Probability of completing a specific set of cards

Question

A deck of cards from a kids game contains 108 cards in total.

There are 12 different categories of card in total, and each category has a certain number of cards within the deck.

Category A: 14 cards. Cat B: 14 cards. Cat C: 14 cards. Cat D: 12 cards. Cat E: 8 cards. Cat F: 6 cards. Cat G: 10 cards. Cat H: 5 cards. Cat I: 5 cards. Cat J: 10 cards. Cat K: 6 cards. Cat L: 4 cards.

The game begins and 3 players share 27 cards, aiming to complete various category ‘sets’ for points. The 27 cards are dealt from the top of the shuffled 108-card deck.

If a player collects 1x Category H card AND 1x Category K card then they score big points!

What is the probability that the 27 dealt cards will contain BOTH of those cards?

There is some discussion in our house about the probability of getting those 2 cards dealt together ie is it a set worth chasing! I’m afraid we don’t know the formula to give us a definitive answer! Thx

Answer 1

Like many card questions, this ultimately boils down to the multivariate hypergeometric distribution. Specifically, in this case, the chance of drawing exactly $h$ Hs and $k$ Ks is:

$$ \frac{\binom{5}{h}\binom{6}{k}\binom{108 - 5 - 6}{27 - h - k}}{\binom{108}{27}} $$

Then you need to sum over all values of $h, k \geq 1$.

I've been working on Icepool, a Python library that can answer these sorts of questions more easily. Here's an example script:

from icepool import Deck

deck = Deck({
    'a' : 14,
    'b' : 14,
    'c' : 14,
    'd' : 12,
    'e' : 8,
    'f' : 6,
    'g' : 10,
    'h' : 5,
    'i' : 5,
    'j' : 10,
    'k' : 6,
    'l' : 4,
})

output(deck.deal(27) >= ['h', 'k'])

Here the >= represents $\supseteq$, the superset comparator on multisets.

You can run the script online here.

The result:

Die with denominator 20984707951701767172932872

Outcome	Quantity	Probability
False	7644946318316705155562148	36.431035%
True	13339761633385062017370724	63.568965%