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

[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.

 

[mmwave]

Thank you for reaching out for help with your workshop. I am happy to assist you with your calculations for the probabilities of drawing at least x cards of type A in n draws.

Firstly, your calculation for x=1 and n=7 is indeed correct. However, for values of x>1, we need to use a different approach. Instead of shifting the numbers of cards in the deck, we can use a combination formula to calculate the probabilities.

The formula for calculating the probability of drawing exactly x cards of type A in n draws is:

P(x) = (nCx)(p^x)(q^(n-x))

Where:
n is the total number of draws
x is the number of cards of type A we want to draw
p is the probability of drawing a card of type A (in your example, 18/40)
q is the probability of drawing a card of type B (in your example, 22/40)
nCx is the combination formula, which can be calculated as n!/(x!(n-x)!), where n! is the factorial of n.

To calculate the probability of drawing at least x cards of type A, we need to sum up the probabilities for all possible values of x. So for x=2 and n=8, the calculation would be:

P(2) = (8C2)(18/40)^2(22/40)^6 + (8C3)(18/40)^3(22/40)^5 + (8C4)(18/40)^4(22/40)^4 + (8C5)(18/40)^5(22/40)^3 + (8C6)(18/40)^6(22/40)^2 + (8C7)(18/40)^7(22/40)^1 + (8C8)(18/40)^8(22/40)^0

This may seem like a lot of calculations, but there are online calculators that can help you with the combination formula and you can also use a spreadsheet to make the calculations easier.

I hope this helps you with your workshop and I wish you all the best in your presentation. Feel free to reach out if you have any further questions or if you need any additional help.