The card after the first ace

Idea. The information “the first ace is 14th” says that the first 13 cards are non-aces. The king of hearts is either among them, or among the 38 cards after the ace, each position equally likely.

Step 1: not used up early. Given the condition, the first 13 cards form a uniformly random set of 13 of the 48 non-aces. The king of hearts is not among them with probability $$\frac{48-13}{48}=\frac{35}{48}.$$

Step 2: then position 15. In that case the king of hearts is among the 38 cards after position 14, and by symmetry it is equally likely to be in any of those positions. So it is the 15th card with probability $\frac{1}{38}$.

Step 3. $$\frac{35}{48}\cdot \frac{1}{38}=\frac{35}{1824}\approx \mathrm{0.0192.}$$

Compare. Without any condition, the card right after the first ace is the king of hearts with probability exactly $\frac{1}{52}\approx 0.01923$. Knowing that the first ace came late makes it slightly less likely, $\frac{35}{1824}\approx 0.01919$, because the king of hearts may already be among the 13 cards before it.