The 52 cards of a standard playing card deck are randomly distributed to two persons: 26 cards to each person. Find the probability that the first person receives all four Kings. Note: The 52 cards include four Kings.
I had this question in my probability exam and my answer was $$ \frac{ {4 \choose 4} . {48 \choose 22}}{52 \choose 26} $$
However the teaching assistant's answer was $(\frac{1}{2})^4 = \frac{1}{16}$ as each card has a probability $\frac{1}{2}$ to go to either of the 2 persons
Which answer is correct?
As Angina Sing noted, your TA is assuming that the $4$ events are independent, but they are not. If we correct his calculation, we arrive at the same answer you did. The probability that the first player is dealt the King of Spades is $\frac{26}{52}$. Once he has the King of Spades, what is the probability that he also receives the King of Hearts? There are $25$ spots remaining in his hand, and $51$ spots overall, so the probability that he is dealt both Aces is $\frac{26}{52}\cdot\frac{25}{51}$. (If you have learned about conditional probability, this is simply the fact that $\Pr(S\cap H)=\Pr(S)\Pr(H|S)$.) Continuing in this manner, we get that the probability that he gets all $4$ Aces is $$\frac{26\cdot25\cdot24\cdot23}{52\cdot51\cdot50\cdot49}$$ the same answer you got.