Randomization and probability with constraints

Question

The probability of getting a specific suit out of a deck is 13/52. Once we have one suit (color) selected the probability is 12/51 for that suit and 13/51 for others. This much is obvious.

Now suppose we want a hand with 2-4 spades, and 4 hearts. The rest is totally random.

So the question is how to calculate probability when suits have constraints like this.

Thanks

Answer 1

I believe a computation with straightforward combinations suffices. There are $d = {52 \choose 13}$ ways to deal a 13-card hand from a 52-card deck.

The number of ways to get exactly 2 spades and exactly 4 hearts is $n_2 = {13 \choose 2}{13 \choose 4}{26 \choose 7}.$ The last factor is necessary to make sure the other cards are all chosen from among the other suites.

Then $n_3$ and $n_4$ can be found similarly, and the required probability is $(n_2 + n_3 + n_4)/d = 0.1848679.$

Addendum: A simulation of 1 million hands gives the same result within simulation error:

 m = 10^6;  s = h = numeric(m)
 deck = rep(1:4, times = 13)  # 1's are hearts; 3's are spades
 for (i in 1:m) {
   hand = sample(deck, 13)
   s[i] = sum(hand==3);  h[i] = sum(hand==1) }
 mean(((s==2)|(s==3)|(s==4)) & (h==4))
 ## 0.184742