# Combinatorics applied to a playing card game

I want to analyze the features of a deck of Magic the Gathering cards game, which many of you know for sure.
My goal now is to create an Excel sheet, where you can enter the composition of the deck (each card with its data), to get statistics, such as the probabilities of playing a certain card in the first round and in round X, mana curves...

The basic rules you need to know for these calcs:
Each card with an active role can be played, in general, paying its cost (mana) with 'land' cards on the field. You can put into the field at most a land.
The starting hand consists of 7 cards drawn from the deck at the same time as soon as the game starts, for the player who goes first, while the second will draw a card, in addition to the initial 7, during his turn.
When the turn starts, the active player draws a card.

Problem 1:
Knowing the number of land cards in the deck and that there is only one card with the cost of 1,
calculate the probability of playing at least one card in the first round (playing the only card with cost 1), which corresponds to probability of having in the first hand at least one land card and the card that only costs one.
With a Python simulation on 100.000 starting hands, I got as the average result, 11.5%.

General problem:
Knowing the number of land cards in the deck and the number of cards with cost 0, of those with cost 1, with cost 2 and so on,
calculate the probability of playing at least one card in round X,
calculate the probability of playing N cards in round X.
The player put a land into the field, if he has it one in the hand.

Who can tell me how to do these calculations and which theorems I need? Thank you all.

• Do you have a single copy of the 1-mana card, or for copies of it? Also worth mentioning, a Magic: the Gathering deck typically has $60$ cards. – Arthur Oct 21 '16 at 10:59
• @Arthur Sure, typically it has 60 cards and in the first problem I have a single copy of the 1-mana card, but I'd like to get a general formula. – JumpAlways Oct 21 '16 at 13:18
• We can say: N = L + A, where L is the number of the land cards and A is the number of the cards you have to pay for and A = c + h, where c is the number of the cards that respect the condition, for example in the first problem, cards which have a cost <= 1. – JumpAlways Oct 21 '16 at 13:23