Probability in cards that $4$ people each get queen and king? A $52$ card deck is shuffled and then dealt out to $4$ people $($each person gets $13$ cards$)$. What is the chance that each person gets an a Queen & a King?
My attempt: I know there's $\binom{52}{13}$ different hands that can be shuffled out. The chances of getting a Queen is $4/52$ and the chances of getting a King is $4/52$. This makes the chance of having both $16/(52*52)$ = $16/2704$. Since there are $4$ people, does that mean the overall chance of  each person having a Queen and King in their $13$ card deck $(\frac{16}{2704})^4$?

There are $\binom{52}{13}\binom{52-13}{13}\binom{52-13-13}{13}$ ways of dealing 4 hands of 13 cards each.
Given that each hand has to have one king and queen, there are $\binom{52-8}{13-2}\binom{52-13-2-8}{13-2}\binom{52-13-2-13-2-8}{13-2}$ for the remaining cards, and $(4\cdot 3\cdot 2)\cdot (4\cdot 3\cdot 2)$ ways for the king and queen to go, So probability is $\frac{24\cdot 24\cdot \binom{52-8}{13-2}\binom{52-13-2-8}{13-2}\binom{52-13-2-13-2-8}{13-2}}{\binom{52}{13}\binom{52-13}{13}\binom{52-13-13}{13}}$
