1. First, let us find the probability that the first card is a diamond.
Now, since we are given that there are four suits and there are, assumably, an equal number of cards in each suit, we can say that the probability of choosing a diamond card is 1/4. We can also write this out as such, where D = Diamond:
Pr(D) = no. of diamond cards / total number of cards
There are 52 cards in a deck, and 13 cards of each suit, thus:
Pr(D) = 13/52 = 1/4
2. Now we need to calculate the probability of not choosing a diamond as the second card.
In many cases, when given a problem that requires you to find the probability of something not happening, it may be easier to set it out as such:
Pr(A') = 1 - Pr(A)
ie. Pr(A not happening, or not A) = 1 - Pr(A happening, or A)
This works because the total probability is always 1 (100%), and it makes sense that to find the probability of A not happening, we take the total probability and subtract the probability of A actually happening.
Thus, given that we have already calculated that the probability of choosing a Diamond is 1/4, we can now set this out as such:
Pr(D') = 1 - Pr(D)
Pr(D') = 1 - 1/4
Pr(D') = 3/4
3. Now we come to the final step. To find the probability of something and then something else happening, we must multiply the two probabilities together. Thus, given that Pr(D) = 1/4 and Pr(D') = 3/4, we get:
Pr(D)*Pr(D') = (1/4)*(3/4)
= 3/16
Thus, the probability of choosing a diamond as the first card and then not choosing a diamond as the second card is 3/16.
. The polynomial remainder theorem guarantees that the value of 
is the remainder upon dividing 
by 
, but I digress...
, as expected)