12...........23
3.........................................12..................................4.......................13...........19...........34
3.........................................14..................................6.......................19...........27...........45
Table 3 - 2-card combos where only one is tutorable and 3-card combos where two of them are tutorable.
What are my chances of seeing (by turn x) 2 or 3 specific cards for a combo, and then what are my chances of seeing one of those and the other(s) or a tutor for it given "x" number of tutors.
Cards/Tutors -- % by turn 4 -- turn 6 -- turn 10
2/0........................<1..............1............3
2/3..........................4..............5............9
2/5..........................5..............7............11
2/7..........................7..............9............13
3/0.........................<1.............<1..........<1
3/3..........................1...............1............3
3/5..........................2...............3............5
3/7..........................3...............4............8
Table 4 - Manabase - This checks how many mana sources in the deck against the % chance of having "x" of them in your opening hand
Sources........# in opening 7 cards..... 2 or more...3 or more...4 or more
34...................................................76%..........45%...........18%
35...................................................78............48..............20
36...................................................80............50..............22
37...................................................81............52..............23
38...................................................83............55..............25
39...................................................84............57..............27
40...................................................86............59..............29
41...................................................87............62..............31
It should be noted that with all the number of mana sources listed, the chance of getting 6 or more in your opening hand was no more than 2%.
Banner by Nakamura, Thanks!
EDH Math
EDH Decks:
Ghost Council: The Magic Mafia of Orzhova
BB Drana: Down with the Sickness
Rasputin: Reality is Broken
Vish Kal Bleeder: Bloody Kisses
Teysa, Orzhov Dominatrix
Stonebrow: Breaking Things
BWR Kaalia Punisher: Heaven's on Fire
Grimgrin: Dead Reckoning
Drawing a discrete number of card of a particular characteristic from a finite population (without replacement) closely describes a hypergeometric distribution where drawing card(s) of the particular characteristic is described as a "success" while the inverse -- not drawing the card(s) of particular characteristic should be deemed as a failure.
As such, the probability of drawing k cards of a particular characteristic can be found from the hypergeometric distribution with: -
Read a more eloquent explanation here.
To put it into practice, say there are 37 lands in your deck and you want to know what is the probability that you draw the following amount of lands in your starting 7 (i.e. N = 99, n = 7, m = 37): -
P(k = 0) = 3.3%
P(k = 1) = 15.3%
P(k = 2) = 28.9%
P(k = 3) = 29.1%
P(k = 4) = 16.8%
P(k = 5) = 5.5%
P(k = 6) = 1.0%
P(k = 7) = 0.1%
Rounding error may explain why the above figures do not add up to 100% (not that I checked if they did or not). How did I get the above figures? I used a software to calculate the above but you could easily find a hypergeometric table somewhere or an online calculator to get more precise figures (I rounded the above figures, in case it isn't obvious). You will note that my answers approximates the numbers generated by MWS (MWS figures are rounded to percentages in whole numbers).
Here's hoping that my rambling helped (or at the very least, it isn't misguided, inaccurate, wrong or misleading)
k = number of desired cards in hand
s = hand size
d = deck size
c = copies of desired card in deck
w = ways to have k desired cards in s card hand =
(s Choose k) * (c! / (c - k)!)
wo = ways to draw all other (undesired) cards in hand =
(d - c)! / (d - c - (s - k) )!
t = total ways to drawing s cards in a deck of d cards =
d! / (d - s)!
p = probability of drawing k desired cards =
w * wo / t
Thus for practical use, c can be replaced with redundancies like "lands, removal, ramp, combo pieces", etc.
EDIT: and yes, my numbers match fzian's figures
What does "s Choose k" mean?
It means combination
http://en.wikipedia.org/wiki/Combination
or the number of ways of of choosing k items out of s total items where the order doesn't matter.
Here is a real puzzle for you:
a) A deck has a 5 card combo, and 7 tutors.
b) Beginning anywhere between turns 4- 7 (6 is usually the latest), the deck loses life every turn, beginning with 2, and doubling every turn. (Glacial Chasm)
c) The deck can reliably draw at least 1 extra card per turn beginning, generally, a few turns after Glacial Chasm. (Say, 2 turns)
d) By the time Glacial hits, life total averages 34.
e) Glacial can be reset once.
f) When the combo is assembled, you win.
EDIT: g) Glacial is NOT part of the combo.
What are the odds of winning by turns: 5, 10, 13, 15, 17, 20?
More power to you if you can do this one....b t dubs if you explain the math you use, the explanation will be read and appreciated, at least by myself:)
I LOVE YOU! Hi, I'm a math freak.
I'll get you your answer as soon as possible. Though do keep in mind that I can only work with pencil, paper, and calculator so it'll take a while. Thanks for the opportunity.
EDH
Maelstrom Wanderer
I kinda have to wonder how people end up bringing threads this old up sometimes, do they search for them, do they just go pages and pages back and ignore the dates...?
UBBreya's Toybox (Competitive, Combo)WR
RGodzilla, King of the MonstersG
-Retired Decks-
UBLazav, Dimir Mastermind (Competitive, UB Voltron/Control)UB
"Knowledge is such a burden. Release it. Release all your fears to me."
—Ashiok, Nightmare Weaver
Thanks to Heroes of the Plane for the awesome Sig.
Currently Playing- EDH
GGGOmnath, Locus of the LifestreamGGG
BBBShirei, Lord of PoniesBBB
UWRasputin Dreamweaver, Russia's Greatest Love MachineUW
UBWZur, Killer of FunUBW
UGWTreva, Princess of CanterlotUGW
RWTajic, Master of the Reverse BladeRW
RRRZirilan, How to Train Your DragonRRR
PDH Decks
Gelectrode
Ascended Lawmage
Blaze Commando
WUB Sharuum the Hegemon, the Destroyer of Darksteel
BGW Teneb, the Harvester, my Pimped Out Reanimator
GUB The Mimeoplasm Ooze-Mill
GWU Rafiq the Exalted
Multani, Maro-Sorcerer - Group Hug
Aurelia, the Warleader - Multi-Attack step aggro
Karona, False God - Your spells are my spells
Glissa, the Traitor - Artifact Toolbox
540 Peasant cube- Gold EditionSomething SpicyI was actually just looking for an Orzhov EDH deck and I found this in Blackjack's Sig. Also, ya I do just ignore how old it is since math is one of those things that just can't change.
If you guys have any Math problems I'd be more than a little excited to work on them (after the first one of course).
EDH
Maelstrom Wanderer
Thanks, Heroes of The Planes! You guys are great!
Actual Truth:
Belongs in all new player advice columns EVER.
:symu::symr: Melek WheelStorm
:symw::symg: Trostani Enchantress (updated 6/5)
:symg::symr::symu: Unexpected Results.dec
Thada Adel Stax WIP
Always good to know where the math guys are, just in case. Math hurts my brain.
BTW, if you (or anyone) come up with any tables that would be useful for deckbuilding, post it, and I'll copy it to the op.
Banner by Nakamura, Thanks!
EDH Math
EDH Decks:
Ghost Council: The Magic Mafia of Orzhova
BB Drana: Down with the Sickness
Rasputin: Reality is Broken
Vish Kal Bleeder: Bloody Kisses
Teysa, Orzhov Dominatrix
Stonebrow: Breaking Things
BWR Kaalia Punisher: Heaven's on Fire
Grimgrin: Dead Reckoning
That said, theoretically speaking: -
Thanks for this thread. I've used it for all my EDH decks. I'm also a fan of your Orzhov lists, and I'm brewing on my own super-secret Teysa Combo list. Your contributions are appreciated.