If four decks of cards are shuffled together, what is the probability of dealing a 13-card hand you have exactly two black Queens?

Question

a little confused with this question any help would be appreciated!

Answer 1

First calculate how many different 13-card hands there are when taken from 4 decks.  In 4 decks there are 4 x 52 = 208 cards.  So the number of different 13 card hands is ₁₃C₂₀₈.  Let's leave it this way for now.

 

Now, how many of these hands will have exactly 2 black queens? 

 

We know that each deck has 4 queens, of which 2 are black, so there are 8 black queens in 208 cards, and 200 cards that are not black queens.

 

2 of the cards in our 13-card hands have to be black queens, so there are ₂C₈ ways to get 2 black queens.  And the rest of the cards in the hands, 11, have to be not black queens, so there are ₁₁C₂₀₀ ways to get 11 cards that are not black queens.

 

So there are a total of ₂C₈ * ₁₁C₂₀₀ ways to get exactly 2 black queens in 13 cards.  So to find the probability of exactly 2 black queens, we just divide this by the total number of possible hands, ₁₃C₂₀₈. 

 

₂C₈ * ₁₁C₂₀₀ / ₁₃C₂₀₈ =

 

(8! / 2! 6!) (200! / 11! 189!) / (208! / 13! 195!) =

 

28 * 387790074428411200 / 149389005978091284720 = 0.0727 = 7.27%