I have two fair cubical dice with faces numbered 1 to 6. One is green and one is red. I throw both dice and get a score of p on the green one and q on the red one.

I notice that I can find integer solutions to the quadratic equation x² – px + q = 0.

What is the probability that neither p nor q is prime?

What if the two dice used were octahedral, each with their eight faces numbered 1 to 8?