# acdbrn2000

#### A study of Probability in CCGs. (Specifically in Shadow Era)

by

, 07-20-2011 at 10:59 AM (5886 Views)
Wow so I guess if you are reading this then you were not scared away by the title. Congratulation, hopefully unlike most math this will actually be useful to you.

For this study I will look at 30 and 40 card decks including the hero and also going for both the first player and second player draw.

30 card deck = 29 drawable cards

40 card deck = 39 drawable cards

first player draw = 6 default cards

second player draw = 7 cards total / 6 default cards + 1 drawn card

Deck size vs Drawing a specific card / both initial draw and on an individual draw

the easy one an individual draw during a game

C = number of a specific card left in the deck so never more then 4 and obviously if 0 are left then there is )% chance of drawing it

D = number of cards left in the deck

the equation to figure out the chance of any one card being pulled then is C/D

example:

4 retreats and 20 cards left in the deck the odds of pulling a retreat is 4/20 or 1/5 or 20%

the harder version multiple draws

H (n) = C (X, n) * C (Y - X, Z - n) / C (Y, Z)

X stands for the number of a certain card that you have in the deck.

Y is the number of cards in the deck.

Z is the number of cards you are drawing.

N is the number you are checking for.

this is a hypergeometric distribution. It is used to determine the probability of certain sets of occurrences when extracting elements without replacement. Hypergeometric distribution may seem like a odd phrase but really all it is simply is the odds of getting something specific from a group of items assuming you don't replace what you are taking out. Like drawing cards out of a deck and not replacing them. This formula can be used to determine how often you draw certain cards from a deck of cards.

This can be used for a single card draw but the easy version works as well. this can also be used to figure out the odds of getting 1 or multiple of any card in your opening hand.

perhaps an example would help

4 retreats and 20 cards left in the deck the odds of pulling a retreat from 2 cards drawn

H (1) = C (4, 1) * C (20 - 4, 2 - 1) / C (20, 2)

I could write out all the math involved in solving this equation but a hypergeometric distribution calculator can be easily found on the web or using excel you could do it to. (also large factorials are annoying to work with)

excel syntax: HYPGEOMDIST (N, Z, X, Y)

for this problem however the answer would be 33.68% of drawing exactly 1 retreat but if you look also at the odds of drawing 2 which using this formula will find as a a bad probability then the odds of drawing 1 or 2 retreats is 36.84% this is the cumulative answer and the more accurate one to what we care about which is getting a retreat in our hands. this is simply adding the probability of both getting 1 retreat + 2 retreats since either probability is fine for us.

Done with all that math just give me the odds

30 card deck (29 cards since the hero comes out automatically) player goes first so 6 card hand

4 copies = 62.72% of at least one of these cards in the hand

3 copies = 51.53% of at least one of these cards in the hand

2 copies = 37.68% of at least one of these cards in the hand

1 copy = 20.69% chance of drawing this card

40 card deck (39 cards since the hero comes out automatically) player goes first so 6 card hand

4 copies = 50.24% of at least one of these cards in the hand

3 copies = 40.30% of at least one of these cards in the hand

2 copies = 28.74% of at least one of these cards in the hand

1 copy = 15.38% chance of drawing this card

now these 2 sets tells a few things to keep in mind when creating a deck

1. 10 cards more in a deck sees a large drop in your odds of drawing any one particular card

2. the gains for each extra copy of a card lessen the more you already had

3. never count on a lucky draw with only one copy of a card the odds are stacked against you.

Practical advice from these stats

Having a bigger deck dose not make you a better player, unless your deck strategy is to try to deck your opponent(kill them with draw burn). All it dose is insure you have lousy odds for getting a card you want.

4 of a card is not always ideal unless you either must have that card or it is a card that you will play more then one of. 3 Aldons are usually just as good as having 4 and sometimes better if it allows you to have an extra copy of something you can play more then one at a time.

Adding just one card almost never helps you all you are doing is reducing chances for other cards.

second important question is going first really the greatest thing ever while board control is easier if you get a ally out first lets look at the drawing odds for the second player.

30 card deck (29 cards since the hero comes out automatically) player goes second so 7 card hand (default 6 + 1 draw)

4 copies = 69.20% of at least one of these cards in the hand

3 copies = 57.85% of at least one of these cards in the hand

2 copies = 43.10% of at least one of these cards in the hand

1 copy = 24.14% chance of drawing this card

here we see gains across the board for all numbers of copies but the greatest gain is given to having 4 copies of a card. So going second while it puts you behind the curve in resources and usually allies. It can also help insure you have the cards to fight back from that position.

This was a little more long winded then I was hoping to make it and I hope it will be of some use to everyone building a deck. If you have any questions feel free to pm me and I will try to help.

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