If you draw two cards, what is the probability that the second card is a queen?

There are two cases here:

Case 1: First card chosen is a queen

$$\frac{4}{52}*\frac{3}{51}=\frac{1}{221}$$

Case 2: First card chosen is not a queen.

$$\frac{48}{52}*\frac{4}{51}=\frac{16}{221}$$

Adding both the cases, we get $\frac{17}{221}$ = $\frac{4}{52}$ = $\frac{1}{13}$

Think about it this way: Shuffle a deck of cards randomly. The probability of drawing a queen as your second card is the same as the probability that the second card in the deck is a queen, which is clearly 4/52.

A slightly more intuitive way of looking at this:

The probability that the second card is a queen should be the same as the probability that the second card is an ace, and the same as the probability that the second card is a 2 etc. There are $ 13 $ possibilities for the card number/letter, so the answer is $ \frac{1}{13} $