jmc

The number $ad-bc$ is even if and only if $ad$ and $bc$ are both odd or are both even. Each of $ad$ and $bc$ is odd if both of its factors are odd, and even otherwise. Exactly half of the integers from 0 to 2007 are odd, so each of $ad$ and $bc$ is odd with probability $(1/2)\cdot (1/2)=1/4$ and are even with probability $3/4$. Hence the probability that $ad-bc$ is even is $$\frac{1}{4}\cdot \frac{1}{4}+\frac{3}{4}\cdot \frac{3}{4}=\boxed{\frac{5}{8}}.$$