Why is the probability of drawing a king and then a heart the same as drawing the king of hearts? I calculated that the probability of drawing a king and then a heart from a deck of cards is 
$\frac{1(12) + 3(13)}{\text{Permutation}(52,2)}=\frac{1}{52}$
However, I also noticed that this is the same as the probability of drawing the king of hearts, which is also $\frac{1}{52}$
Is this just a coincidence or is there a reason why this happens?
 A: One possible interpretation of your question is as follows. Let $A$ be the event the first draw results in a King, and $B$ be the event the second draw results in a heart. Why are the events $A$ and $B$ independent?
That they are independent can be verified by the cases computation in the post above. But let us see why the result is intuitively clear.
The probability of a heart on the second draw, is, like the probability of a heart of the first draw, or the seventeenth, equal to $\frac{1}{4}$.
Now suppose that we are told the first card drawn was a King. Should that change our estimate of the probability that the second is a heart? If so, being told the first draw was a Jack should change our estimate in exactly the same way, as should being told that the first draw was a $9$. 
Since all ranks are equally likely to be drawn first, the conditional probability that the second is a heart given the first is of a specified rank is the same as the plain unconditional probability that the second card is a heart. 
This says that $A$ and $B$ are independent, that is, $\Pr(A\cap B)=\frac{1}{13}\cdot\frac{1}{4}$. 
A: Let's say you have a different deck.  One where there are $x$ values and $y$ suits and therefore $xy$ cards.  Let's ask what the probability of drawing a particular value (numbers of suits divided by total cards) followed by a particular suit (number of cards of that suit divided by left over cards).  This will be be:
$$
\frac{y}{xy} \left(\frac{1}{y}\frac{x - 1}{xy - 1} + \frac{y - 1}{y}\frac{x}{xy - 1}\right) = \frac{x - 1 + x(y - 1)}{xy(xy - 1)} = \frac{x - 1 + xy - x}{xy(xy - 1)} = \frac{xy - 1}{xy(xy - 1)} = \frac{1}{xy}
$$
I think André Nicolas gave a much better answer, but here is a more general case which shows that this isn't just a coincidence.
