Let's begin by modeling the procedure described. This is the key to solving any problem in probability. If we can understand in words what is being done, then we can convert our description into a mathematical representation and solve.
One card is removed at random from the deck. This may be a black king or it may be any other card. (Since our goal in the end is to draw a black king, it will be important to know what card is drawn so we know how many black kings remain during the second step. We obviously can't know what card is drawn, we must consider both possibilities).
PART 2: Fifty-one cards remain and a card is picked. It may be a black king, or it may be any other card.
PUTTING IT ALL TOGETHER
Picking a black king in the end can happen in one of two ways:
(First draw = BLACK KING) AND (Second draw = BLACK KING)
(First draw = NOT BLACK KING) AND (Second draw = BLACK KING)
The problem contains 4 situations (each in its own parentheses above) who's probabilities we must model and connect appropriately. Let's model them first and then we will connect them.
Probability of (First draw = BLACK KING):
In the numerator we want to write the number of ways our special case can occur. Specifically, how many ways are there in which a black king can be drawn from a deck of 52 cards. Two! (There are two black kings in each deck.)
In the denominator we will write the number of ways there are of drawing the first card. Fifty-two! (There are 52 cards we could pick.)
So, P(First draw = BLACK KING) = 2/52
Probability of (Second draw = BLACK KING), given (First draw = BLACK KING):
We are only left with one black king in the deck, and the deck now has only 51 cards.
Therefore, P(Second draw = BLACK KING) = 1/51
Probability of (First draw = NOT BLACK KING):
In this case, we have 50 cards we can choose that are not a black king, and 52 possible ways of choosing one card. So,
P(First draw = NOT BLACK KING) = 50/52
Probability of (Second draw = BLACK KING), given (First draw = NOT BLACK KING):
In this case, both black kings are still in the deck (2 ways of picking a black king) and there are only 51 total cards remaining in the deck (51 possible ways of choosing one card).
P(Second draw = BLACK KING) = 2/51
CONNECTING ALL THE PIECES:
Whenever we say AND in our model, we should think multiplication.
Whenever we say OR in our model, we should think addition.
So, we have for the probability of picking a black card from a deck after throwing out one card at random:
P = (2/52)*(1/51) + (50/52)*(2/51)
Now just evaluate the model and you're done!
P = 0.0385
Hope this helps! I know it's a long answer, but hopefully talking out the whole process helps you with future problems like this one.