At least one by turn k (and other data)

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So I was wondering what my chances would be of having a certain card by turn k.  Instead of counting turns, I count draws (so draw 7 means your initial hand and draw 10 could mean the third turn).

Suppose I have a 60 card deck and I have n copies of a certain card in my deck.  What is the probability that I will have at least one copy of this card by draw k? (of course assuming that all cards drawn are available to me)  In the following graph, the colored numbers represent the number of copies of the chosen card in your 60 card deck. k is the x-axis.

IMAGE(http://img820.imageshack.us/img820/3402/atleast1bydrawk.png)

Now suppose you have 4 copies of a card in a 60 card deck.  The following graph shows the chances of getting at least n of them by draw k.  The x-axis is the k and the colored numbers are the n's.
IMAGE(http://img692.imageshack.us/img692/5076/atleastnbydrawk.png)
Now if you only had 3, 2 or 1 copies of the card in your deck:
IMAGE(http://img252.imageshack.us/img252/4373/atleastnbydrawk2.png)
IMAGE(http://img857.imageshack.us/img857/2313/atleastnbydrawk3.png)
IMAGE(http://img841.imageshack.us/img841/4490/atleastnbydrawk4.png)

-Jingbo
That's pretty awesome.  What equasions did you use to figure these out?
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oh god math
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57817638 wrote:
I like storm crow because I really like crows in real life, as an animal, and the card isn't terribly stupid, but packs a good deal of nostalgia and also a chunck of the game's history. So it's perhaps one of the cards I have most affection to, but not because "lol storm crow is bad hurr hurr durr".
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56747598 wrote:
57295478 wrote:
Although I do assume you deliberately refer to them (DCI) as The Grand Imperial Convocation of Evil just for the purposes of making them sound like an ancient and terrible conspiracy.
Now, now. 1994 doesn't quite qualify as "ancient".
56734518 wrote:
Oh, it's a brilliant plan. You see, Bolas was travelling through shadowmoor, causing trouble, when he saw a Wickerbough Elder with its stylin' dead scarecrow hat. Now, Bolas being Bolas took the awesome hat and he put it on his head, but even with all his titanic powers of magic he couldn't make it fit. He grabbed some more scarecrows, but then a little kithkin girl asked if he was trying to build a toupee. "BY ALL THE POWERS IN THE MULTIVERSE!" he roared, "I WILL HAVE A HAT WORTHY OF MY GLORY." and so he went through his Dark Lore of Doom (tm) looking for something he could make into a hat that would look as stylish on him as a scarecrow does on a treefolk. He thought about the Phyrexians, but they were covered in goopy oil that would make his nonexistant hair greasy. He Tried out angels for a while but they didn't sit quite right. Then, he looked under "e" (because in the Elder Draconic alphabet, "e" for Eldrazi is right next to "h" for Hat) in his Dark Lore of Doom and saw depictions of the Eldrazi, and all their forms. "THIS SHALL BE MY HAT!" he declared, poking a picture of Emrakul, "AND WITH IT I WILL USHER IN A NEW AGE OF DARKNESS -- ER, I MEAN A NEW AGE OF FASHION!" And so Nicol Bolas masterminded the release of the Eldrazi.
57864098 wrote:
Rhox War Monk just flips pancakes, and if games have told us anything, it's that food = life.
56747598 wrote:
76973988 wrote:
This thread has gotten creepy. XP
Really? Really? The last couple days have been roughly every perverse fetish imaginable, but it only got "creepy" when speculation on Mother of Runes's mob affiliation came up?
76672808 wrote:
57864098 wrote:
57531048 wrote:
Nice mana base. Not really.
Yeah, really. If my deck was going to cost $1000+, I'd at least make it good.
99812049 wrote:
I like to think up what I consider clever names for my decks, only later to be laughed at by my wife. It kills me a little on the inside, but thats what marriage is about.
56816728 wrote:
56854588 wrote:
Of course, the best use [of tolaria west] is transmuting for the real Tolaria. ;)
Absolutely. I used to loose to my buddy's Banding deck for ages, it was then that I found out about Tolaria, and I was finally able win my first game.
70246459 wrote:
WOAH wait wait wait
56957928 wrote:
You know, being shallow and jusdgmental aside, "I later found out that Jon infiltrated his way into OKCupid dates with at least two other people"
56957928 wrote:
"I later found out that Jon infiltrated his way into OKCupid dates with at least two other people"
56957928 wrote:
Jon infiltrated his way into OKCupid dates
56957928 wrote:
OH MY GOD
109874309 wrote:
The only way I'd cast this card is into a bonfire.
82032421 wrote:
The short answer is that there's no rule barring annoying people from posting, but there a rule barring us from harassing them about it.
56747598 wrote:
Browbeat is a card that is an appropriate deck choice when there's no better idea available. "No better idea available" was pretty much the running theme of Odyssey era.
56874518 wrote:
Or perhaps it was a more straightforward comment indicating a wish for you to be bitten (Perhaps repeatedly) by a small yet highly venomous arachnid.
70246459 wrote:
58280208 wrote:
You're an idiot, and I'm in no mood for silliness.
57817638 wrote:
57145078 wrote:
You just... Vektor it.
That's the answer to everything.
70246459 wrote:
58347268 wrote:
I think the problem is that you don't exist.
This would sound great out of context!
56965458 wrote:
Modern is like playing a new tournament every time : you build a deck, you win with it, don't bother keeping it. Just build another, its key pieces will get banned.
57864098 wrote:
57309598 wrote:
I specifically remember posting a thread when I was just a witty bitty noob.
You make it sound like that's still not the case.
58325628 wrote:
Rap is what happens when the c from crap is taken away.
Doug Beyer:
But sometimes it's also challenging. Because sometimes OH MY GOD, WHAT THE HELL IS THIS THING?
141434757 wrote:
Flashforward five thousand years (Click for atmosphere) :
57927608 wrote:
to paraphrase Jeff Goldblum, Vektor finds a way.
58347268 wrote:
when in rome **** AND PILLAGE
143229641 wrote:
I always find it helpful when im angry to dress up in an owl costume and rub pennies all over my body in front of a full body mirror next to the window.
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Playing Magic without Blue is like sleeping without any sheets or blankets. You can do it...but why?
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Just do it like Yu-Gi-Oh or monkeys: throw all the crap you got at them and hope it works or else the by-standers (or opponents) just get dirty and pissed.
57471038 wrote:
58258708 wrote:
It's true that Alpha and Beta didn't contain any cards like Tarmogoyf, Darksteel Colossus, or Platinum Angel. It just contained weak, insignificant cards like Black Lotus, Mox Sapphire, and Time Walk.
Normally it's difficult to pick up on your jokes/sarcasm. But this one's pretty much out there. Good progress. You have moved up to Humanoid. You'll be Human in no time.
91893448 wrote:
94618431 wrote:
I didn't know Samurai were known to be able to cut down whole armies...
They can when they're using lightsabers!
57129358 wrote:
97980259 wrote:
My wife brought home a baby black squirrel they found on a horse track and cared for it for a few days. We named it Grixis, but it died.
Unearth it!
70246459 wrote:
[/spoiler] And I'm on Magic Arcana. How about you? Oh, by the way, I'm also on From the Lab now. Twice, actually. And now with my own submited decklist!
If you pull the graphs back pre draw #7 you can account for how mulliganing will play into the chances.  i.e. if you have 4 of a card in your deck, don't get it in your opening hand, what chance do you have of seeing it on the mulligan, etc.

I've enjoyed your graph posts, btw -- I think they're cool, and I love visualizing applied mathematics.

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@Cathaldus: I used elementary combinatorics.  I have no problem disclosing the formulas, but it wouldn't be very useful unless you knew the math already.

@FirstTurnKill:  You can sorta guess what would happen there.  You can see that there is some type of symmetry in the graphs.  The first graph would turn out looking like a bow or a taco heh.  I think it would be most interesting if I could have a graph for "chance I would have at least x lands and at least one of a chosen card by turn k" because of course we can't play the chosen card without having the lands available (in general... pretend Bloom Tender doesn't exist etc.).  However, this is a complicated problem that I could best solve with Mathematica which I do not have at the moment.  I know how to calculate these probabilities and would make a set of graphs for lands up to 6 say, but I would need to either think of a quicker way to approximate it with calculus or create/find an algorithm to find the exact answers by giving me the right arrays at each step.

There may be some interesting applications of Game Theory too... Perhaps it would be useful to take a famous magic deck and explore the mathematical reasons it works so well.  Then we could learn from this and emulate its greatness in our own decks.
If you pull the graphs back pre draw #7 you can account for how mulliganing will play into the chances.  i.e. if you have 4 of a card in your deck, don't get it in your opening hand, what chance do you have of seeing it on the mulligan, etc.

I've enjoyed your graph posts, btw -- I think they're cool, and I love visualizing applied mathematics.



you would have actually less of a chance seeing it after the mulligan since you are drawing 6 cards and not 7 because since the original hand is shuffled in it basically resets everything.  Unless of course you follow the whole gamblers fallacy thing.
I like fun, but competitive decks. So I might not play what is optimal but they have normally been tested to have a 2/3 winrate.
you would have actually less of a chance seeing it after the mulligan since you are drawing 6 cards and not 7 because since the original hand is shuffled in it basically resets everything.


Yes, I realize that, that doesn't mean knowing the additional probability is valueless.  It's still applicable to see what those probabilities are for total draws less than 7, even though they will be independently lower.

You can still use that additional probability to determine the odds that a particular card is drawn in any one of the set of chances [first hand, first mulligan, second mulligan, etc].  If, in 4 out of 10 games you see the card in your opening hand, but in ~3.5 out of 10 of the 6 out of 10 other games (when you don't see the card, so you mulligan), you see the card in your first mulligan, then the overall chances of seeing the card *by* the first mulligan is greater than seeing in only your first hand.  I think that's worth knowing when it comes to asking the question "is it worth mulliganing for a particular card or set of cards?"

A simple thought experiment is flipping a penny to get heads.  If heads shows up on the first flip, don't flip again.  If tails shows up, flip only once more.  Though the second flip still only has a 50% chance of being heads, you will see a heads flip show up an overall 75% of all games.

Somnia, the Evanescent Plane -- A 3-set Block
Set 1 — Somnia
Set 2 — TBD
Set 3 — TBD
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Build Around This
A weekly MTG Cards and Combos forum game.
Build Around This #1 - Sage's Starfish Wish
BAT #1 was built using the Legacy format with Spiny Starfish, Sage's Knowledge, and Make a Wish. Winner: Dilleux_Lepaire with Fishy Starfishies. Runner-Up: JBTM
If you pull the graphs back pre draw #7 you can account for how mulliganing will play into the chances.  i.e. if you have 4 of a card in your deck, don't get it in your opening hand, what chance do you have of seeing it on the mulligan, etc.

I've enjoyed your graph posts, btw -- I think they're cool, and I love visualizing applied mathematics.



you would have actually less of a chance seeing it after the mulligan since you are drawing 6 cards and not 7 because since the original hand is shuffled in it basically resets everything.  Unless of course you follow the whole gamblers fallacy thing.


If you've already drawn your 7 and did not draw the card you want, then you're not talking probability anymore.  There is zero chance to get the card you want, and we can know this because you have already drawn your cards and did not get the one you want.  Shuffling up and trying again can't make this chance less than zero.  He just wants the graphs extended backwards to show probabilities after 6, 5, 4 draws etc. too instead of only 7+.
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IMAGE(http://img213.imageshack.us/img213/9458/atleast1bydrawk2.png)

I didn't save the code for the other graphs since I just modified one file to do them all as I went along.
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