Step 1:

Let $P(R_1)$ be the probability of getting a red queen in the I draw =$\large\frac{2}{52}$

Let $P(K_2)$ be the probability of getting a king of black in the II draw $=\large\frac{2}{51}$

Let $P(R_2)$ be the probability of getting a red queen in the II draw $=\large\frac{2}{51}$

Let $P(K_1)$ be the probability of getting a black king in the I draw $=\large\frac{2}{52}$

Step 2:

Required probability is $P(R_1).P(\large\frac{K_2}{R_1})$$+P(K_1).P(\large\frac{R_2}{K_1})$

$\Rightarrow \large\frac{2}{52}\times \frac{2}{51}+\frac{2}{52}+\frac{2}{51}$

$\Rightarrow \large\frac{2}{663}$