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November 10th, 2013, 06:56 PM   #1
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Magic The Gathering card game calculation I need help with

(first post!)

So I play a collectible card game called Magic The Gathering. It's a hobby of mine.

In one format of this card game, players create 40 card decks and duel against each other. To begin, each player shuffles their deck and draws a 7 card hand. Each player respectively looks at their hand and decides whether to keep that hand and begin the game, or shuffle those 7 cards back into the deck and redraw a new hand - but this time with only 6 cards. This is called a "mulligan". The players then chooses to "mulligan" down to 5, 4, 3, 2, or even 1 card in their starting hand. Once both players have agreed to "keep" their starting hands, the game begins. The players takes turns playing cards from their hand and drawing cards from their deck, until win conditions are met and the game ends in a win, loss, or draw.

In the current format of Magic The Gathering, there is a specific play that if made on the first and second turns of the game, is devastating to the opponent and very hard to recover from.

So I've been thinking about how to create a deck that has a good chance of having a specific set of cards in it's starting hand that will satisfy the requirements to make this play. Once I can figure out how to calculate the odds correctly, I can change the variables to compare between deck construction options and find what I consider to be the "ideal" deck, based on hard numbers.

There are 4 types of cards that matter to this issue. All of the other cards in the 40 card deck can be ignored, as they pertain to the first two turns of the game.

I'll start out with what some inputs for these 4 card types that I think I'll be able to regularly put into my deck construction, and we can go from there.

Deck Composition:
Card Mountain, or "m". 8 of the 40 cards in the deck are "m"
Card Plains, or "p". 8 of the 40 cards in the deck are "p"
Card Ordeal, or "d". 4 of the 40 cards in the deck are "d"
Card Crusader, or "c". 4 of the 40 cards in the deck are "c"

I need to know this:
If I start the game with the combination of cards in it listed below, I'll consider that a "success". If I start the game without the specific combination of cards, I'll consider that "failure". Based on the deck composition above, and using the conditions listed below, what is the rate of "success" I can expect if I take the following actions: I draw a 7 card hand, look at it, notice the conditions are met and keep it, OR notice that the below conditions are not met, "Mulligan" to 6, look at the 6 card hand, look at it, notice the conditions are met and keep it, OR notice that the below conditions are not met, "Mulligan" to 5, look at it, notice the conditions are met and keep it, OR notice that the below conditions are not met, "Mulligan" to 4, and then keep the 4 card hand regardless of the composition of the hand.

Conditions:
no less than 1 Mountain, AND
no less than 1 Plains, AND
no less than 1 Ordeal, AND
no less than 1 Crusader


Thanks in advance for any help you can offer!
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November 16th, 2013, 09:48 PM   #2
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Re: Magic The Gathering card game calculation I need help wi

CLUE - I found this comment somewhere on the web:

"Binomial coefficients arethe key functions for all kinds of probability computations in MtG..."
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