If you draw two cards in consecutively in a standard deck of 52 cards, what is the probability of getting black on the second draw? This is my thought process:
$P(2^{nd}\ \text{black}) = P(2^{nd}\ \text{black} \mid 1^{st}\ \text{red}) P(1^{st}\ \text{red}) + P(2^{nd}\ \text{black} \mid 1^{st}\ \text{black}) P(1^{st} \ \text{black}) $ 
$\frac{26}{51}*\frac{26}{52} + \frac{25}{51}*\frac{26}{52} = \frac{1}{2} $
Is this correct?

You are correct... unless the colour of the first card is known, in which case it would be either:
$\dfrac{25}{51}$ if the first card was black, or
$\dfrac{26}{51}$ if the first card was red
