### Instruction: Two cards are dealt at random from a standard deck of 52 cards.  What is the probability that the first card is a King and the second card is a $\heartsuit$?### Response: I need to find the fraction of all possible outcomes that satisfy the given condition.
To do that, I need to multiply the probability of the first card being a King by the probability of the second card being a $\heartsuit$, given that the first card was a King.
The probability of the first card being a King is 4/52, since there are four Kings in the deck and 52 cards total.
The probability of the second card being a $\heartsuit$, given that the first card was a King, depends on whether the first King was a $\heartsuit$ or not.
If the first King was a $\heartsuit$, then there are 12 $\heartsuit$ cards left out of 51, since one card was removed from the deck.
If the first King was not a $\heartsuit$, then there are 13 $\heartsuit$ cards left out of 51.
So, I need to account for both cases and add their probabilities.
The probability of the first King being a $\heartsuit$ is 1/4, since there is one $\heartsuit$ King out of four Kings.
The probability of the first King not being a $\heartsuit$ is 3/4, since there are three non-$\heartsuit$ Kings out of four.
Therefore, the probability of the second card being a $\heartsuit$, given that the first card was a King, is (1/4)(12/51) + (3/4)(13/51).
Simplifying, this is (12/204) + (39/204) = (51/204) = (1/4).
Finally, I multiply the probabilities of the first and second cards to get the final answer.
The probability of the first card being a King and the second card being a $\heartsuit$ is (4/52)(1/4) = (1/13)(1/4) = 1/52.