"Given that a diamond was drawn on the first draw and not replaced" means we know 100% that there are P = 13 - 1 = 12 diamonds left out of Q = 52-1 = 51 cards overall.
The probability of picking another diamond card is therefore P/Q = 12/51
This only works if the card is not replaced. If the card is replaced, then the probability would be 13/52 = 1/4.
