probability of picking a specific card from a deck

Your thinking is correct, though let me provide another way of looking at the problem that might make its structure clearer:

• There are ${51\choose 4}$ ways to pick a hand that include the Jack of Hearts (because once we've picked the Jack, we can choose 4 other cards from the remaining 51)
• There are ${52\choose 5}$ ways to pick a hand, with no restrictions.

Therefore the probability of getting a hand with the Jack of Hearts is

$$\frac{51\choose 4}{52\choose 5} = \frac{51!5!47!}{4!47!52!} = \frac{5}{52}$$

You can check that the obvious generalization is, in fact, true: the probability of drawing a particular card in a hand of $m$ cards with a deck of size $n$ is $m/n$.

Alternatively, there are $\binom{51}{5}$ ways of picking a hand that does not have the Jack of Hearts. There are a total of $\binom{52}{5}$ ways of picking $5$ cards, so the probability of choosing a hand with the Jack of Hearts is: $$1 - \frac{\binom{51}{5}}{\binom{52}{5}} = 1-\frac{47}{52} = \frac{5}{52}$$

The probability you draw the jack of hearts is the same as the probability of drawing any other particular card. Since you draw 5 cards, the 52 individual probabilities have to add up to 5, so each probability is 5/52. In particular, the probability of drawing the jack of hearts is 5/52.