2007 AMC 12A Problem 12

Exactly half of the integers from $0$ to $2007$ are odd. A product $ad$ is odd only when both factors are odd, with probability $\tfrac12\cdot\tfrac12=\tfrac14,$ and even with probability $\tfrac34.$ The same holds for $bc.$ Then $ad-bc$ is even when both products are odd or both are even: $$\tfrac14\cdot\tfrac14+\tfrac34\cdot\tfrac34=\tfrac{10}{16}=\tfrac58.$$ Thus, the correct answer is E.