# Probability of drawing at least x cards of type A in n draws

1. Apr 25, 2012

### LeePerry

I've been asked to give a workshop at my local youth center about a certain card game (MtG for those that know it). The last time I did this was a few years ago and I lost some of my notes in the mean time.

I always presented the attendants with a little spreadsheet with a variety of useful statistics for the game. But I'm having some trouble to recreate my files for one specific data-point. And I need a little bit of help getting the correct formula, since the probabilities I found seemed to be way off with my intuition of what they should be.

This is the setup and the probabilities I want to calculate.

Imagine a deck of cards with two types of cards A & B, the division between A & B is uneven (for example 18 A & 22 B, for a total of 40 cards)
The question now is what are the probabilities of drawing at least x (1,2,3,...) cards of type A in n (1,2,3,...) draws.

for x=1 and n=7, I use the following calculation (which I'm pretty sure is correct):

1-((22/40)(21/39)(20/38)...(16/34)) and if I want to adjust the value of n I can just add more terms (if n=8 then the last term would be (15/33), etc...)

my problem comes if I want to calculate the probability for values of x > 1

at first I tried out shifting the numbers of cards in the deck (as if I had taken out 1 card A of the deck and then used the same method to calculate the probability for at least (x-1) in my next (n-1) draws. leading to the following

if x=2 and n=8 : 1-((22/39)(21/38)...(15/33))

I can give the workshop without this data, but I've always got positive reactions to the fact that I could explain certain 'strategic' rules from the game based on actual numbers, and I'm feeling a bit of a klutz right now because I can't even figure out these numbers on my own at the moment.

so if anyone could give me some pointers as to how I would go about to calculate these odds with x > 1 and n > 7, I would be very much obliged.