I just started experimenting with a half deck in elo. I was assuming it should be about 50% 2/2 (2 cards from each clan I mean). In 48 fights I've only gotten 2/2 11 times, or about 23%. I know 48 isn't a large sample but it's consistently hovering around 21%. Is there some card selection process I don't know about, or am I just having bad beginner's luck?
First off, we find the number of possible combinations in an eight card deck. Since order doesn't matter and since a card can't be picked twice, we find the binomial distribution and get:
Where n is the number of cards in a deck ( and k is the number of cards in a hand (4):
Number of combinations = n!/(k!(n-k)!) = 8!/(4!(8-4)!)
Now, we need to know the amount of combinations that add up to two cards from each clan. Once again, we find the binomial distribution but this time with the number of cards in one hand (4) and the number of cards from one specific clan in a 2/2 hand (2).
Number of combinations = n!/(k!(n-k)!) = 4!/(2!(4-2)!) = 6
But remember: For each of the six combinations in one clan, there are also six possible combination for the other clan. For this reason, we square 6.
6^2 = 36
We divide the number of 2/2 hands by the total number of hands and get:
36/70 = 0.514 = 51.4%
Statistics are fun
Mathematical chance is supposing there is no bias towards selection but it its totally random.
Since we are not certain if there is bias we cannot use mathematical model, but have to rely on repeated tests.
Thats why rivalry01 brought this up, UR does NOT promise that draws are made purely of luck, therefore to test against bias you need real tests and not mathematical theory.