AMC 12 2006b Problem 17

For a particular peculiar pair of dice, the probabilities of rolling $1$, $2$, $3$, $4$, $5$ and $6$ on each die are in the ratio $1:2:3:4:5:6$. What is the probability of rolling a total of $7$ on the two dice?

$\mathrm{(A)}\ \frac 4{63} \mathrm{(B)}\ \frac 18 \mathrm{(C)}\ \frac 8{63} \mathrm{(D)}\ \frac 16 \mathrm{(E)}\ \frac 27$